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Re: HIRLAM and -ftree-loop-linear
- From: Toon Moene <toon at moene dot indiv dot nluug dot nl>
- To: Dorit Nuzman <DORIT at il dot ibm dot com>
- Cc: gcc mailing list <gcc at gcc dot gnu dot org>, Sebastian Pop <sebpop at gmail dot com>
- Date: Mon, 17 Dec 2007 22:18:45 +0100
- Subject: Re: HIRLAM and -ftree-loop-linear
- References: <OF624D922F.D6F9EB25-ONC22573B3.004F3439-C22573B3.004F6278@il.ibm.com>
Dorit Nuzman wrote:
any chance you kept the dumps and can report which loops were not
vectorized/recognized with -ftree-loop-linear (so we could see if these
represent missed vectorization opportunities?)
It's a bit much to send you everything, so I'll just send you the diff +
two routines (from blas/lapack) that had differences (one positive, one
negative).
The sum total of 'LOOP VECTORIZED' messages:
$ wc -l vect.*
5651 vect.lin # -ftree-loop-linear
5673 vect.nolin # no -ftree-loop-linear
$ diff -cp vect.nolin vect.lin # attached
Hope this is useful,
--
Toon Moene - e-mail: toon@moene.indiv.nluug.nl - phone: +31 346 214290
Saturnushof 14, 3738 XG Maartensdijk, The Netherlands
At home: http://moene.indiv.nluug.nl/~toon/
GNU Fortran's path to Fortran 2003: http://gcc.gnu.org/wiki/Fortran2003
*** vect.nolin 2007-12-17 22:08:13.000000000 +0100
--- vect.lin 2007-12-17 22:05:06.000000000 +0100
*************** abgcri.f:147: note: LOOP VECTORIZED.
*** 2,8 ****
abgcri.f:159: note: LOOP VECTORIZED.
abgcri.f:170: note: LOOP VECTORIZED.
acconvad.f90:1076: note: LOOP VECTORIZED.
- acconvad.f90:1235: note: LOOP VECTORIZED.
acconvad.f90:1250: note: LOOP VECTORIZED.
acconvad.f90:1321: note: LOOP VECTORIZED.
acconvad.f90:1322: note: LOOP VECTORIZED.
--- 2,7 ----
*************** acconvad.f90:1368: note: LOOP VECTORIZED
*** 53,59 ****
acconvad.f90:1483: note: LOOP VECTORIZED.
acconvad.f90:2418: note: LOOP VECTORIZED.
acconvad.f90:2593: note: LOOP VECTORIZED.
- acconvad.f90:2719: note: LOOP VECTORIZED.
acconvad.f90:509: note: LOOP VECTORIZED.
acconvad.f90:784: note: LOOP VECTORIZED.
acconv.f90:355: note: LOOP VECTORIZED.
--- 52,57 ----
*************** bndrel.f:295: note: LOOP VECTORIZED.
*** 720,725 ****
--- 718,724 ----
bndrel.f:307: note: LOOP VECTORIZED.
bndrel.f:323: note: LOOP VECTORIZED.
bndrel.f:342: note: LOOP VECTORIZED.
+ b_obspace.f:1262: note: LOOP VECTORIZED.
b_obspace.f:1539: note: LOOP VECTORIZED.
b_obspace.f:1546: note: LOOP VECTORIZED.
b_obspace.f:815: note: LOOP VECTORIZED.
*************** bucomp.f:116: note: LOOP VECTORIZED.
*** 734,739 ****
--- 733,739 ----
bucomp.f:94: note: LOOP VECTORIZED.
bucrekey.f:90: note: LOOP VECTORIZED.
bucrekey.f:98: note: LOOP VECTORIZED.
+ bucrkey.f:95: note: LOOP VECTORIZED.
buedd.f:202: note: LOOP VECTORIZED.
buedd.f:215: note: LOOP VECTORIZED.
buedd.f:228: note: LOOP VECTORIZED.
*************** buepwt.f:170: note: LOOP VECTORIZED.
*** 756,761 ****
--- 756,763 ----
buepwt.f:184: note: LOOP VECTORIZED.
buepwt.f:255: note: LOOP VECTORIZED.
buepwt.f:270: note: LOOP VECTORIZED.
+ buetab.f:231: note: LOOP VECTORIZED.
+ buetab.f:246: note: LOOP VECTORIZED.
buetab.f:331: note: LOOP VECTORIZED.
buetab.f:336: note: LOOP VECTORIZED.
buetab.f:346: note: LOOP VECTORIZED.
*************** bufrsort.f90:879: note: LOOP VECTORIZED.
*** 807,812 ****
--- 809,816 ----
bufrsort_prepare.f90:197: note: LOOP VECTORIZED.
bufrthin.f:138: note: LOOP VECTORIZED.
bufr_traffic_info.f90:173: note: LOOP VECTORIZED.
+ bugbts.f:253: note: LOOP VECTORIZED.
+ bugbts.f:260: note: LOOP VECTORIZED.
bugbts.f:349: note: LOOP VECTORIZED.
bugbts.f:354: note: LOOP VECTORIZED.
bugbts.f:356: note: LOOP VECTORIZED.
*************** bustdr.f:120: note: LOOP VECTORIZED.
*** 863,868 ****
--- 867,873 ----
bustdr.f:133: note: LOOP VECTORIZED.
bustdr.f:149: note: LOOP VECTORIZED.
bustdr.f:164: note: LOOP VECTORIZED.
+ buta.f:261: note: LOOP VECTORIZED.
buta.f:470: note: LOOP VECTORIZED.
buta.f:476: note: LOOP VECTORIZED.
buta.f:672: note: LOOP VECTORIZED.
*************** cal_cov_multi.f:580: note: LOOP VECTORIZ
*** 904,910 ****
cal_cov_multi.f:941: note: LOOP VECTORIZED.
cal_cov_multi.f:942: note: LOOP VECTORIZED.
cal_cov_multi.f:943: note: LOOP VECTORIZED.
- cal_cov_multi.f:944: note: LOOP VECTORIZED.
cal_cov_multi.f:945: note: LOOP VECTORIZED.
cal_cov_multi.f:946: note: LOOP VECTORIZED.
cal_cov_multi.f:947: note: LOOP VECTORIZED.
--- 909,914 ----
*************** dgeev.f:6347: note: LOOP VECTORIZED.
*** 1337,1342 ****
--- 1341,1348 ----
dgeev.f:6391: note: LOOP VECTORIZED.
dgeev.f:6504: note: LOOP VECTORIZED.
dgeev.f:6525: note: LOOP VECTORIZED.
+ dgeev.f:6783: note: LOOP VECTORIZED.
+ dgeev.f:6822: note: LOOP VECTORIZED.
dgeev.f:772: note: LOOP VECTORIZED.
dgeev.f:843: note: LOOP VECTORIZED.
dgemm.f:231: note: LOOP VECTORIZED.
*************** extrob.f:390: note: LOOP VECTORIZED.
*** 1645,1652 ****
extrob.f:393: note: LOOP VECTORIZED.
extzer.f:132: note: LOOP VECTORIZED.
extzer.f:143: note: LOOP VECTORIZED.
- f02age.f:56: note: LOOP VECTORIZED.
- f02age.f:62: note: LOOP VECTORIZED.
fdvrfy.f:165: note: LOOP VECTORIZED.
fdvrfy.f:271: note: LOOP VECTORIZED.
fdvrfy.f:307: note: LOOP VECTORIZED.
--- 1651,1656 ----
*************** gemini_ad.f:739: note: LOOP VECTORIZED.
*** 1810,1815 ****
--- 1814,1820 ----
gemini_ad.f:812: note: LOOP VECTORIZED.
gemini.f:1317: note: LOOP VECTORIZED.
gemini.f:1647: note: LOOP VECTORIZED.
+ gemini.f:1664: note: LOOP VECTORIZED.
gemini.f:1884: note: LOOP VECTORIZED.
gemini.f:1923: note: LOOP VECTORIZED.
gemini.f:1943: note: LOOP VECTORIZED.
*************** gribiohixb.f:1849: note: LOOP VECTORIZED
*** 2023,2028 ****
--- 2028,2034 ----
gribiohixb.f:1926: note: LOOP VECTORIZED.
gribiohixb.f:231: note: LOOP VECTORIZED.
gribiohixb.f:2392: note: LOOP VECTORIZED.
+ grid2obspoints.f:95: note: LOOP VECTORIZED.
grid2obspointsfgat.f:129: note: LOOP VECTORIZED.
grid2obspointsfgat.f:249: note: LOOP VECTORIZED.
grid2obspointsfgat.f:307: note: LOOP VECTORIZED.
*************** mrtmsys.f:879: note: LOOP VECTORIZED.
*** 2778,2783 ****
--- 2784,2790 ----
mrtmsys.f:880: note: LOOP VECTORIZED.
mrtmsys.f:881: note: LOOP VECTORIZED.
mrtmsys.f:971: note: LOOP VECTORIZED.
+ mrtmsys.f:972: note: LOOP VECTORIZED.
mrtmsys.f:974: note: LOOP VECTORIZED.
mrtmsys.f:975: note: LOOP VECTORIZED.
mrtmsys.f:978: note: LOOP VECTORIZED.
*************** npsol.f:8203: note: LOOP VECTORIZED.
*** 3012,3018 ****
npsol.f:8328: note: LOOP VECTORIZED.
npsol.f:8333: note: LOOP VECTORIZED.
npsol.f:8540: note: LOOP VECTORIZED.
- npsol.f:8566: note: LOOP VECTORIZED.
npsol.f:8592: note: LOOP VECTORIZED.
npsol.f:8736: note: LOOP VECTORIZED.
npsol.f:8772: note: LOOP VECTORIZED.
--- 3019,3024 ----
*************** postpp.f:752: note: LOOP VECTORIZED.
*** 3353,3359 ****
postpp.f:790: note: LOOP VECTORIZED.
prcbuo.f:832: note: LOOP VECTORIZED.
prcshp.f:821: note: LOOP VECTORIZED.
- prcsyn.f:821: note: LOOP VECTORIZED.
precheck_cma.f90:152: note: LOOP VECTORIZED.
precip.f:56: note: LOOP VECTORIZED.
precond.f:355: note: LOOP VECTORIZED.
--- 3359,3364 ----
*************** prepare_scatt_work.f:714: note: LOOP VEC
*** 3408,3413 ****
--- 3413,3420 ----
prepare_seawinds_work.f:324: note: LOOP VECTORIZED.
prepare_ship_work.f:296: note: LOOP VECTORIZED.
prepare_ship_work.f:357: note: LOOP VECTORIZED.
+ prepare_ssmi_work.f:446: note: LOOP VECTORIZED.
+ prepare_ssmi_work.f:450: note: LOOP VECTORIZED.
prepare_synop_work.f:295: note: LOOP VECTORIZED.
prepare_synop_work.f:357: note: LOOP VECTORIZED.
prepare_temp_work.f:290: note: LOOP VECTORIZED.
*************** rttov_aitosu_ad.f90:492: note: LOOP VECT
*** 3908,3926 ****
rttov_aitosu_ad.f90:493: note: LOOP VECTORIZED.
rttov_aitosu_ad.f90:494: note: LOOP VECTORIZED.
rttov_aitosu_ad.f90:495: note: LOOP VECTORIZED.
- rttov_aitosu_ad.f90:603: note: LOOP VECTORIZED.
rttov_aitosu_ad.f90:608: note: LOOP VECTORIZED.
rttov_aitosu_ad.f90:609: note: LOOP VECTORIZED.
rttov_aitosu_ad.f90:610: note: LOOP VECTORIZED.
rttov_aitosu_ad.f90:611: note: LOOP VECTORIZED.
- rttov_aitosu.f90:132: note: LOOP VECTORIZED.
- rttov_aitosu.f90:133: note: LOOP VECTORIZED.
- rttov_aitosu.f90:134: note: LOOP VECTORIZED.
- rttov_aitosu.f90:136: note: LOOP VECTORIZED.
- rttov_aitosu.f90:189: note: LOOP VECTORIZED.
- rttov_aitosu.f90:190: note: LOOP VECTORIZED.
- rttov_aitosu.f90:191: note: LOOP VECTORIZED.
- rttov_aitosu.f90:193: note: LOOP VECTORIZED.
rttov_aitosu_tl.f90:182: note: LOOP VECTORIZED.
rttov_aitosu_tl.f90:183: note: LOOP VECTORIZED.
rttov_aitosu_tl.f90:184: note: LOOP VECTORIZED.
--- 3915,3924 ----
*************** rttov_integratesource_ad.f90:108: note:
*** 4187,4193 ****
rttov_integratesource_ad.f90:128: note: LOOP VECTORIZED.
rttov_integratesource_ad.f90:129: note: LOOP VECTORIZED.
rttov_integratesource_ad.f90:130: note: LOOP VECTORIZED.
- rttov_integratesource_ad.f90:131: note: LOOP VECTORIZED.
rttov_integratesource_ad.f90:132: note: LOOP VECTORIZED.
rttov_integratesource_ad.f90:133: note: LOOP VECTORIZED.
rttov_integratesource_ad.f90:134: note: LOOP VECTORIZED.
--- 4185,4190 ----
*************** rttov_intex_tl.f90:83: note: LOOP VECTOR
*** 4231,4236 ****
--- 4228,4234 ----
rttov_intext_prof.f90:156: note: LOOP VECTORIZED.
rttov_intext_prof.f90:186: note: LOOP VECTORIZED.
rttov_intext_prof.f90:192: note: LOOP VECTORIZED.
+ rttov_intext_prof.f90:251: note: LOOP VECTORIZED.
rttov_intext_prof.f90:253: note: LOOP VECTORIZED.
rttov_intext_prof.f90:259: note: LOOP VECTORIZED.
rttov_intext_prof.f90:273: note: LOOP VECTORIZED.
*************** rttov_k.f90:1096: note: LOOP VECTORIZED.
*** 4246,4257 ****
rttov_k.f90:1106: note: LOOP VECTORIZED.
rttov_k.f90:1123: note: LOOP VECTORIZED.
rttov_k.f90:1166: note: LOOP VECTORIZED.
- rttov_k.f90:1213: note: LOOP VECTORIZED.
- rttov_k.f90:1234: note: LOOP VECTORIZED.
- rttov_k.f90:1246: note: LOOP VECTORIZED.
- rttov_k.f90:1258: note: LOOP VECTORIZED.
- rttov_k.f90:1267: note: LOOP VECTORIZED.
- rttov_k.f90:1326: note: LOOP VECTORIZED.
rttov_k.f90:1381: note: LOOP VECTORIZED.
rttov_k.f90:1403: note: LOOP VECTORIZED.
rttov_k.f90:1413: note: LOOP VECTORIZED.
--- 4244,4249 ----
*************** signr.f:27: note: LOOP VECTORIZED.
*** 4647,4653 ****
simconv.f90:379: note: LOOP VECTORIZED.
simconv.f90:415: note: LOOP VECTORIZED.
simload.f90:216: note: LOOP VECTORIZED.
- simobs_diagnose.f:11: note: LOOP VECTORIZED.
sirhs.f:103: note: LOOP VECTORIZED.
sirhs.f:184: note: LOOP VECTORIZED.
sirhs.f:197: note: LOOP VECTORIZED.
--- 4639,4644 ----
*************** sposv.f:1197: note: LOOP VECTORIZED.
*** 4902,4924 ****
sposv.f:1202: note: LOOP VECTORIZED.
sposv.f:1204: note: LOOP VECTORIZED.
sposv.f:1210: note: LOOP VECTORIZED.
- sposv.f:1212: note: LOOP VECTORIZED.
sposv.f:1216: note: LOOP VECTORIZED.
- sposv.f:1218: note: LOOP VECTORIZED.
sposv.f:1223: note: LOOP VECTORIZED.
- sposv.f:1225: note: LOOP VECTORIZED.
sposv.f:1237: note: LOOP VECTORIZED.
- sposv.f:1239: note: LOOP VECTORIZED.
sposv.f:1244: note: LOOP VECTORIZED.
- sposv.f:1246: note: LOOP VECTORIZED.
sposv.f:1250: note: LOOP VECTORIZED.
- sposv.f:1252: note: LOOP VECTORIZED.
sposv.f:1259: note: LOOP VECTORIZED.
- sposv.f:1261: note: LOOP VECTORIZED.
sposv.f:1266: note: LOOP VECTORIZED.
- sposv.f:1268: note: LOOP VECTORIZED.
sposv.f:1272: note: LOOP VECTORIZED.
- sposv.f:1274: note: LOOP VECTORIZED.
sposv.f:792: note: LOOP VECTORIZED.
sposv.f:794: note: LOOP VECTORIZED.
sposv.f:798: note: LOOP VECTORIZED.
--- 4893,4906 ----
*************** ssyev.f:6339: note: LOOP VECTORIZED.
*** 5034,5050 ****
ssyev.f:6358: note: LOOP VECTORIZED.
ssyev.f:6358: note: LOOP VECTORIZED.
ssyev.f:635: note: LOOP VECTORIZED.
- ssyev.f:635: note: LOOP VECTORIZED.
ssyev.f:6363: note: LOOP VECTORIZED.
ssyev.f:6363: note: LOOP VECTORIZED.
ssyev.f:6408: note: LOOP VECTORIZED.
- ssyev.f:6408: note: LOOP VECTORIZED.
- ssyev.f:682: note: LOOP VECTORIZED.
ssyev.f:682: note: LOOP VECTORIZED.
ssyev.f:7075: note: LOOP VECTORIZED.
ssyev.f:7075: note: LOOP VECTORIZED.
ssyev.f:7338: note: LOOP VECTORIZED.
- ssyev.f:7338: note: LOOP VECTORIZED.
ssyev.f:7718: note: LOOP VECTORIZED.
ssyev.f:7718: note: LOOP VECTORIZED.
ssyev.f:7722: note: LOOP VECTORIZED.
--- 5016,5028 ----
*************** ssyev.f:9234: note: LOOP VECTORIZED.
*** 5094,5100 ****
ssyev.f:9243: note: LOOP VECTORIZED.
ssyev.f:9243: note: LOOP VECTORIZED.
ssyev.f:9426: note: LOOP VECTORIZED.
- ssyev.f:9426: note: LOOP VECTORIZED.
ssyev.f:953: note: LOOP VECTORIZED.
ssyev.f:953: note: LOOP VECTORIZED.
ssyev.f:959: note: LOOP VECTORIZED.
--- 5072,5077 ----
*************** stamic.f:1458: note: LOOP VECTORIZED.
*** 5150,5156 ****
stamic.f:1484: note: LOOP VECTORIZED.
stamic.f:1509: note: LOOP VECTORIZED.
stamic.f:1524: note: LOOP VECTORIZED.
- stamic.f:1537: note: LOOP VECTORIZED.
stamic.f:1563: note: LOOP VECTORIZED.
stamic.f:685: note: LOOP VECTORIZED.
stamic.f:757: note: LOOP VECTORIZED.
--- 5127,5132 ----
*************** syminv.f:207: note: LOOP VECTORIZED.
*** 5327,5336 ****
--- 5303,5314 ----
sync_timeslot_data.f90:326: note: LOOP VECTORIZED.
sync_timeslot_data.f90:341: note: LOOP VECTORIZED.
sync_timeslot_data.f90:434: note: LOOP VECTORIZED.
+ tables36.f:245: note: LOOP VECTORIZED.
tables36.f:437: note: LOOP VECTORIZED.
tables36.f:443: note: LOOP VECTORIZED.
tables36.f:642: note: LOOP VECTORIZED.
tables36.f:648: note: LOOP VECTORIZED.
+ tables.f:258: note: LOOP VECTORIZED.
tables.f:455: note: LOOP VECTORIZED.
tables.f:461: note: LOOP VECTORIZED.
tables.f:660: note: LOOP VECTORIZED.
SUBROUTINE SPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
* -- LAPACK driver routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* SPOSV computes the solution to a real system of linear equations
* A * X = B,
* where A is an N-by-N symmetric positive definite matrix and X and B
* are N-by-NRHS matrices.
*
* The Cholesky decomposition is used to factor A as
* A = U**T* U, if UPLO = 'U', or
* A = L * L**T, if UPLO = 'L',
* where U is an upper triangular matrix and L is a lower triangular
* matrix. The factored form of A is then used to solve the system of
* equations A * X = B.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
*
* On exit, if INFO = 0, the factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the leading minor of order i of A is not
* positive definite, so the factorization could not be
* completed, and the solution has not been computed.
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SPOTRF, SPOTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPOSV ', -INFO )
RETURN
END IF
*
* Compute the Cholesky factorization A = U'*U or A = L*L'.
*
CALL SPOTRF( UPLO, N, A, LDA, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL SPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
END IF
RETURN
*
* End of SPOSV
*
END
SUBROUTINE SPOTF2( UPLO, N, A, LDA, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
REAL A( LDA, * )
* ..
*
* Purpose
* =======
*
* SPOTF2 computes the Cholesky factorization of a real symmetric
* positive definite matrix A.
*
* The factorization has the form
* A = U' * U , if UPLO = 'U', or
* A = L * L', if UPLO = 'L',
* where U is an upper triangular matrix and L is lower triangular.
*
* This is the unblocked version of the algorithm, calling Level 2 BLAS.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* symmetric matrix A is stored.
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* n by n upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading n by n lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
*
* On exit, if INFO = 0, the factor U or L from the Cholesky
* factorization A = U'*U or A = L*L'.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
* > 0: if INFO = k, the leading minor of order k is not
* positive definite, and the factorization could not be
* completed.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J
REAL AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SDOT
EXTERNAL LSAME, SDOT
* ..
* .. External Subroutines ..
EXTERNAL SGEMV, SSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPOTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U'*U.
*
DO 10 J = 1, N
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = A( J, J ) - SDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 )
IF( AJJ.LE.ZERO ) THEN
A( J, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of row J.
*
IF( J.LT.N ) THEN
CALL SGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ),
$ LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
CALL SSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L'.
*
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = A( J, J ) - SDOT( J-1, A( J, 1 ), LDA, A( J, 1 ),
$ LDA )
IF( AJJ.LE.ZERO ) THEN
A( J, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
A( J, J ) = AJJ
*
* Compute elements J+1:N of column J.
*
IF( J.LT.N ) THEN
CALL SGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1, 1 ),
$ LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
CALL SSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
END IF
20 CONTINUE
END IF
GO TO 40
*
30 CONTINUE
INFO = J
*
40 CONTINUE
RETURN
*
* End of SPOTF2
*
END
SUBROUTINE SPOTRF( UPLO, N, A, LDA, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
REAL A( LDA, * )
* ..
*
* Purpose
* =======
*
* SPOTRF computes the Cholesky factorization of a real symmetric
* positive definite matrix A.
*
* The factorization has the form
* A = U**T * U, if UPLO = 'U', or
* A = L * L**T, if UPLO = 'L',
* where U is an upper triangular matrix and L is lower triangular.
*
* This is the block version of the algorithm, calling Level 3 BLAS.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
*
* On exit, if INFO = 0, the factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the leading minor of order i is not
* positive definite, and the factorization could not be
* completed.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, JB, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL SGEMM, SPOTF2, SSYRK, STRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPOTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'SPOTRF', UPLO, N, -1, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
* Use unblocked code.
*
CALL SPOTF2( UPLO, N, A, LDA, INFO )
ELSE
*
* Use blocked code.
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U'*U.
*
DO 10 J = 1, N, NB
*
* Update and factorize the current diagonal block and test
* for non-positive-definiteness.
*
JB = MIN( NB, N-J+1 )
CALL SSYRK( 'Upper', 'Transpose', JB, J-1, -ONE,
$ A( 1, J ), LDA, ONE, A( J, J ), LDA )
CALL SPOTF2( 'Upper', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( J+JB.LE.N ) THEN
*
* Compute the current block row.
*
CALL SGEMM( 'Transpose', 'No transpose', JB, N-J-JB+1,
$ J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ),
$ LDA, ONE, A( J, J+JB ), LDA )
CALL STRSM( 'Left', 'Upper', 'Transpose', 'Non-unit',
$ JB, N-J-JB+1, ONE, A( J, J ), LDA,
$ A( J, J+JB ), LDA )
END IF
10 CONTINUE
*
ELSE
*
* Compute the Cholesky factorization A = L*L'.
*
DO 20 J = 1, N, NB
*
* Update and factorize the current diagonal block and test
* for non-positive-definiteness.
*
JB = MIN( NB, N-J+1 )
CALL SSYRK( 'Lower', 'No transpose', JB, J-1, -ONE,
$ A( J, 1 ), LDA, ONE, A( J, J ), LDA )
CALL SPOTF2( 'Lower', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( J+JB.LE.N ) THEN
*
* Compute the current block column.
*
CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
$ J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ),
$ LDA, ONE, A( J+JB, J ), LDA )
CALL STRSM( 'Right', 'Lower', 'Transpose', 'Non-unit',
$ N-J-JB+1, JB, ONE, A( J, J ), LDA,
$ A( J+JB, J ), LDA )
END IF
20 CONTINUE
END IF
END IF
GO TO 40
*
30 CONTINUE
INFO = INFO + J - 1
*
40 CONTINUE
RETURN
*
* End of SPOTRF
*
END
SUBROUTINE SPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* SPOTRS solves a system of linear equations A*X = B with a symmetric
* positive definite matrix A using the Cholesky factorization
* A = U**T*U or A = L*L**T computed by SPOTRF.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* A (input) REAL array, dimension (LDA,N)
* The triangular factor U or L from the Cholesky factorization
* A = U**T*U or A = L*L**T, as computed by SPOTRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, the solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL STRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SPOTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B where A = U'*U.
*
* Solve U'*X = B, overwriting B with X.
*
CALL STRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS,
$ ONE, A, LDA, B, LDB )
*
* Solve U*X = B, overwriting B with X.
*
CALL STRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, A, LDA, B, LDB )
ELSE
*
* Solve A*X = B where A = L*L'.
*
* Solve L*X = B, overwriting B with X.
*
CALL STRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, A, LDA, B, LDB )
*
* Solve L'*X = B, overwriting B with X.
*
CALL STRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N, NRHS,
$ ONE, A, LDA, B, LDB )
END IF
*
RETURN
*
* End of SPOTRS
*
END
SUBROUTINE SSYRK ( UPLO, TRANS, N, K, ALPHA, A, LDA,
$ BETA, C, LDC )
* .. Scalar Arguments ..
CHARACTER*1 UPLO, TRANS
INTEGER N, K, LDA, LDC
REAL ALPHA, BETA
* .. Array Arguments ..
REAL A( LDA, * ), C( LDC, * )
* ..
*
* Purpose
* =======
*
* SSYRK performs one of the symmetric rank k operations
*
* C := alpha*A*A' + beta*C,
*
* or
*
* C := alpha*A'*A + beta*C,
*
* where alpha and beta are scalars, C is an n by n symmetric matrix
* and A is an n by k matrix in the first case and a k by n matrix
* in the second case.
*
* Parameters
* ==========
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the upper or lower
* triangular part of the array C is to be referenced as
* follows:
*
* UPLO = 'U' or 'u' Only the upper triangular part of C
* is to be referenced.
*
* UPLO = 'L' or 'l' Only the lower triangular part of C
* is to be referenced.
*
* Unchanged on exit.
*
* TRANS - CHARACTER*1.
* On entry, TRANS specifies the operation to be performed as
* follows:
*
* TRANS = 'N' or 'n' C := alpha*A*A' + beta*C.
*
* TRANS = 'T' or 't' C := alpha*A'*A + beta*C.
*
* TRANS = 'C' or 'c' C := alpha*A'*A + beta*C.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix C. N must be
* at least zero.
* Unchanged on exit.
*
* K - INTEGER.
* On entry with TRANS = 'N' or 'n', K specifies the number
* of columns of the matrix A, and on entry with
* TRANS = 'T' or 't' or 'C' or 'c', K specifies the number
* of rows of the matrix A. K must be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, ka ), where ka is
* k when TRANS = 'N' or 'n', and is n otherwise.
* Before entry with TRANS = 'N' or 'n', the leading n by k
* part of the array A must contain the matrix A, otherwise
* the leading k by n part of the array A must contain the
* matrix A.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When TRANS = 'N' or 'n'
* then LDA must be at least max( 1, n ), otherwise LDA must
* be at least max( 1, k ).
* Unchanged on exit.
*
* BETA - REAL .
* On entry, BETA specifies the scalar beta.
* Unchanged on exit.
*
* C - REAL array of DIMENSION ( LDC, n ).
* Before entry with UPLO = 'U' or 'u', the leading n by n
* upper triangular part of the array C must contain the upper
* triangular part of the symmetric matrix and the strictly
* lower triangular part of C is not referenced. On exit, the
* upper triangular part of the array C is overwritten by the
* upper triangular part of the updated matrix.
* Before entry with UPLO = 'L' or 'l', the leading n by n
* lower triangular part of the array C must contain the lower
* triangular part of the symmetric matrix and the strictly
* upper triangular part of C is not referenced. On exit, the
* lower triangular part of the array C is overwritten by the
* lower triangular part of the updated matrix.
*
* LDC - INTEGER.
* On entry, LDC specifies the first dimension of C as declared
* in the calling (sub) program. LDC must be at least
* max( 1, n ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, INFO, J, L, NROWA
REAL TEMP
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
IF( LSAME( TRANS, 'N' ) )THEN
NROWA = N
ELSE
NROWA = K
END IF
UPPER = LSAME( UPLO, 'U' )
*
INFO = 0
IF( ( .NOT.UPPER ).AND.
$ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN
INFO = 1
ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND.
$ ( .NOT.LSAME( TRANS, 'T' ) ).AND.
$ ( .NOT.LSAME( TRANS, 'C' ) ) )THEN
INFO = 2
ELSE IF( N .LT.0 )THEN
INFO = 3
ELSE IF( K .LT.0 )THEN
INFO = 4
ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
INFO = 7
ELSE IF( LDC.LT.MAX( 1, N ) )THEN
INFO = 10
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SSYRK ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( N.EQ.0 ).OR.
$ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
$ RETURN
*
* And when alpha.eq.zero.
*
IF( ALPHA.EQ.ZERO )THEN
IF( UPPER )THEN
IF( BETA.EQ.ZERO )THEN
DO 20, J = 1, N
DO 10, I = 1, J
C( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40, J = 1, N
DO 30, I = 1, J
C( I, J ) = BETA*C( I, J )
30 CONTINUE
40 CONTINUE
END IF
ELSE
IF( BETA.EQ.ZERO )THEN
DO 60, J = 1, N
DO 50, I = J, N
C( I, J ) = ZERO
50 CONTINUE
60 CONTINUE
ELSE
DO 80, J = 1, N
DO 70, I = J, N
C( I, J ) = BETA*C( I, J )
70 CONTINUE
80 CONTINUE
END IF
END IF
RETURN
END IF
*
* Start the operations.
*
IF( LSAME( TRANS, 'N' ) )THEN
*
* Form C := alpha*A*A' + beta*C.
*
IF( UPPER )THEN
DO 130, J = 1, N
IF( BETA.EQ.ZERO )THEN
DO 90, I = 1, J
C( I, J ) = ZERO
90 CONTINUE
ELSE IF( BETA.NE.ONE )THEN
DO 100, I = 1, J
C( I, J ) = BETA*C( I, J )
100 CONTINUE
END IF
DO 120, L = 1, K
IF( A( J, L ).NE.ZERO )THEN
TEMP = ALPHA*A( J, L )
DO 110, I = 1, J
C( I, J ) = C( I, J ) + TEMP*A( I, L )
110 CONTINUE
END IF
120 CONTINUE
130 CONTINUE
ELSE
DO 180, J = 1, N
IF( BETA.EQ.ZERO )THEN
DO 140, I = J, N
C( I, J ) = ZERO
140 CONTINUE
ELSE IF( BETA.NE.ONE )THEN
DO 150, I = J, N
C( I, J ) = BETA*C( I, J )
150 CONTINUE
END IF
DO 170, L = 1, K
IF( A( J, L ).NE.ZERO )THEN
TEMP = ALPHA*A( J, L )
DO 160, I = J, N
C( I, J ) = C( I, J ) + TEMP*A( I, L )
160 CONTINUE
END IF
170 CONTINUE
180 CONTINUE
END IF
ELSE
*
* Form C := alpha*A'*A + beta*C.
*
IF( UPPER )THEN
DO 210, J = 1, N
DO 200, I = 1, J
TEMP = ZERO
DO 190, L = 1, K
TEMP = TEMP + A( L, I )*A( L, J )
190 CONTINUE
IF( BETA.EQ.ZERO )THEN
C( I, J ) = ALPHA*TEMP
ELSE
C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
END IF
200 CONTINUE
210 CONTINUE
ELSE
DO 240, J = 1, N
DO 230, I = J, N
TEMP = ZERO
DO 220, L = 1, K
TEMP = TEMP + A( L, I )*A( L, J )
220 CONTINUE
IF( BETA.EQ.ZERO )THEN
C( I, J ) = ALPHA*TEMP
ELSE
C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
END IF
230 CONTINUE
240 CONTINUE
END IF
END IF
*
RETURN
*
* End of SSYRK .
*
END
SUBROUTINE STRSM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA,
$ B, LDB )
* .. Scalar Arguments ..
CHARACTER*1 SIDE, UPLO, TRANSA, DIAG
INTEGER M, N, LDA, LDB
REAL ALPHA
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* STRSM solves one of the matrix equations
*
* op( A )*X = alpha*B, or X*op( A ) = alpha*B,
*
* where alpha is a scalar, X and B are m by n matrices, A is a unit, or
* non-unit, upper or lower triangular matrix and op( A ) is one of
*
* op( A ) = A or op( A ) = A'.
*
* The matrix X is overwritten on B.
*
* Parameters
* ==========
*
* SIDE - CHARACTER*1.
* On entry, SIDE specifies whether op( A ) appears on the left
* or right of X as follows:
*
* SIDE = 'L' or 'l' op( A )*X = alpha*B.
*
* SIDE = 'R' or 'r' X*op( A ) = alpha*B.
*
* Unchanged on exit.
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the matrix A is an upper or
* lower triangular matrix as follows:
*
* UPLO = 'U' or 'u' A is an upper triangular matrix.
*
* UPLO = 'L' or 'l' A is a lower triangular matrix.
*
* Unchanged on exit.
*
* TRANSA - CHARACTER*1.
* On entry, TRANSA specifies the form of op( A ) to be used in
* the matrix multiplication as follows:
*
* TRANSA = 'N' or 'n' op( A ) = A.
*
* TRANSA = 'T' or 't' op( A ) = A'.
*
* TRANSA = 'C' or 'c' op( A ) = A'.
*
* Unchanged on exit.
*
* DIAG - CHARACTER*1.
* On entry, DIAG specifies whether or not A is unit triangular
* as follows:
*
* DIAG = 'U' or 'u' A is assumed to be unit triangular.
*
* DIAG = 'N' or 'n' A is not assumed to be unit
* triangular.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of B. M must be at
* least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of B. N must be
* at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha. When alpha is
* zero then A is not referenced and B need not be set before
* entry.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, k ), where k is m
* when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
* Before entry with UPLO = 'U' or 'u', the leading k by k
* upper triangular part of the array A must contain the upper
* triangular matrix and the strictly lower triangular part of
* A is not referenced.
* Before entry with UPLO = 'L' or 'l', the leading k by k
* lower triangular part of the array A must contain the lower
* triangular matrix and the strictly upper triangular part of
* A is not referenced.
* Note that when DIAG = 'U' or 'u', the diagonal elements of
* A are not referenced either, but are assumed to be unity.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When SIDE = 'L' or 'l' then
* LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
* then LDA must be at least max( 1, n ).
* Unchanged on exit.
*
* B - REAL array of DIMENSION ( LDB, n ).
* Before entry, the leading m by n part of the array B must
* contain the right-hand side matrix B, and on exit is
* overwritten by the solution matrix X.
*
* LDB - INTEGER.
* On entry, LDB specifies the first dimension of B as declared
* in the calling (sub) program. LDB must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Local Scalars ..
LOGICAL LSIDE, NOUNIT, UPPER
INTEGER I, INFO, J, K, NROWA
REAL TEMP
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
LSIDE = LSAME( SIDE , 'L' )
IF( LSIDE )THEN
NROWA = M
ELSE
NROWA = N
END IF
NOUNIT = LSAME( DIAG , 'N' )
UPPER = LSAME( UPLO , 'U' )
*
INFO = 0
IF( ( .NOT.LSIDE ).AND.
$ ( .NOT.LSAME( SIDE , 'R' ) ) )THEN
INFO = 1
ELSE IF( ( .NOT.UPPER ).AND.
$ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN
INFO = 2
ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND.
$ ( .NOT.LSAME( TRANSA, 'T' ) ).AND.
$ ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN
INFO = 3
ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND.
$ ( .NOT.LSAME( DIAG , 'N' ) ) )THEN
INFO = 4
ELSE IF( M .LT.0 )THEN
INFO = 5
ELSE IF( N .LT.0 )THEN
INFO = 6
ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
INFO = 9
ELSE IF( LDB.LT.MAX( 1, M ) )THEN
INFO = 11
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'STRSM ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* And when alpha.eq.zero.
*
IF( ALPHA.EQ.ZERO )THEN
DO 20, J = 1, N
DO 10, I = 1, M
B( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
RETURN
END IF
*
* Start the operations.
*
IF( LSIDE )THEN
IF( LSAME( TRANSA, 'N' ) )THEN
*
* Form B := alpha*inv( A )*B.
*
IF( UPPER )THEN
DO 60, J = 1, N
IF( ALPHA.NE.ONE )THEN
DO 30, I = 1, M
B( I, J ) = ALPHA*B( I, J )
30 CONTINUE
END IF
DO 50, K = M, 1, -1
IF( B( K, J ).NE.ZERO )THEN
IF( NOUNIT )
$ B( K, J ) = B( K, J )/A( K, K )
DO 40, I = 1, K - 1
B( I, J ) = B( I, J ) - B( K, J )*A( I, K )
40 CONTINUE
END IF
50 CONTINUE
60 CONTINUE
ELSE
DO 100, J = 1, N
IF( ALPHA.NE.ONE )THEN
DO 70, I = 1, M
B( I, J ) = ALPHA*B( I, J )
70 CONTINUE
END IF
DO 90 K = 1, M
IF( B( K, J ).NE.ZERO )THEN
IF( NOUNIT )
$ B( K, J ) = B( K, J )/A( K, K )
DO 80, I = K + 1, M
B( I, J ) = B( I, J ) - B( K, J )*A( I, K )
80 CONTINUE
END IF
90 CONTINUE
100 CONTINUE
END IF
ELSE
*
* Form B := alpha*inv( A' )*B.
*
IF( UPPER )THEN
DO 130, J = 1, N
DO 120, I = 1, M
TEMP = ALPHA*B( I, J )
DO 110, K = 1, I - 1
TEMP = TEMP - A( K, I )*B( K, J )
110 CONTINUE
IF( NOUNIT )
$ TEMP = TEMP/A( I, I )
B( I, J ) = TEMP
120 CONTINUE
130 CONTINUE
ELSE
DO 160, J = 1, N
DO 150, I = M, 1, -1
TEMP = ALPHA*B( I, J )
DO 140, K = I + 1, M
TEMP = TEMP - A( K, I )*B( K, J )
140 CONTINUE
IF( NOUNIT )
$ TEMP = TEMP/A( I, I )
B( I, J ) = TEMP
150 CONTINUE
160 CONTINUE
END IF
END IF
ELSE
IF( LSAME( TRANSA, 'N' ) )THEN
*
* Form B := alpha*B*inv( A ).
*
IF( UPPER )THEN
DO 210, J = 1, N
IF( ALPHA.NE.ONE )THEN
DO 170, I = 1, M
B( I, J ) = ALPHA*B( I, J )
170 CONTINUE
END IF
DO 190, K = 1, J - 1
IF( A( K, J ).NE.ZERO )THEN
DO 180, I = 1, M
B( I, J ) = B( I, J ) - A( K, J )*B( I, K )
180 CONTINUE
END IF
190 CONTINUE
IF( NOUNIT )THEN
TEMP = ONE/A( J, J )
DO 200, I = 1, M
B( I, J ) = TEMP*B( I, J )
200 CONTINUE
END IF
210 CONTINUE
ELSE
DO 260, J = N, 1, -1
IF( ALPHA.NE.ONE )THEN
DO 220, I = 1, M
B( I, J ) = ALPHA*B( I, J )
220 CONTINUE
END IF
DO 240, K = J + 1, N
IF( A( K, J ).NE.ZERO )THEN
DO 230, I = 1, M
B( I, J ) = B( I, J ) - A( K, J )*B( I, K )
230 CONTINUE
END IF
240 CONTINUE
IF( NOUNIT )THEN
TEMP = ONE/A( J, J )
DO 250, I = 1, M
B( I, J ) = TEMP*B( I, J )
250 CONTINUE
END IF
260 CONTINUE
END IF
ELSE
*
* Form B := alpha*B*inv( A' ).
*
IF( UPPER )THEN
DO 310, K = N, 1, -1
IF( NOUNIT )THEN
TEMP = ONE/A( K, K )
DO 270, I = 1, M
B( I, K ) = TEMP*B( I, K )
270 CONTINUE
END IF
DO 290, J = 1, K - 1
IF( A( J, K ).NE.ZERO )THEN
TEMP = A( J, K )
DO 280, I = 1, M
B( I, J ) = B( I, J ) - TEMP*B( I, K )
280 CONTINUE
END IF
290 CONTINUE
IF( ALPHA.NE.ONE )THEN
DO 300, I = 1, M
B( I, K ) = ALPHA*B( I, K )
300 CONTINUE
END IF
310 CONTINUE
ELSE
DO 360, K = 1, N
IF( NOUNIT )THEN
TEMP = ONE/A( K, K )
DO 320, I = 1, M
B( I, K ) = TEMP*B( I, K )
320 CONTINUE
END IF
DO 340, J = K + 1, N
IF( A( J, K ).NE.ZERO )THEN
TEMP = A( J, K )
DO 330, I = 1, M
B( I, J ) = B( I, J ) - TEMP*B( I, K )
330 CONTINUE
END IF
340 CONTINUE
IF( ALPHA.NE.ONE )THEN
DO 350, I = 1, M
B( I, K ) = ALPHA*B( I, K )
350 CONTINUE
END IF
360 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of STRSM .
*
END
INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3,
$ N4 )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, N1, N2, N3, N4
* ..
*
* Purpose
* =======
*
* ILAENV is called from the LAPACK routines to choose problem-dependent
* parameters for the local environment. See ISPEC for a description of
* the parameters.
*
* This version provides a set of parameters which should give good,
* but not optimal, performance on many of the currently available
* computers. Users are encouraged to modify this subroutine to set
* the tuning parameters for their particular machine using the option
* and problem size information in the arguments.
*
* This routine will not function correctly if it is converted to all
* lower case. Converting it to all upper case is allowed.
*
* Arguments
* =========
*
* ISPEC (input) INTEGER
* Specifies the parameter to be returned as the value of
* ILAENV.
* = 1: the optimal blocksize; if this value is 1, an unblocked
* algorithm will give the best performance.
* = 2: the minimum block size for which the block routine
* should be used; if the usable block size is less than
* this value, an unblocked routine should be used.
* = 3: the crossover point (in a block routine, for N less
* than this value, an unblocked routine should be used)
* = 4: the number of shifts, used in the nonsymmetric
* eigenvalue routines
* = 5: the minimum column dimension for blocking to be used;
* rectangular blocks must have dimension at least k by m,
* where k is given by ILAENV(2,...) and m by ILAENV(5,...)
* = 6: the crossover point for the SVD (when reducing an m by n
* matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
* this value, a QR factorization is used first to reduce
* the matrix to a triangular form.)
* = 7: the number of processors
* = 8: the crossover point for the multishift QR and QZ methods
* for nonsymmetric eigenvalue problems.
*
* NAME (input) CHARACTER*(*)
* The name of the calling subroutine, in either upper case or
* lower case.
*
* OPTS (input) CHARACTER*(*)
* The character options to the subroutine NAME, concatenated
* into a single character string. For example, UPLO = 'U',
* TRANS = 'T', and DIAG = 'N' for a triangular routine would
* be specified as OPTS = 'UTN'.
*
* N1 (input) INTEGER
* N2 (input) INTEGER
* N3 (input) INTEGER
* N4 (input) INTEGER
* Problem dimensions for the subroutine NAME; these may not all
* be required.
*
* (ILAENV) (output) INTEGER
* >= 0: the value of the parameter specified by ISPEC
* < 0: if ILAENV = -k, the k-th argument had an illegal value.
*
* Further Details
* ===============
*
* The following conventions have been used when calling ILAENV from the
* LAPACK routines:
* 1) OPTS is a concatenation of all of the character options to
* subroutine NAME, in the same order that they appear in the
* argument list for NAME, even if they are not used in determining
* the value of the parameter specified by ISPEC.
* 2) The problem dimensions N1, N2, N3, N4 are specified in the order
* that they appear in the argument list for NAME. N1 is used
* first, N2 second, and so on, and unused problem dimensions are
* passed a value of -1.
* 3) The parameter value returned by ILAENV is checked for validity in
* the calling subroutine. For example, ILAENV is used to retrieve
* the optimal blocksize for STRTRI as follows:
*
* NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
* IF( NB.LE.1 ) NB = MAX( 1, N )
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL CNAME, SNAME
CHARACTER*1 C1
CHARACTER*2 C2, C4
CHARACTER*3 C3
CHARACTER*6 SUBNAM
INTEGER I, IC, IZ, NB, NBMIN, NX
* ..
* .. Intrinsic Functions ..
INTRINSIC CHAR, ICHAR, INT, MIN, REAL
* ..
* .. Executable Statements ..
*
GO TO ( 100, 100, 100, 400, 500, 600, 700, 800 ) ISPEC
*
* Invalid value for ISPEC
*
ILAENV = -1
RETURN
*
100 CONTINUE
*
* Convert NAME to upper case if the first character is lower case.
*
ILAENV = 1
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1:1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1:1 ) = CHAR( IC-32 )
DO 10 I = 2, 6
IC = ICHAR( SUBNAM( I:I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I:I ) = CHAR( IC-32 )
10 CONTINUE
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1:1 ) = CHAR( IC+64 )
DO 20 I = 2, 6
IC = ICHAR( SUBNAM( I:I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )
$ SUBNAM( I:I ) = CHAR( IC+64 )
20 CONTINUE
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1:1 ) = CHAR( IC-32 )
DO 30 I = 2, 6
IC = ICHAR( SUBNAM( I:I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I:I ) = CHAR( IC-32 )
30 CONTINUE
END IF
END IF
*
C1 = SUBNAM( 1:1 )
SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
IF( .NOT.( CNAME .OR. SNAME ) )
$ RETURN
C2 = SUBNAM( 2:3 )
C3 = SUBNAM( 4:6 )
C4 = C3( 2:3 )
*
GO TO ( 110, 200, 300 ) ISPEC
*
110 CONTINUE
*
* ISPEC = 1: block size
*
* In these examples, separate code is provided for setting NB for
* real and complex. We assume that NB will take the same value in
* single or double precision.
*
NB = 1
*
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'PO' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NB = 1
ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRF' ) THEN
NB = 64
ELSE IF( C3.EQ.'TRD' ) THEN
NB = 1
ELSE IF( C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1:1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NB = 32
END IF
ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NB = 32
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1:1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NB = 32
END IF
ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'GB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'PB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'TR' ) THEN
IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'LA' ) THEN
IF( C3.EQ.'UUM' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
IF( C3.EQ.'EBZ' ) THEN
NB = 1
END IF
END IF
ILAENV = NB
RETURN
*
200 CONTINUE
*
* ISPEC = 2: minimum block size
*
NBMIN = 2
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NBMIN = 8
ELSE
NBMIN = 8
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1:1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NBMIN = 2
END IF
ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NBMIN = 2
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1:1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NBMIN = 2
END IF
ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NBMIN = 2
END IF
END IF
END IF
ILAENV = NBMIN
RETURN
*
300 CONTINUE
*
* ISPEC = 3: crossover point
*
NX = 0
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NX = 1
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NX = 1
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1:1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NX = 128
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1:1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
$ C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
$ C4.EQ.'BR' ) THEN
NX = 128
END IF
END IF
END IF
ILAENV = NX
RETURN
*
400 CONTINUE
*
* ISPEC = 4: number of shifts (used by xHSEQR)
*
ILAENV = 6
RETURN
*
500 CONTINUE
*
* ISPEC = 5: minimum column dimension (not used)
*
ILAENV = 2
RETURN
*
600 CONTINUE
*
* ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD)
*
ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
RETURN
*
700 CONTINUE
*
* ISPEC = 7: number of processors (not used)
*
ILAENV = 1
RETURN
*
800 CONTINUE
*
* ISPEC = 8: crossover point for multishift (used by xHSEQR)
*
ILAENV = 50
RETURN
*
* End of ILAENV
*
END
LOGICAL FUNCTION LSAME( CA, CB )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER CA, CB
* ..
*
* Purpose
* =======
*
* LSAME returns .TRUE. if CA is the same letter as CB regardless of
* case.
*
* Arguments
* =========
*
* CA (input) CHARACTER*1
* CB (input) CHARACTER*1
* CA and CB specify the single characters to be compared.
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC ICHAR
* ..
* .. Local Scalars ..
INTEGER INTA, INTB, ZCODE
* ..
* .. Executable Statements ..
*
* Test if the characters are equal
*
LSAME = CA.EQ.CB
IF( LSAME )
$ RETURN
*
* Now test for equivalence if both characters are alphabetic.
*
ZCODE = ICHAR( 'Z' )
*
* Use 'Z' rather than 'A' so that ASCII can be detected on Prime
* machines, on which ICHAR returns a value with bit 8 set.
* ICHAR('A') on Prime machines returns 193 which is the same as
* ICHAR('A') on an EBCDIC machine.
*
INTA = ICHAR( CA )
INTB = ICHAR( CB )
*
IF( ZCODE.EQ.90 .OR. ZCODE.EQ.122 ) THEN
*
* ASCII is assumed - ZCODE is the ASCII code of either lower or
* upper case 'Z'.
*
IF( INTA.GE.97 .AND. INTA.LE.122 ) INTA = INTA - 32
IF( INTB.GE.97 .AND. INTB.LE.122 ) INTB = INTB - 32
*
ELSE IF( ZCODE.EQ.233 .OR. ZCODE.EQ.169 ) THEN
*
* EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or
* upper case 'Z'.
*
IF( INTA.GE.129 .AND. INTA.LE.137 .OR.
$ INTA.GE.145 .AND. INTA.LE.153 .OR.
$ INTA.GE.162 .AND. INTA.LE.169 ) INTA = INTA + 64
IF( INTB.GE.129 .AND. INTB.LE.137 .OR.
$ INTB.GE.145 .AND. INTB.LE.153 .OR.
$ INTB.GE.162 .AND. INTB.LE.169 ) INTB = INTB + 64
*
ELSE IF( ZCODE.EQ.218 .OR. ZCODE.EQ.250 ) THEN
*
* ASCII is assumed, on Prime machines - ZCODE is the ASCII code
* plus 128 of either lower or upper case 'Z'.
*
IF( INTA.GE.225 .AND. INTA.LE.250 ) INTA = INTA - 32
IF( INTB.GE.225 .AND. INTB.LE.250 ) INTB = INTB - 32
END IF
LSAME = INTA.EQ.INTB
*
* RETURN
*
* End of LSAME
*
END
subroutine saxpy(n,sa,sx,incx,sy,incy)
c
c constant times a vector plus a vector.
c uses unrolled loop for increments equal to one.
c jack dongarra, linpack, 3/11/78.
c modified 12/3/93, array(1) declarations changed to array(*)
c
real sx(*),sy(*),sa
integer i,incx,incy,ix,iy,m,mp1,n
c
if(n.le.0)return
if (sa .eq. 0.0) return
if(incx.eq.1.and.incy.eq.1)go to 20
c
c code for unequal increments or equal increments
c not equal to 1
c
ix = 1
iy = 1
if(incx.lt.0)ix = (-n+1)*incx + 1
if(incy.lt.0)iy = (-n+1)*incy + 1
do 10 i = 1,n
sy(iy) = sy(iy) + sa*sx(ix)
ix = ix + incx
iy = iy + incy
10 continue
return
c
c code for both increments equal to 1
c
c
c clean-up loop
c
20 m = mod(n,4)
if( m .eq. 0 ) go to 40
do 30 i = 1,m
sy(i) = sy(i) + sa*sx(i)
30 continue
if( n .lt. 4 ) return
40 mp1 = m + 1
do 50 i = mp1,n,4
sy(i) = sy(i) + sa*sx(i)
sy(i + 1) = sy(i + 1) + sa*sx(i + 1)
sy(i + 2) = sy(i + 2) + sa*sx(i + 2)
sy(i + 3) = sy(i + 3) + sa*sx(i + 3)
50 continue
return
end
subroutine scopy(n,sx,incx,sy,incy)
c
c copies a vector, x, to a vector, y.
c uses unrolled loops for increments equal to 1.
c jack dongarra, linpack, 3/11/78.
c modified 12/3/93, array(1) declarations changed to array(*)
c
real sx(*),sy(*)
integer i,incx,incy,ix,iy,m,mp1,n
c
if(n.le.0)return
if(incx.eq.1.and.incy.eq.1)go to 20
c
c code for unequal increments or equal increments
c not equal to 1
c
ix = 1
iy = 1
if(incx.lt.0)ix = (-n+1)*incx + 1
if(incy.lt.0)iy = (-n+1)*incy + 1
do 10 i = 1,n
sy(iy) = sx(ix)
ix = ix + incx
iy = iy + incy
10 continue
return
c
c code for both increments equal to 1
c
c
c clean-up loop
c
20 m = mod(n,7)
if( m .eq. 0 ) go to 40
do 30 i = 1,m
sy(i) = sx(i)
30 continue
if( n .lt. 7 ) return
40 mp1 = m + 1
do 50 i = mp1,n,7
sy(i) = sx(i)
sy(i + 1) = sx(i + 1)
sy(i + 2) = sx(i + 2)
sy(i + 3) = sx(i + 3)
sy(i + 4) = sx(i + 4)
sy(i + 5) = sx(i + 5)
sy(i + 6) = sx(i + 6)
50 continue
return
end
real function sdot(n,sx,incx,sy,incy)
c
c forms the dot product of two vectors.
c uses unrolled loops for increments equal to one.
c jack dongarra, linpack, 3/11/78.
c modified 12/3/93, array(1) declarations changed to array(*)
c
real sx(*),sy(*),stemp
integer i,incx,incy,ix,iy,m,mp1,n
c
stemp = 0.0e0
sdot = 0.0e0
if(n.le.0)return
if(incx.eq.1.and.incy.eq.1)go to 20
c
c code for unequal increments or equal increments
c not equal to 1
c
ix = 1
iy = 1
if(incx.lt.0)ix = (-n+1)*incx + 1
if(incy.lt.0)iy = (-n+1)*incy + 1
do 10 i = 1,n
stemp = stemp + sx(ix)*sy(iy)
ix = ix + incx
iy = iy + incy
10 continue
sdot = stemp
return
c
c code for both increments equal to 1
c
c
c clean-up loop
c
20 m = mod(n,5)
if( m .eq. 0 ) go to 40
do 30 i = 1,m
stemp = stemp + sx(i)*sy(i)
30 continue
if( n .lt. 5 ) go to 60
40 mp1 = m + 1
do 50 i = mp1,n,5
stemp = stemp + sx(i)*sy(i) + sx(i + 1)*sy(i + 1) +
* sx(i + 2)*sy(i + 2) + sx(i + 3)*sy(i + 3) + sx(i + 4)*sy(i + 4)
50 continue
60 sdot = stemp
return
end
SUBROUTINE SGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB,
$ BETA, C, LDC )
* .. Scalar Arguments ..
CHARACTER*1 TRANSA, TRANSB
INTEGER M, N, K, LDA, LDB, LDC
REAL ALPHA, BETA
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * ), C( LDC, * )
* ..
*
* Purpose
* =======
*
* SGEMM performs one of the matrix-matrix operations
*
* C := alpha*op( A )*op( B ) + beta*C,
*
* where op( X ) is one of
*
* op( X ) = X or op( X ) = X',
*
* alpha and beta are scalars, and A, B and C are matrices, with op( A )
* an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
*
* Parameters
* ==========
*
* TRANSA - CHARACTER*1.
* On entry, TRANSA specifies the form of op( A ) to be used in
* the matrix multiplication as follows:
*
* TRANSA = 'N' or 'n', op( A ) = A.
*
* TRANSA = 'T' or 't', op( A ) = A'.
*
* TRANSA = 'C' or 'c', op( A ) = A'.
*
* Unchanged on exit.
*
* TRANSB - CHARACTER*1.
* On entry, TRANSB specifies the form of op( B ) to be used in
* the matrix multiplication as follows:
*
* TRANSB = 'N' or 'n', op( B ) = B.
*
* TRANSB = 'T' or 't', op( B ) = B'.
*
* TRANSB = 'C' or 'c', op( B ) = B'.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of the matrix
* op( A ) and of the matrix C. M must be at least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of the matrix
* op( B ) and the number of columns of the matrix C. N must be
* at least zero.
* Unchanged on exit.
*
* K - INTEGER.
* On entry, K specifies the number of columns of the matrix
* op( A ) and the number of rows of the matrix op( B ). K must
* be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, ka ), where ka is
* k when TRANSA = 'N' or 'n', and is m otherwise.
* Before entry with TRANSA = 'N' or 'n', the leading m by k
* part of the array A must contain the matrix A, otherwise
* the leading k by m part of the array A must contain the
* matrix A.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When TRANSA = 'N' or 'n' then
* LDA must be at least max( 1, m ), otherwise LDA must be at
* least max( 1, k ).
* Unchanged on exit.
*
* B - REAL array of DIMENSION ( LDB, kb ), where kb is
* n when TRANSB = 'N' or 'n', and is k otherwise.
* Before entry with TRANSB = 'N' or 'n', the leading k by n
* part of the array B must contain the matrix B, otherwise
* the leading n by k part of the array B must contain the
* matrix B.
* Unchanged on exit.
*
* LDB - INTEGER.
* On entry, LDB specifies the first dimension of B as declared
* in the calling (sub) program. When TRANSB = 'N' or 'n' then
* LDB must be at least max( 1, k ), otherwise LDB must be at
* least max( 1, n ).
* Unchanged on exit.
*
* BETA - REAL .
* On entry, BETA specifies the scalar beta. When BETA is
* supplied as zero then C need not be set on input.
* Unchanged on exit.
*
* C - REAL array of DIMENSION ( LDC, n ).
* Before entry, the leading m by n part of the array C must
* contain the matrix C, except when beta is zero, in which
* case C need not be set on entry.
* On exit, the array C is overwritten by the m by n matrix
* ( alpha*op( A )*op( B ) + beta*C ).
*
* LDC - INTEGER.
* On entry, LDC specifies the first dimension of C as declared
* in the calling (sub) program. LDC must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Local Scalars ..
LOGICAL NOTA, NOTB
INTEGER I, INFO, J, L, NCOLA, NROWA, NROWB
REAL TEMP
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Executable Statements ..
*
* Set NOTA and NOTB as true if A and B respectively are not
* transposed and set NROWA, NCOLA and NROWB as the number of rows
* and columns of A and the number of rows of B respectively.
*
NOTA = LSAME( TRANSA, 'N' )
NOTB = LSAME( TRANSB, 'N' )
IF( NOTA )THEN
NROWA = M
NCOLA = K
ELSE
NROWA = K
NCOLA = M
END IF
IF( NOTB )THEN
NROWB = K
ELSE
NROWB = N
END IF
*
* Test the input parameters.
*
INFO = 0
IF( ( .NOT.NOTA ).AND.
$ ( .NOT.LSAME( TRANSA, 'C' ) ).AND.
$ ( .NOT.LSAME( TRANSA, 'T' ) ) )THEN
INFO = 1
ELSE IF( ( .NOT.NOTB ).AND.
$ ( .NOT.LSAME( TRANSB, 'C' ) ).AND.
$ ( .NOT.LSAME( TRANSB, 'T' ) ) )THEN
INFO = 2
ELSE IF( M .LT.0 )THEN
INFO = 3
ELSE IF( N .LT.0 )THEN
INFO = 4
ELSE IF( K .LT.0 )THEN
INFO = 5
ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
INFO = 8
ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN
INFO = 10
ELSE IF( LDC.LT.MAX( 1, M ) )THEN
INFO = 13
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SGEMM ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
$ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
$ RETURN
*
* And if alpha.eq.zero.
*
IF( ALPHA.EQ.ZERO )THEN
IF( BETA.EQ.ZERO )THEN
DO 20, J = 1, N
DO 10, I = 1, M
C( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40, J = 1, N
DO 30, I = 1, M
C( I, J ) = BETA*C( I, J )
30 CONTINUE
40 CONTINUE
END IF
RETURN
END IF
*
* Start the operations.
*
IF( NOTB )THEN
IF( NOTA )THEN
*
* Form C := alpha*A*B + beta*C.
*
DO 90, J = 1, N
IF( BETA.EQ.ZERO )THEN
DO 50, I = 1, M
C( I, J ) = ZERO
50 CONTINUE
ELSE IF( BETA.NE.ONE )THEN
DO 60, I = 1, M
C( I, J ) = BETA*C( I, J )
60 CONTINUE
END IF
DO 80, L = 1, K
IF( B( L, J ).NE.ZERO )THEN
TEMP = ALPHA*B( L, J )
DO 70, I = 1, M
C( I, J ) = C( I, J ) + TEMP*A( I, L )
70 CONTINUE
END IF
80 CONTINUE
90 CONTINUE
ELSE
*
* Form C := alpha*A'*B + beta*C
*
DO 120, J = 1, N
DO 110, I = 1, M
TEMP = ZERO
DO 100, L = 1, K
TEMP = TEMP + A( L, I )*B( L, J )
100 CONTINUE
IF( BETA.EQ.ZERO )THEN
C( I, J ) = ALPHA*TEMP
ELSE
C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
END IF
110 CONTINUE
120 CONTINUE
END IF
ELSE
IF( NOTA )THEN
*
* Form C := alpha*A*B' + beta*C
*
DO 170, J = 1, N
IF( BETA.EQ.ZERO )THEN
DO 130, I = 1, M
C( I, J ) = ZERO
130 CONTINUE
ELSE IF( BETA.NE.ONE )THEN
DO 140, I = 1, M
C( I, J ) = BETA*C( I, J )
140 CONTINUE
END IF
DO 160, L = 1, K
IF( B( J, L ).NE.ZERO )THEN
TEMP = ALPHA*B( J, L )
DO 150, I = 1, M
C( I, J ) = C( I, J ) + TEMP*A( I, L )
150 CONTINUE
END IF
160 CONTINUE
170 CONTINUE
ELSE
*
* Form C := alpha*A'*B' + beta*C
*
DO 200, J = 1, N
DO 190, I = 1, M
TEMP = ZERO
DO 180, L = 1, K
TEMP = TEMP + A( L, I )*B( J, L )
180 CONTINUE
IF( BETA.EQ.ZERO )THEN
C( I, J ) = ALPHA*TEMP
ELSE
C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
END IF
190 CONTINUE
200 CONTINUE
END IF
END IF
*
RETURN
*
* End of SGEMM .
*
END
SUBROUTINE SGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
* .. Scalar Arguments ..
REAL ALPHA, BETA
INTEGER INCX, INCY, LDA, M, N
CHARACTER*1 TRANS
* .. Array Arguments ..
REAL A( LDA, * ), X( * ), Y( * )
* ..
*
* Purpose
* =======
*
* SGEMV performs one of the matrix-vector operations
*
* y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y,
*
* where alpha and beta are scalars, x and y are vectors and A is an
* m by n matrix.
*
* Parameters
* ==========
*
* TRANS - CHARACTER*1.
* On entry, TRANS specifies the operation to be performed as
* follows:
*
* TRANS = 'N' or 'n' y := alpha*A*x + beta*y.
*
* TRANS = 'T' or 't' y := alpha*A'*x + beta*y.
*
* TRANS = 'C' or 'c' y := alpha*A'*x + beta*y.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of the matrix A.
* M must be at least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, n ).
* Before entry, the leading m by n part of the array A must
* contain the matrix of coefficients.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, m ).
* Unchanged on exit.
*
* X - REAL array of DIMENSION at least
* ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
* and at least
* ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
* Before entry, the incremented array X must contain the
* vector x.
* Unchanged on exit.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
* BETA - REAL .
* On entry, BETA specifies the scalar beta. When BETA is
* supplied as zero then Y need not be set on input.
* Unchanged on exit.
*
* Y - REAL array of DIMENSION at least
* ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
* and at least
* ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
* Before entry with BETA non-zero, the incremented array Y
* must contain the vector y. On exit, Y is overwritten by the
* updated vector y.
*
* INCY - INTEGER.
* On entry, INCY specifies the increment for the elements of
* Y. INCY must not be zero.
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* .. Local Scalars ..
REAL TEMP
INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF ( .NOT.LSAME( TRANS, 'N' ).AND.
$ .NOT.LSAME( TRANS, 'T' ).AND.
$ .NOT.LSAME( TRANS, 'C' ) )THEN
INFO = 1
ELSE IF( M.LT.0 )THEN
INFO = 2
ELSE IF( N.LT.0 )THEN
INFO = 3
ELSE IF( LDA.LT.MAX( 1, M ) )THEN
INFO = 6
ELSE IF( INCX.EQ.0 )THEN
INFO = 8
ELSE IF( INCY.EQ.0 )THEN
INFO = 11
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SGEMV ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
$ ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) )
$ RETURN
*
* Set LENX and LENY, the lengths of the vectors x and y, and set
* up the start points in X and Y.
*
IF( LSAME( TRANS, 'N' ) )THEN
LENX = N
LENY = M
ELSE
LENX = M
LENY = N
END IF
IF( INCX.GT.0 )THEN
KX = 1
ELSE
KX = 1 - ( LENX - 1 )*INCX
END IF
IF( INCY.GT.0 )THEN
KY = 1
ELSE
KY = 1 - ( LENY - 1 )*INCY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through A.
*
* First form y := beta*y.
*
IF( BETA.NE.ONE )THEN
IF( INCY.EQ.1 )THEN
IF( BETA.EQ.ZERO )THEN
DO 10, I = 1, LENY
Y( I ) = ZERO
10 CONTINUE
ELSE
DO 20, I = 1, LENY
Y( I ) = BETA*Y( I )
20 CONTINUE
END IF
ELSE
IY = KY
IF( BETA.EQ.ZERO )THEN
DO 30, I = 1, LENY
Y( IY ) = ZERO
IY = IY + INCY
30 CONTINUE
ELSE
DO 40, I = 1, LENY
Y( IY ) = BETA*Y( IY )
IY = IY + INCY
40 CONTINUE
END IF
END IF
END IF
IF( ALPHA.EQ.ZERO )
$ RETURN
IF( LSAME( TRANS, 'N' ) )THEN
*
* Form y := alpha*A*x + y.
*
JX = KX
IF( INCY.EQ.1 )THEN
DO 60, J = 1, N
IF( X( JX ).NE.ZERO )THEN
TEMP = ALPHA*X( JX )
DO 50, I = 1, M
Y( I ) = Y( I ) + TEMP*A( I, J )
50 CONTINUE
END IF
JX = JX + INCX
60 CONTINUE
ELSE
DO 80, J = 1, N
IF( X( JX ).NE.ZERO )THEN
TEMP = ALPHA*X( JX )
IY = KY
DO 70, I = 1, M
Y( IY ) = Y( IY ) + TEMP*A( I, J )
IY = IY + INCY
70 CONTINUE
END IF
JX = JX + INCX
80 CONTINUE
END IF
ELSE
*
* Form y := alpha*A'*x + y.
*
JY = KY
IF( INCX.EQ.1 )THEN
DO 100, J = 1, N
TEMP = ZERO
DO 90, I = 1, M
TEMP = TEMP + A( I, J )*X( I )
90 CONTINUE
Y( JY ) = Y( JY ) + ALPHA*TEMP
JY = JY + INCY
100 CONTINUE
ELSE
DO 120, J = 1, N
TEMP = ZERO
IX = KX
DO 110, I = 1, M
TEMP = TEMP + A( I, J )*X( IX )
IX = IX + INCX
110 CONTINUE
Y( JY ) = Y( JY ) + ALPHA*TEMP
JY = JY + INCY
120 CONTINUE
END IF
END IF
*
RETURN
*
* End of SGEMV .
*
END
SUBROUTINE SGER ( M, N, ALPHA, X, INCX, Y, INCY, A, LDA )
* .. Scalar Arguments ..
REAL ALPHA
INTEGER INCX, INCY, LDA, M, N
* .. Array Arguments ..
REAL A( LDA, * ), X( * ), Y( * )
* ..
*
* Purpose
* =======
*
* SGER performs the rank 1 operation
*
* A := alpha*x*y' + A,
*
* where alpha is a scalar, x is an m element vector, y is an n element
* vector and A is an m by n matrix.
*
* Parameters
* ==========
*
* M - INTEGER.
* On entry, M specifies the number of rows of the matrix A.
* M must be at least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* X - REAL array of dimension at least
* ( 1 + ( m - 1 )*abs( INCX ) ).
* Before entry, the incremented array X must contain the m
* element vector x.
* Unchanged on exit.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
* Y - REAL array of dimension at least
* ( 1 + ( n - 1 )*abs( INCY ) ).
* Before entry, the incremented array Y must contain the n
* element vector y.
* Unchanged on exit.
*
* INCY - INTEGER.
* On entry, INCY specifies the increment for the elements of
* Y. INCY must not be zero.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, n ).
* Before entry, the leading m by n part of the array A must
* contain the matrix of coefficients. On exit, A is
* overwritten by the updated matrix.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* .. Local Scalars ..
REAL TEMP
INTEGER I, INFO, IX, J, JY, KX
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF ( M.LT.0 )THEN
INFO = 1
ELSE IF( N.LT.0 )THEN
INFO = 2
ELSE IF( INCX.EQ.0 )THEN
INFO = 5
ELSE IF( INCY.EQ.0 )THEN
INFO = 7
ELSE IF( LDA.LT.MAX( 1, M ) )THEN
INFO = 9
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SGER ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) )
$ RETURN
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through A.
*
IF( INCY.GT.0 )THEN
JY = 1
ELSE
JY = 1 - ( N - 1 )*INCY
END IF
IF( INCX.EQ.1 )THEN
DO 20, J = 1, N
IF( Y( JY ).NE.ZERO )THEN
TEMP = ALPHA*Y( JY )
DO 10, I = 1, M
A( I, J ) = A( I, J ) + X( I )*TEMP
10 CONTINUE
END IF
JY = JY + INCY
20 CONTINUE
ELSE
IF( INCX.GT.0 )THEN
KX = 1
ELSE
KX = 1 - ( M - 1 )*INCX
END IF
DO 40, J = 1, N
IF( Y( JY ).NE.ZERO )THEN
TEMP = ALPHA*Y( JY )
IX = KX
DO 30, I = 1, M
A( I, J ) = A( I, J ) + X( IX )*TEMP
IX = IX + INCX
30 CONTINUE
END IF
JY = JY + INCY
40 CONTINUE
END IF
*
RETURN
*
* End of SGER .
*
END
SUBROUTINE SLAE2( A, B, C, RT1, RT2 )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
REAL A, B, C, RT1, RT2
* ..
*
* Purpose
* =======
*
* SLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix
* [ A B ]
* [ B C ].
* On return, RT1 is the eigenvalue of larger absolute value, and RT2
* is the eigenvalue of smaller absolute value.
*
* Arguments
* =========
*
* A (input) REAL
* The (1,1) element of the 2-by-2 matrix.
*
* B (input) REAL
* The (1,2) and (2,1) elements of the 2-by-2 matrix.
*
* C (input) REAL
* The (2,2) element of the 2-by-2 matrix.
*
* RT1 (output) REAL
* The eigenvalue of larger absolute value.
*
* RT2 (output) REAL
* The eigenvalue of smaller absolute value.
*
* Further Details
* ===============
*
* RT1 is accurate to a few ulps barring over/underflow.
*
* RT2 may be inaccurate if there is massive cancellation in the
* determinant A*C-B*B; higher precision or correctly rounded or
* correctly truncated arithmetic would be needed to compute RT2
* accurately in all cases.
*
* Overflow is possible only if RT1 is within a factor of 5 of overflow.
* Underflow is harmless if the input data is 0 or exceeds
* underflow_threshold / macheps.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E0 )
REAL TWO
PARAMETER ( TWO = 2.0E0 )
REAL ZERO
PARAMETER ( ZERO = 0.0E0 )
REAL HALF
PARAMETER ( HALF = 0.5E0 )
* ..
* .. Local Scalars ..
REAL AB, ACMN, ACMX, ADF, DF, RT, SM, TB
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
END IF
RETURN
*
* End of SLAE2
*
END
SUBROUTINE SLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
REAL A, B, C, CS1, RT1, RT2, SN1
* ..
*
* Purpose
* =======
*
* SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
* [ A B ]
* [ B C ].
* On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
* eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
* eigenvector for RT1, giving the decomposition
*
* [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ]
* [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
*
* Arguments
* =========
*
* A (input) REAL
* The (1,1) element of the 2-by-2 matrix.
*
* B (input) REAL
* The (1,2) element and the conjugate of the (2,1) element of
* the 2-by-2 matrix.
*
* C (input) REAL
* The (2,2) element of the 2-by-2 matrix.
*
* RT1 (output) REAL
* The eigenvalue of larger absolute value.
*
* RT2 (output) REAL
* The eigenvalue of smaller absolute value.
*
* CS1 (output) REAL
* SN1 (output) REAL
* The vector (CS1, SN1) is a unit right eigenvector for RT1.
*
* Further Details
* ===============
*
* RT1 is accurate to a few ulps barring over/underflow.
*
* RT2 may be inaccurate if there is massive cancellation in the
* determinant A*C-B*B; higher precision or correctly rounded or
* correctly truncated arithmetic would be needed to compute RT2
* accurately in all cases.
*
* CS1 and SN1 are accurate to a few ulps barring over/underflow.
*
* Overflow is possible only if RT1 is within a factor of 5 of overflow.
* Underflow is harmless if the input data is 0 or exceeds
* underflow_threshold / macheps.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E0 )
REAL TWO
PARAMETER ( TWO = 2.0E0 )
REAL ZERO
PARAMETER ( ZERO = 0.0E0 )
REAL HALF
PARAMETER ( HALF = 0.5E0 )
* ..
* .. Local Scalars ..
INTEGER SGN1, SGN2
REAL AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
$ TB, TN
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
SGN1 = -1
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
SGN1 = 1
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
SGN1 = 1
END IF
*
* Compute the eigenvector
*
IF( DF.GE.ZERO ) THEN
CS = DF + RT
SGN2 = 1
ELSE
CS = DF - RT
SGN2 = -1
END IF
ACS = ABS( CS )
IF( ACS.GT.AB ) THEN
CT = -TB / CS
SN1 = ONE / SQRT( ONE+CT*CT )
CS1 = CT*SN1
ELSE
IF( AB.EQ.ZERO ) THEN
CS1 = ONE
SN1 = ZERO
ELSE
TN = -CS / TB
CS1 = ONE / SQRT( ONE+TN*TN )
SN1 = TN*CS1
END IF
END IF
IF( SGN1.EQ.SGN2 ) THEN
TN = CS1
CS1 = -SN1
SN1 = TN
END IF
RETURN
*
* End of SLAEV2
*
END
REAL FUNCTION SLAMCH( CMACH )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER CMACH
* ..
*
* Purpose
* =======
*
* SLAMCH determines single precision machine parameters.
*
* Arguments
* =========
*
* CMACH (input) CHARACTER*1
* Specifies the value to be returned by SLAMCH:
* = 'E' or 'e', SLAMCH := eps
* = 'S' or 's , SLAMCH := sfmin
* = 'B' or 'b', SLAMCH := base
* = 'P' or 'p', SLAMCH := eps*base
* = 'N' or 'n', SLAMCH := t
* = 'R' or 'r', SLAMCH := rnd
* = 'M' or 'm', SLAMCH := emin
* = 'U' or 'u', SLAMCH := rmin
* = 'L' or 'l', SLAMCH := emax
* = 'O' or 'o', SLAMCH := rmax
*
* where
*
* eps = relative machine precision
* sfmin = safe minimum, such that 1/sfmin does not overflow
* base = base of the machine
* prec = eps*base
* t = number of (base) digits in the mantissa
* rnd = 1.0 when rounding occurs in addition, 0.0 otherwise
* emin = minimum exponent before (gradual) underflow
* rmin = underflow threshold - base**(emin-1)
* emax = largest exponent before overflow
* rmax = overflow threshold - (base**emax)*(1-eps)
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL FIRST, LRND
INTEGER BETA, IMAX, IMIN, IT
REAL BASE, EMAX, EMIN, EPS, PREC, RMACH, RMAX, RMIN,
$ RND, SFMIN, SMALL, T
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SLAMC2
* ..
* .. Save statement ..
SAVE FIRST, EPS, SFMIN, BASE, T, RND, EMIN, RMIN,
$ EMAX, RMAX, PREC
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
CALL SLAMC2( BETA, IT, LRND, EPS, IMIN, RMIN, IMAX, RMAX )
BASE = BETA
T = IT
IF( LRND ) THEN
RND = ONE
EPS = ( BASE**( 1-IT ) ) / 2
ELSE
RND = ZERO
EPS = BASE**( 1-IT )
END IF
PREC = EPS*BASE
EMIN = IMIN
EMAX = IMAX
SFMIN = RMIN
SMALL = ONE / RMAX
IF( SMALL.GE.SFMIN ) THEN
*
* Use SMALL plus a bit, to avoid the possibility of rounding
* causing overflow when computing 1/sfmin.
*
SFMIN = SMALL*( ONE+EPS )
END IF
END IF
*
IF( LSAME( CMACH, 'E' ) ) THEN
RMACH = EPS
ELSE IF( LSAME( CMACH, 'S' ) ) THEN
RMACH = SFMIN
ELSE IF( LSAME( CMACH, 'B' ) ) THEN
RMACH = BASE
ELSE IF( LSAME( CMACH, 'P' ) ) THEN
RMACH = PREC
ELSE IF( LSAME( CMACH, 'N' ) ) THEN
RMACH = T
ELSE IF( LSAME( CMACH, 'R' ) ) THEN
RMACH = RND
ELSE IF( LSAME( CMACH, 'M' ) ) THEN
RMACH = EMIN
ELSE IF( LSAME( CMACH, 'U' ) ) THEN
RMACH = RMIN
ELSE IF( LSAME( CMACH, 'L' ) ) THEN
RMACH = EMAX
ELSE IF( LSAME( CMACH, 'O' ) ) THEN
RMACH = RMAX
END IF
*
SLAMCH = RMACH
RETURN
*
* End of SLAMCH
*
END
*
************************************************************************
*
SUBROUTINE SLAMC1( BETA, T, RND, IEEE1 )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL IEEE1, RND
INTEGER BETA, T
* ..
*
* Purpose
* =======
*
* SLAMC1 determines the machine parameters given by BETA, T, RND, and
* IEEE1.
*
* Arguments
* =========
*
* BETA (output) INTEGER
* The base of the machine.
*
* T (output) INTEGER
* The number of ( BETA ) digits in the mantissa.
*
* RND (output) LOGICAL
* Specifies whether proper rounding ( RND = .TRUE. ) or
* chopping ( RND = .FALSE. ) occurs in addition. This may not
* be a reliable guide to the way in which the machine performs
* its arithmetic.
*
* IEEE1 (output) LOGICAL
* Specifies whether rounding appears to be done in the IEEE
* 'round to nearest' style.
*
* Further Details
* ===============
*
* The routine is based on the routine ENVRON by Malcolm and
* incorporates suggestions by Gentleman and Marovich. See
*
* Malcolm M. A. (1972) Algorithms to reveal properties of
* floating-point arithmetic. Comms. of the ACM, 15, 949-951.
*
* Gentleman W. M. and Marovich S. B. (1974) More on algorithms
* that reveal properties of floating point arithmetic units.
* Comms. of the ACM, 17, 276-277.
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL FIRST, LIEEE1, LRND
INTEGER LBETA, LT
REAL A, B, C, F, ONE, QTR, SAVEC, T1, T2
* ..
* .. External Functions ..
REAL SLAMC3
EXTERNAL SLAMC3
* ..
* .. Save statement ..
SAVE FIRST, LIEEE1, LBETA, LRND, LT
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
ONE = 1
*
* LBETA, LIEEE1, LT and LRND are the local values of BETA,
* IEEE1, T and RND.
*
* Throughout this routine we use the function SLAMC3 to ensure
* that relevant values are stored and not held in registers, or
* are not affected by optimizers.
*
* Compute a = 2.0**m with the smallest positive integer m such
* that
*
* fl( a + 1.0 ) = a.
*
A = 1
C = 1
*
*+ WHILE( C.EQ.ONE )LOOP
10 CONTINUE
IF( C.EQ.ONE ) THEN
A = 2*A
C = SLAMC3( A, ONE )
C = SLAMC3( C, -A )
GO TO 10
END IF
*+ END WHILE
*
* Now compute b = 2.0**m with the smallest positive integer m
* such that
*
* fl( a + b ) .gt. a.
*
B = 1
C = SLAMC3( A, B )
*
*+ WHILE( C.EQ.A )LOOP
20 CONTINUE
IF( C.EQ.A ) THEN
B = 2*B
C = SLAMC3( A, B )
GO TO 20
END IF
*+ END WHILE
*
* Now compute the base. a and c are neighbouring floating point
* numbers in the interval ( beta**t, beta**( t + 1 ) ) and so
* their difference is beta. Adding 0.25 to c is to ensure that it
* is truncated to beta and not ( beta - 1 ).
*
QTR = ONE / 4
SAVEC = C
C = SLAMC3( C, -A )
LBETA = C + QTR
*
* Now determine whether rounding or chopping occurs, by adding a
* bit less than beta/2 and a bit more than beta/2 to a.
*
B = LBETA
F = SLAMC3( B / 2, -B / 100 )
C = SLAMC3( F, A )
IF( C.EQ.A ) THEN
LRND = .TRUE.
ELSE
LRND = .FALSE.
END IF
F = SLAMC3( B / 2, B / 100 )
C = SLAMC3( F, A )
IF( ( LRND ) .AND. ( C.EQ.A ) )
$ LRND = .FALSE.
*
* Try and decide whether rounding is done in the IEEE 'round to
* nearest' style. B/2 is half a unit in the last place of the two
* numbers A and SAVEC. Furthermore, A is even, i.e. has last bit
* zero, and SAVEC is odd. Thus adding B/2 to A should not change
* A, but adding B/2 to SAVEC should change SAVEC.
*
T1 = SLAMC3( B / 2, A )
T2 = SLAMC3( B / 2, SAVEC )
LIEEE1 = ( T1.EQ.A ) .AND. ( T2.GT.SAVEC ) .AND. LRND
*
* Now find the mantissa, t. It should be the integer part of
* log to the base beta of a, however it is safer to determine t
* by powering. So we find t as the smallest positive integer for
* which
*
* fl( beta**t + 1.0 ) = 1.0.
*
LT = 0
A = 1
C = 1
*
*+ WHILE( C.EQ.ONE )LOOP
30 CONTINUE
IF( C.EQ.ONE ) THEN
LT = LT + 1
A = A*LBETA
C = SLAMC3( A, ONE )
C = SLAMC3( C, -A )
GO TO 30
END IF
*+ END WHILE
*
END IF
*
BETA = LBETA
T = LT
RND = LRND
IEEE1 = LIEEE1
RETURN
*
* End of SLAMC1
*
END
*
************************************************************************
*
SUBROUTINE SLAMC2( BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL RND
INTEGER BETA, EMAX, EMIN, T
REAL EPS, RMAX, RMIN
* ..
*
* Purpose
* =======
*
* SLAMC2 determines the machine parameters specified in its argument
* list.
*
* Arguments
* =========
*
* BETA (output) INTEGER
* The base of the machine.
*
* T (output) INTEGER
* The number of ( BETA ) digits in the mantissa.
*
* RND (output) LOGICAL
* Specifies whether proper rounding ( RND = .TRUE. ) or
* chopping ( RND = .FALSE. ) occurs in addition. This may not
* be a reliable guide to the way in which the machine performs
* its arithmetic.
*
* EPS (output) REAL
* The smallest positive number such that
*
* fl( 1.0 - EPS ) .LT. 1.0,
*
* where fl denotes the computed value.
*
* EMIN (output) INTEGER
* The minimum exponent before (gradual) underflow occurs.
*
* RMIN (output) REAL
* The smallest normalized number for the machine, given by
* BASE**( EMIN - 1 ), where BASE is the floating point value
* of BETA.
*
* EMAX (output) INTEGER
* The maximum exponent before overflow occurs.
*
* RMAX (output) REAL
* The largest positive number for the machine, given by
* BASE**EMAX * ( 1 - EPS ), where BASE is the floating point
* value of BETA.
*
* Further Details
* ===============
*
* The computation of EPS is based on a routine PARANOIA by
* W. Kahan of the University of California at Berkeley.
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL FIRST, IEEE, IWARN, LIEEE1, LRND
INTEGER GNMIN, GPMIN, I, LBETA, LEMAX, LEMIN, LT,
$ NGNMIN, NGPMIN
REAL A, B, C, HALF, LEPS, LRMAX, LRMIN, ONE, RBASE,
$ SIXTH, SMALL, THIRD, TWO, ZERO
* ..
* .. External Functions ..
REAL SLAMC3
EXTERNAL SLAMC3
* ..
* .. External Subroutines ..
EXTERNAL SLAMC1, SLAMC4, SLAMC5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Save statement ..
SAVE FIRST, IWARN, LBETA, LEMAX, LEMIN, LEPS, LRMAX,
$ LRMIN, LT
* ..
* .. Data statements ..
DATA FIRST / .TRUE. / , IWARN / .FALSE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
ZERO = 0
ONE = 1
TWO = 2
*
* LBETA, LT, LRND, LEPS, LEMIN and LRMIN are the local values of
* BETA, T, RND, EPS, EMIN and RMIN.
*
* Throughout this routine we use the function SLAMC3 to ensure
* that relevant values are stored and not held in registers, or
* are not affected by optimizers.
*
* SLAMC1 returns the parameters LBETA, LT, LRND and LIEEE1.
*
CALL SLAMC1( LBETA, LT, LRND, LIEEE1 )
*
* Start to find EPS.
*
B = LBETA
A = B**( -LT )
LEPS = A
*
* Try some tricks to see whether or not this is the correct EPS.
*
B = TWO / 3
HALF = ONE / 2
SIXTH = SLAMC3( B, -HALF )
THIRD = SLAMC3( SIXTH, SIXTH )
B = SLAMC3( THIRD, -HALF )
B = SLAMC3( B, SIXTH )
B = ABS( B )
IF( B.LT.LEPS )
$ B = LEPS
*
LEPS = 1
*
*+ WHILE( ( LEPS.GT.B ).AND.( B.GT.ZERO ) )LOOP
10 CONTINUE
IF( ( LEPS.GT.B ) .AND. ( B.GT.ZERO ) ) THEN
LEPS = B
C = SLAMC3( HALF*LEPS, ( TWO**5 )*( LEPS**2 ) )
C = SLAMC3( HALF, -C )
B = SLAMC3( HALF, C )
C = SLAMC3( HALF, -B )
B = SLAMC3( HALF, C )
GO TO 10
END IF
*+ END WHILE
*
IF( A.LT.LEPS )
$ LEPS = A
*
* Computation of EPS complete.
*
* Now find EMIN. Let A = + or - 1, and + or - (1 + BASE**(-3)).
* Keep dividing A by BETA until (gradual) underflow occurs. This
* is detected when we cannot recover the previous A.
*
RBASE = ONE / LBETA
SMALL = ONE
DO 20 I = 1, 3
SMALL = SLAMC3( SMALL*RBASE, ZERO )
20 CONTINUE
A = SLAMC3( ONE, SMALL )
CALL SLAMC4( NGPMIN, ONE, LBETA )
CALL SLAMC4( NGNMIN, -ONE, LBETA )
CALL SLAMC4( GPMIN, A, LBETA )
CALL SLAMC4( GNMIN, -A, LBETA )
IEEE = .FALSE.
*
IF( ( NGPMIN.EQ.NGNMIN ) .AND. ( GPMIN.EQ.GNMIN ) ) THEN
IF( NGPMIN.EQ.GPMIN ) THEN
LEMIN = NGPMIN
* ( Non twos-complement machines, no gradual underflow;
* e.g., VAX )
ELSE IF( ( GPMIN-NGPMIN ).EQ.3 ) THEN
LEMIN = NGPMIN - 1 + LT
IEEE = .TRUE.
* ( Non twos-complement machines, with gradual underflow;
* e.g., IEEE standard followers )
ELSE
LEMIN = MIN( NGPMIN, GPMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE IF( ( NGPMIN.EQ.GPMIN ) .AND. ( NGNMIN.EQ.GNMIN ) ) THEN
IF( ABS( NGPMIN-NGNMIN ).EQ.1 ) THEN
LEMIN = MAX( NGPMIN, NGNMIN )
* ( Twos-complement machines, no gradual underflow;
* e.g., CYBER 205 )
ELSE
LEMIN = MIN( NGPMIN, NGNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE IF( ( ABS( NGPMIN-NGNMIN ).EQ.1 ) .AND.
$ ( GPMIN.EQ.GNMIN ) ) THEN
IF( ( GPMIN-MIN( NGPMIN, NGNMIN ) ).EQ.3 ) THEN
LEMIN = MAX( NGPMIN, NGNMIN ) - 1 + LT
* ( Twos-complement machines with gradual underflow;
* no known machine )
ELSE
LEMIN = MIN( NGPMIN, NGNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE
LEMIN = MIN( NGPMIN, NGNMIN, GPMIN, GNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
***
* Comment out this if block if EMIN is ok
IF( IWARN ) THEN
FIRST = .TRUE.
WRITE( 6, FMT = 9999 )LEMIN
END IF
***
*
* Assume IEEE arithmetic if we found denormalised numbers above,
* or if arithmetic seems to round in the IEEE style, determined
* in routine SLAMC1. A true IEEE machine should have both things
* true; however, faulty machines may have one or the other.
*
IEEE = IEEE .OR. LIEEE1
*
* Compute RMIN by successive division by BETA. We could compute
* RMIN as BASE**( EMIN - 1 ), but some machines underflow during
* this computation.
*
LRMIN = 1
DO 30 I = 1, 1 - LEMIN
LRMIN = SLAMC3( LRMIN*RBASE, ZERO )
30 CONTINUE
*
* Finally, call SLAMC5 to compute EMAX and RMAX.
*
CALL SLAMC5( LBETA, LT, LEMIN, IEEE, LEMAX, LRMAX )
END IF
*
BETA = LBETA
T = LT
RND = LRND
EPS = LEPS
EMIN = LEMIN
RMIN = LRMIN
EMAX = LEMAX
RMAX = LRMAX
*
RETURN
*
9999 FORMAT( / / ' WARNING. The value EMIN may be incorrect:-',
$ ' EMIN = ', I8, /
$ ' If, after inspection, the value EMIN looks',
$ ' acceptable please comment out ',
$ / ' the IF block as marked within the code of routine',
$ ' SLAMC2,', / ' otherwise supply EMIN explicitly.', / )
*
* End of SLAMC2
*
END
*
************************************************************************
*
REAL FUNCTION SLAMC3( A, B )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
REAL A, B
* ..
*
* Purpose
* =======
*
* SLAMC3 is intended to force A and B to be stored prior to doing
* the addition of A and B , for use in situations where optimizers
* might hold one of these in a register.
*
* Arguments
* =========
*
* A, B (input) REAL
* The values A and B.
*
* =====================================================================
*
* .. Executable Statements ..
*
SLAMC3 = A + B
*
RETURN
*
* End of SLAMC3
*
END
*
************************************************************************
*
SUBROUTINE SLAMC4( EMIN, START, BASE )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
INTEGER BASE, EMIN
REAL START
* ..
*
* Purpose
* =======
*
* SLAMC4 is a service routine for SLAMC2.
*
* Arguments
* =========
*
* EMIN (output) EMIN
* The minimum exponent before (gradual) underflow, computed by
* setting A = START and dividing by BASE until the previous A
* can not be recovered.
*
* START (input) REAL
* The starting point for determining EMIN.
*
* BASE (input) INTEGER
* The base of the machine.
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I
REAL A, B1, B2, C1, C2, D1, D2, ONE, RBASE, ZERO
* ..
* .. External Functions ..
REAL SLAMC3
EXTERNAL SLAMC3
* ..
* .. Executable Statements ..
*
A = START
ONE = 1
RBASE = ONE / BASE
ZERO = 0
EMIN = 1
B1 = SLAMC3( A*RBASE, ZERO )
C1 = A
C2 = A
D1 = A
D2 = A
*+ WHILE( ( C1.EQ.A ).AND.( C2.EQ.A ).AND.
* $ ( D1.EQ.A ).AND.( D2.EQ.A ) )LOOP
10 CONTINUE
IF( ( C1.EQ.A ) .AND. ( C2.EQ.A ) .AND. ( D1.EQ.A ) .AND.
$ ( D2.EQ.A ) ) THEN
EMIN = EMIN - 1
A = B1
B1 = SLAMC3( A / BASE, ZERO )
C1 = SLAMC3( B1*BASE, ZERO )
D1 = ZERO
DO 20 I = 1, BASE
D1 = D1 + B1
20 CONTINUE
B2 = SLAMC3( A*RBASE, ZERO )
C2 = SLAMC3( B2 / RBASE, ZERO )
D2 = ZERO
DO 30 I = 1, BASE
D2 = D2 + B2
30 CONTINUE
GO TO 10
END IF
*+ END WHILE
*
RETURN
*
* End of SLAMC4
*
END
*
************************************************************************
*
SUBROUTINE SLAMC5( BETA, P, EMIN, IEEE, EMAX, RMAX )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER BETA, EMAX, EMIN, P
REAL RMAX
* ..
*
* Purpose
* =======
*
* SLAMC5 attempts to compute RMAX, the largest machine floating-point
* number, without overflow. It assumes that EMAX + abs(EMIN) sum
* approximately to a power of 2. It will fail on machines where this
* assumption does not hold, for example, the Cyber 205 (EMIN = -28625,
* EMAX = 28718). It will also fail if the value supplied for EMIN is
* too large (i.e. too close to zero), probably with overflow.
*
* Arguments
* =========
*
* BETA (input) INTEGER
* The base of floating-point arithmetic.
*
* P (input) INTEGER
* The number of base BETA digits in the mantissa of a
* floating-point value.
*
* EMIN (input) INTEGER
* The minimum exponent before (gradual) underflow.
*
* IEEE (input) LOGICAL
* A logical flag specifying whether or not the arithmetic
* system is thought to comply with the IEEE standard.
*
* EMAX (output) INTEGER
* The largest exponent before overflow
*
* RMAX (output) REAL
* The largest machine floating-point number.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
INTEGER EXBITS, EXPSUM, I, LEXP, NBITS, TRY, UEXP
REAL OLDY, RECBAS, Y, Z
* ..
* .. External Functions ..
REAL SLAMC3
EXTERNAL SLAMC3
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* First compute LEXP and UEXP, two powers of 2 that bound
* abs(EMIN). We then assume that EMAX + abs(EMIN) will sum
* approximately to the bound that is closest to abs(EMIN).
* (EMAX is the exponent of the required number RMAX).
*
LEXP = 1
EXBITS = 1
10 CONTINUE
TRY = LEXP*2
IF( TRY.LE.( -EMIN ) ) THEN
LEXP = TRY
EXBITS = EXBITS + 1
GO TO 10
END IF
IF( LEXP.EQ.-EMIN ) THEN
UEXP = LEXP
ELSE
UEXP = TRY
EXBITS = EXBITS + 1
END IF
*
* Now -LEXP is less than or equal to EMIN, and -UEXP is greater
* than or equal to EMIN. EXBITS is the number of bits needed to
* store the exponent.
*
IF( ( UEXP+EMIN ).GT.( -LEXP-EMIN ) ) THEN
EXPSUM = 2*LEXP
ELSE
EXPSUM = 2*UEXP
END IF
*
* EXPSUM is the exponent range, approximately equal to
* EMAX - EMIN + 1 .
*
EMAX = EXPSUM + EMIN - 1
NBITS = 1 + EXBITS + P
*
* NBITS is the total number of bits needed to store a
* floating-point number.
*
IF( ( MOD( NBITS, 2 ).EQ.1 ) .AND. ( BETA.EQ.2 ) ) THEN
*
* Either there are an odd number of bits used to store a
* floating-point number, which is unlikely, or some bits are
* not used in the representation of numbers, which is possible,
* (e.g. Cray machines) or the mantissa has an implicit bit,
* (e.g. IEEE machines, Dec Vax machines), which is perhaps the
* most likely. We have to assume the last alternative.
* If this is true, then we need to reduce EMAX by one because
* there must be some way of representing zero in an implicit-bit
* system. On machines like Cray, we are reducing EMAX by one
* unnecessarily.
*
EMAX = EMAX - 1
END IF
*
IF( IEEE ) THEN
*
* Assume we are on an IEEE machine which reserves one exponent
* for infinity and NaN.
*
EMAX = EMAX - 1
END IF
*
* Now create RMAX, the largest machine number, which should
* be equal to (1.0 - BETA**(-P)) * BETA**EMAX .
*
* First compute 1.0 - BETA**(-P), being careful that the
* result is less than 1.0 .
*
RECBAS = ONE / BETA
Z = BETA - ONE
Y = ZERO
DO 20 I = 1, P
Z = Z*RECBAS
IF( Y.LT.ONE )
$ OLDY = Y
Y = SLAMC3( Y, Z )
20 CONTINUE
IF( Y.GE.ONE )
$ Y = OLDY
*
* Now multiply by BETA**EMAX to get RMAX.
*
DO 30 I = 1, EMAX
Y = SLAMC3( Y*BETA, ZERO )
30 CONTINUE
*
RMAX = Y
RETURN
*
* End of SLAMC5
*
END
REAL FUNCTION SLANST( NORM, N, D, E )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER NORM
INTEGER N
* ..
* .. Array Arguments ..
REAL D( * ), E( * )
* ..
*
* Purpose
* =======
*
* SLANST returns the value of the one norm, or the Frobenius norm, or
* the infinity norm, or the element of largest absolute value of a
* real symmetric tridiagonal matrix A.
*
* Description
* ===========
*
* SLANST returns the value
*
* SLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm'
* (
* ( norm1(A), NORM = '1', 'O' or 'o'
* (
* ( normI(A), NORM = 'I' or 'i'
* (
* ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*
* where norm1 denotes the one norm of a matrix (maximum column sum),
* normI denotes the infinity norm of a matrix (maximum row sum) and
* normF denotes the Frobenius norm of a matrix (square root of sum of
* squares). Note that max(abs(A(i,j))) is not a matrix norm.
*
* Arguments
* =========
*
* NORM (input) CHARACTER*1
* Specifies the value to be returned in SLANST as described
* above.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0. When N = 0, SLANST is
* set to zero.
*
* D (input) REAL array, dimension (N)
* The diagonal elements of A.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) sub-diagonal or super-diagonal elements of A.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I
REAL ANORM, SCALE, SUM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
IF( N.LE.0 ) THEN
ANORM = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
ANORM = ABS( D( N ) )
DO 10 I = 1, N - 1
ANORM = MAX( ANORM, ABS( D( I ) ) )
ANORM = MAX( ANORM, ABS( E( I ) ) )
10 CONTINUE
ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' .OR.
$ LSAME( NORM, 'I' ) ) THEN
*
* Find norm1(A).
*
IF( N.EQ.1 ) THEN
ANORM = ABS( D( 1 ) )
ELSE
ANORM = MAX( ABS( D( 1 ) )+ABS( E( 1 ) ),
$ ABS( E( N-1 ) )+ABS( D( N ) ) )
DO 20 I = 2, N - 1
ANORM = MAX( ANORM, ABS( D( I ) )+ABS( E( I ) )+
$ ABS( E( I-1 ) ) )
20 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( N.GT.1 ) THEN
CALL SLASSQ( N-1, E, 1, SCALE, SUM )
SUM = 2*SUM
END IF
CALL SLASSQ( N, D, 1, SCALE, SUM )
ANORM = SCALE*SQRT( SUM )
END IF
*
SLANST = ANORM
RETURN
*
* End of SLANST
*
END
REAL FUNCTION SLANSY( NORM, UPLO, N, A, LDA, WORK )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER LDA, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SLANSY returns the value of the one norm, or the Frobenius norm, or
* the infinity norm, or the element of largest absolute value of a
* real symmetric matrix A.
*
* Description
* ===========
*
* SLANSY returns the value
*
* SLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
* (
* ( norm1(A), NORM = '1', 'O' or 'o'
* (
* ( normI(A), NORM = 'I' or 'i'
* (
* ( normF(A), NORM = 'F', 'f', 'E' or 'e'
*
* where norm1 denotes the one norm of a matrix (maximum column sum),
* normI denotes the infinity norm of a matrix (maximum row sum) and
* normF denotes the Frobenius norm of a matrix (square root of sum of
* squares). Note that max(abs(A(i,j))) is not a matrix norm.
*
* Arguments
* =========
*
* NORM (input) CHARACTER*1
* Specifies the value to be returned in SLANSY as described
* above.
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* symmetric matrix A is to be referenced.
* = 'U': Upper triangular part of A is referenced
* = 'L': Lower triangular part of A is referenced
*
* N (input) INTEGER
* The order of the matrix A. N >= 0. When N = 0, SLANSY is
* set to zero.
*
* A (input) REAL array, dimension (LDA,N)
* The symmetric matrix A. If UPLO = 'U', the leading n by n
* upper triangular part of A contains the upper triangular part
* of the matrix A, and the strictly lower triangular part of A
* is not referenced. If UPLO = 'L', the leading n by n lower
* triangular part of A contains the lower triangular part of
* the matrix A, and the strictly upper triangular part of A is
* not referenced.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(N,1).
*
* WORK (workspace) REAL array, dimension (LWORK),
* where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
* WORK is not referenced.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
REAL ABSA, SCALE, SUM, VALUE
* ..
* .. External Subroutines ..
EXTERNAL SLASSQ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
IF( N.EQ.0 ) THEN
VALUE = ZERO
ELSE IF( LSAME( NORM, 'M' ) ) THEN
*
* Find max(abs(A(i,j))).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, J
VALUE = MAX( VALUE, ABS( A( I, J ) ) )
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = J, N
VALUE = MAX( VALUE, ABS( A( I, J ) ) )
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
$ ( NORM.EQ.'1' ) ) THEN
*
* Find normI(A) ( = norm1(A), since A is symmetric).
*
VALUE = ZERO
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
SUM = ZERO
DO 50 I = 1, J - 1
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
50 CONTINUE
WORK( J ) = SUM + ABS( A( J, J ) )
60 CONTINUE
DO 70 I = 1, N
VALUE = MAX( VALUE, WORK( I ) )
70 CONTINUE
ELSE
DO 80 I = 1, N
WORK( I ) = ZERO
80 CONTINUE
DO 100 J = 1, N
SUM = WORK( J ) + ABS( A( J, J ) )
DO 90 I = J + 1, N
ABSA = ABS( A( I, J ) )
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
90 CONTINUE
VALUE = MAX( VALUE, SUM )
100 CONTINUE
END IF
ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
*
* Find normF(A).
*
SCALE = ZERO
SUM = ONE
IF( LSAME( UPLO, 'U' ) ) THEN
DO 110 J = 2, N
CALL SLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
110 CONTINUE
ELSE
DO 120 J = 1, N - 1
CALL SLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
120 CONTINUE
END IF
SUM = 2*SUM
CALL SLASSQ( N, A, LDA+1, SCALE, SUM )
VALUE = SCALE*SQRT( SUM )
END IF
*
SLANSY = VALUE
RETURN
*
* End of SLANSY
*
END
REAL FUNCTION SLAPY2( X, Y )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
REAL X, Y
* ..
*
* Purpose
* =======
*
* SLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary
* overflow.
*
* Arguments
* =========
*
* X (input) REAL
* Y (input) REAL
* X and Y specify the values x and y.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E0 )
REAL ONE
PARAMETER ( ONE = 1.0E0 )
* ..
* .. Local Scalars ..
REAL W, XABS, YABS, Z
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
XABS = ABS( X )
YABS = ABS( Y )
W = MAX( XABS, YABS )
Z = MIN( XABS, YABS )
IF( Z.EQ.ZERO ) THEN
SLAPY2 = W
ELSE
SLAPY2 = W*SQRT( ONE+( Z / W )**2 )
END IF
RETURN
*
* End of SLAPY2
*
END
SUBROUTINE SLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER SIDE
INTEGER INCV, LDC, M, N
REAL TAU
* ..
* .. Array Arguments ..
REAL C( LDC, * ), V( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SLARF applies a real elementary reflector H to a real m by n matrix
* C, from either the left or the right. H is represented in the form
*
* H = I - tau * v * v'
*
* where tau is a real scalar and v is a real vector.
*
* If tau = 0, then H is taken to be the unit matrix.
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* = 'L': form H * C
* = 'R': form C * H
*
* M (input) INTEGER
* The number of rows of the matrix C.
*
* N (input) INTEGER
* The number of columns of the matrix C.
*
* V (input) REAL array, dimension
* (1 + (M-1)*abs(INCV)) if SIDE = 'L'
* or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
* The vector v in the representation of H. V is not used if
* TAU = 0.
*
* INCV (input) INTEGER
* The increment between elements of v. INCV <> 0.
*
* TAU (input) REAL
* The value tau in the representation of H.
*
* C (input/output) REAL array, dimension (LDC,N)
* On entry, the m by n matrix C.
* On exit, C is overwritten by the matrix H * C if SIDE = 'L',
* or C * H if SIDE = 'R'.
*
* LDC (input) INTEGER
* The leading dimension of the array C. LDC >= max(1,M).
*
* WORK (workspace) REAL array, dimension
* (N) if SIDE = 'L'
* or (M) if SIDE = 'R'
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. External Subroutines ..
EXTERNAL SGEMV, SGER
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C
*
IF( TAU.NE.ZERO ) THEN
*
* w := C' * v
*
CALL SGEMV( 'Transpose', M, N, ONE, C, LDC, V, INCV, ZERO,
$ WORK, 1 )
*
* C := C - v * w'
*
CALL SGER( M, N, -TAU, V, INCV, WORK, 1, C, LDC )
END IF
ELSE
*
* Form C * H
*
IF( TAU.NE.ZERO ) THEN
*
* w := C * v
*
CALL SGEMV( 'No transpose', M, N, ONE, C, LDC, V, INCV,
$ ZERO, WORK, 1 )
*
* C := C - w * v'
*
CALL SGER( M, N, -TAU, WORK, 1, V, INCV, C, LDC )
END IF
END IF
RETURN
*
* End of SLARF
*
END
SUBROUTINE SLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV,
$ T, LDT, C, LDC, WORK, LDWORK )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER DIRECT, SIDE, STOREV, TRANS
INTEGER K, LDC, LDT, LDV, LDWORK, M, N
* ..
* .. Array Arguments ..
REAL C( LDC, * ), T( LDT, * ), V( LDV, * ),
$ WORK( LDWORK, * )
* ..
*
* Purpose
* =======
*
* SLARFB applies a real block reflector H or its transpose H' to a
* real m by n matrix C, from either the left or the right.
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* = 'L': apply H or H' from the Left
* = 'R': apply H or H' from the Right
*
* TRANS (input) CHARACTER*1
* = 'N': apply H (No transpose)
* = 'T': apply H' (Transpose)
*
* DIRECT (input) CHARACTER*1
* Indicates how H is formed from a product of elementary
* reflectors
* = 'F': H = H(1) H(2) . . . H(k) (Forward)
* = 'B': H = H(k) . . . H(2) H(1) (Backward)
*
* STOREV (input) CHARACTER*1
* Indicates how the vectors which define the elementary
* reflectors are stored:
* = 'C': Columnwise
* = 'R': Rowwise
*
* M (input) INTEGER
* The number of rows of the matrix C.
*
* N (input) INTEGER
* The number of columns of the matrix C.
*
* K (input) INTEGER
* The order of the matrix T (= the number of elementary
* reflectors whose product defines the block reflector).
*
* V (input) REAL array, dimension
* (LDV,K) if STOREV = 'C'
* (LDV,M) if STOREV = 'R' and SIDE = 'L'
* (LDV,N) if STOREV = 'R' and SIDE = 'R'
* The matrix V. See further details.
*
* LDV (input) INTEGER
* The leading dimension of the array V.
* If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
* if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
* if STOREV = 'R', LDV >= K.
*
* T (input) REAL array, dimension (LDT,K)
* The triangular k by k matrix T in the representation of the
* block reflector.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= K.
*
* C (input/output) REAL array, dimension (LDC,N)
* On entry, the m by n matrix C.
* On exit, C is overwritten by H*C or H'*C or C*H or C*H'.
*
* LDC (input) INTEGER
* The leading dimension of the array C. LDA >= max(1,M).
*
* WORK (workspace) REAL array, dimension (LDWORK,K)
*
* LDWORK (input) INTEGER
* The leading dimension of the array WORK.
* If SIDE = 'L', LDWORK >= max(1,N);
* if SIDE = 'R', LDWORK >= max(1,M).
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
CHARACTER TRANST
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGEMM, STRMM
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( LSAME( TRANS, 'N' ) ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
IF( LSAME( STOREV, 'C' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 ) (first K rows)
* ( V2 )
* where V1 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H' * C where C = ( C1 )
* ( C2 )
*
* W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK)
*
* W := C1'
*
DO 10 J = 1, K
CALL SCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
10 CONTINUE
*
* W := W * V1
*
CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', N,
$ K, ONE, V, LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C2'*V2
*
CALL SGEMM( 'Transpose', 'No transpose', N, K, M-K,
$ ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T' or W * T
*
CALL STRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W'
*
IF( M.GT.K ) THEN
*
* C2 := C2 - V2 * W'
*
CALL SGEMM( 'No transpose', 'Transpose', M-K, N, K,
$ -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1'
*
CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K,
$ ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W'
*
DO 30 J = 1, K
DO 20 I = 1, N
C( J, I ) = C( J, I ) - WORK( I, J )
20 CONTINUE
30 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H' where C = ( C1 C2 )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C1
*
DO 40 J = 1, K
CALL SCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
40 CONTINUE
*
* W := W * V1
*
CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', M,
$ K, ONE, V, LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C2 * V2
*
CALL SGEMM( 'No transpose', 'No transpose', M, K, N-K,
$ ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T'
*
CALL STRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V'
*
IF( N.GT.K ) THEN
*
* C2 := C2 - W * V2'
*
CALL SGEMM( 'No transpose', 'Transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1'
*
CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K,
$ ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 60 J = 1, K
DO 50 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
ELSE
*
* Let V = ( V1 )
* ( V2 ) (last K rows)
* where V2 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H' * C where C = ( C1 )
* ( C2 )
*
* W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK)
*
* W := C2'
*
DO 70 J = 1, K
CALL SCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 )
70 CONTINUE
*
* W := W * V2
*
CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', N,
$ K, ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1'*V1
*
CALL SGEMM( 'Transpose', 'No transpose', N, K, M-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T' or W * T
*
CALL STRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V * W'
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1 * W'
*
CALL SGEMM( 'No transpose', 'Transpose', M-K, N, K,
$ -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC )
END IF
*
* W := W * V2'
*
CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K,
$ ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W'
*
DO 90 J = 1, K
DO 80 I = 1, N
C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
80 CONTINUE
90 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H' where C = ( C1 C2 )
*
* W := C * V = (C1*V1 + C2*V2) (stored in WORK)
*
* W := C2
*
DO 100 J = 1, K
CALL SCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
100 CONTINUE
*
* W := W * V2
*
CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', M,
$ K, ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1
*
CALL SGEMM( 'No transpose', 'No transpose', M, K, N-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T'
*
CALL STRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V'
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1'
*
CALL SGEMM( 'No transpose', 'Transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
END IF
*
* W := W * V2'
*
CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K,
$ ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W
*
DO 120 J = 1, K
DO 110 I = 1, M
C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
110 CONTINUE
120 CONTINUE
END IF
END IF
*
ELSE IF( LSAME( STOREV, 'R' ) ) THEN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
*
* Let V = ( V1 V2 ) (V1: first K columns)
* where V1 is unit upper triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H' * C where C = ( C1 )
* ( C2 )
*
* W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK)
*
* W := C1'
*
DO 130 J = 1, K
CALL SCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 )
130 CONTINUE
*
* W := W * V1'
*
CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K,
$ ONE, V, LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C2'*V2'
*
CALL SGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE,
$ C( K+1, 1 ), LDC, V( 1, K+1 ), LDV, ONE,
$ WORK, LDWORK )
END IF
*
* W := W * T' or W * T
*
CALL STRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V' * W'
*
IF( M.GT.K ) THEN
*
* C2 := C2 - V2' * W'
*
CALL SGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE,
$ V( 1, K+1 ), LDV, WORK, LDWORK, ONE,
$ C( K+1, 1 ), LDC )
END IF
*
* W := W * V1
*
CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', N,
$ K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W'
*
DO 150 J = 1, K
DO 140 I = 1, N
C( J, I ) = C( J, I ) - WORK( I, J )
140 CONTINUE
150 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H' where C = ( C1 C2 )
*
* W := C * V' = (C1*V1' + C2*V2') (stored in WORK)
*
* W := C1
*
DO 160 J = 1, K
CALL SCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 )
160 CONTINUE
*
* W := W * V1'
*
CALL STRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K,
$ ONE, V, LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C2 * V2'
*
CALL SGEMM( 'No transpose', 'Transpose', M, K, N-K,
$ ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV,
$ ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T'
*
CALL STRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C2 := C2 - W * V2
*
CALL SGEMM( 'No transpose', 'No transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V( 1, K+1 ), LDV, ONE,
$ C( 1, K+1 ), LDC )
END IF
*
* W := W * V1
*
CALL STRMM( 'Right', 'Upper', 'No transpose', 'Unit', M,
$ K, ONE, V, LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 180 J = 1, K
DO 170 I = 1, M
C( I, J ) = C( I, J ) - WORK( I, J )
170 CONTINUE
180 CONTINUE
*
END IF
*
ELSE
*
* Let V = ( V1 V2 ) (V2: last K columns)
* where V2 is unit lower triangular.
*
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form H * C or H' * C where C = ( C1 )
* ( C2 )
*
* W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK)
*
* W := C2'
*
DO 190 J = 1, K
CALL SCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 )
190 CONTINUE
*
* W := W * V2'
*
CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K,
$ ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
IF( M.GT.K ) THEN
*
* W := W + C1'*V1'
*
CALL SGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE,
$ C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T' or W * T
*
CALL STRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - V' * W'
*
IF( M.GT.K ) THEN
*
* C1 := C1 - V1' * W'
*
CALL SGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE,
$ V, LDV, WORK, LDWORK, ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', N,
$ K, ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK )
*
* C2 := C2 - W'
*
DO 210 J = 1, K
DO 200 I = 1, N
C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J )
200 CONTINUE
210 CONTINUE
*
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form C * H or C * H' where C = ( C1 C2 )
*
* W := C * V' = (C1*V1' + C2*V2') (stored in WORK)
*
* W := C2
*
DO 220 J = 1, K
CALL SCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 )
220 CONTINUE
*
* W := W * V2'
*
CALL STRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K,
$ ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
IF( N.GT.K ) THEN
*
* W := W + C1 * V1'
*
CALL SGEMM( 'No transpose', 'Transpose', M, K, N-K,
$ ONE, C, LDC, V, LDV, ONE, WORK, LDWORK )
END IF
*
* W := W * T or W * T'
*
CALL STRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K,
$ ONE, T, LDT, WORK, LDWORK )
*
* C := C - W * V
*
IF( N.GT.K ) THEN
*
* C1 := C1 - W * V1
*
CALL SGEMM( 'No transpose', 'No transpose', M, N-K, K,
$ -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC )
END IF
*
* W := W * V2
*
CALL STRMM( 'Right', 'Lower', 'No transpose', 'Unit', M,
$ K, ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK )
*
* C1 := C1 - W
*
DO 240 J = 1, K
DO 230 I = 1, M
C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J )
230 CONTINUE
240 CONTINUE
*
END IF
*
END IF
END IF
*
RETURN
*
* End of SLARFB
*
END
SUBROUTINE SLARFG( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER INCX, N
REAL ALPHA, TAU
* ..
* .. Array Arguments ..
REAL X( * )
* ..
*
* Purpose
* =======
*
* SLARFG generates a real elementary reflector H of order n, such
* that
*
* H * ( alpha ) = ( beta ), H' * H = I.
* ( x ) ( 0 )
*
* where alpha and beta are scalars, and x is an (n-1)-element real
* vector. H is represented in the form
*
* H = I - tau * ( 1 ) * ( 1 v' ) ,
* ( v )
*
* where tau is a real scalar and v is a real (n-1)-element
* vector.
*
* If the elements of x are all zero, then tau = 0 and H is taken to be
* the unit matrix.
*
* Otherwise 1 <= tau <= 2.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the elementary reflector.
*
* ALPHA (input/output) REAL
* On entry, the value alpha.
* On exit, it is overwritten with the value beta.
*
* X (input/output) REAL array, dimension
* (1+(N-2)*abs(INCX))
* On entry, the vector x.
* On exit, it is overwritten with the vector v.
*
* INCX (input) INTEGER
* The increment between elements of X. INCX > 0.
*
* TAU (output) REAL
* The value tau.
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
REAL BETA, RSAFMN, SAFMIN, XNORM
* ..
* .. External Functions ..
REAL SLAMCH, SLAPY2, SNRM2
EXTERNAL SLAMCH, SLAPY2, SNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN
* ..
* .. External Subroutines ..
EXTERNAL SSCAL
* ..
* .. Executable Statements ..
*
IF( N.LE.1 ) THEN
TAU = ZERO
RETURN
END IF
*
XNORM = SNRM2( N-1, X, INCX )
*
IF( XNORM.EQ.ZERO ) THEN
*
* H = I
*
TAU = ZERO
ELSE
*
* general case
*
BETA = -SIGN( SLAPY2( ALPHA, XNORM ), ALPHA )
SAFMIN = SLAMCH( 'S' ) / SLAMCH( 'E' )
IF( ABS( BETA ).LT.SAFMIN ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
RSAFMN = ONE / SAFMIN
KNT = 0
10 CONTINUE
KNT = KNT + 1
CALL SSCAL( N-1, RSAFMN, X, INCX )
BETA = BETA*RSAFMN
ALPHA = ALPHA*RSAFMN
IF( ABS( BETA ).LT.SAFMIN )
$ GO TO 10
*
* New BETA is at most 1, at least SAFMIN
*
XNORM = SNRM2( N-1, X, INCX )
BETA = -SIGN( SLAPY2( ALPHA, XNORM ), ALPHA )
TAU = ( BETA-ALPHA ) / BETA
CALL SSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
* If ALPHA is subnormal, it may lose relative accuracy
*
ALPHA = BETA
DO 20 J = 1, KNT
ALPHA = ALPHA*SAFMIN
20 CONTINUE
ELSE
TAU = ( BETA-ALPHA ) / BETA
CALL SSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
ALPHA = BETA
END IF
END IF
*
RETURN
*
* End of SLARFG
*
END
SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
REAL T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* Purpose
* =======
*
* SLARFT forms the triangular factor T of a real block reflector H
* of order n, which is defined as a product of k elementary reflectors.
*
* If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*
* If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*
* If STOREV = 'C', the vector which defines the elementary reflector
* H(i) is stored in the i-th column of the array V, and
*
* H = I - V * T * V'
*
* If STOREV = 'R', the vector which defines the elementary reflector
* H(i) is stored in the i-th row of the array V, and
*
* H = I - V' * T * V
*
* Arguments
* =========
*
* DIRECT (input) CHARACTER*1
* Specifies the order in which the elementary reflectors are
* multiplied to form the block reflector:
* = 'F': H = H(1) H(2) . . . H(k) (Forward)
* = 'B': H = H(k) . . . H(2) H(1) (Backward)
*
* STOREV (input) CHARACTER*1
* Specifies how the vectors which define the elementary
* reflectors are stored (see also Further Details):
* = 'C': columnwise
* = 'R': rowwise
*
* N (input) INTEGER
* The order of the block reflector H. N >= 0.
*
* K (input) INTEGER
* The order of the triangular factor T (= the number of
* elementary reflectors). K >= 1.
*
* V (input/output) REAL array, dimension
* (LDV,K) if STOREV = 'C'
* (LDV,N) if STOREV = 'R'
* The matrix V. See further details.
*
* LDV (input) INTEGER
* The leading dimension of the array V.
* If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*
* TAU (input) REAL array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i).
*
* T (output) REAL array, dimension (LDT,K)
* The k by k triangular factor T of the block reflector.
* If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
* lower triangular. The rest of the array is not used.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= K.
*
* Further Details
* ===============
*
* The shape of the matrix V and the storage of the vectors which define
* the H(i) is best illustrated by the following example with n = 5 and
* k = 3. The elements equal to 1 are not stored; the corresponding
* array elements are modified but restored on exit. The rest of the
* array is not used.
*
* DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*
* V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
* ( v1 1 ) ( 1 v2 v2 v2 )
* ( v1 v2 1 ) ( 1 v3 v3 )
* ( v1 v2 v3 )
* ( v1 v2 v3 )
*
* DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*
* V = ( v1 v2 v3 ) V = ( v1 v1 1 )
* ( v1 v2 v3 ) ( v2 v2 v2 1 )
* ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
* ( 1 v3 )
* ( 1 )
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
REAL VII
* ..
* .. External Subroutines ..
EXTERNAL SGEMV, STRMV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 20 I = 1, K
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO 10 J = 1, I
T( J, I ) = ZERO
10 CONTINUE
ELSE
*
* general case
*
VII = V( I, I )
V( I, I ) = ONE
IF( LSAME( STOREV, 'C' ) ) THEN
*
* T(1:i-1,i) := - tau(i) * V(i:n,1:i-1)' * V(i:n,i)
*
CALL SGEMV( 'Transpose', N-I+1, I-1, -TAU( I ),
$ V( I, 1 ), LDV, V( I, I ), 1, ZERO,
$ T( 1, I ), 1 )
ELSE
*
* T(1:i-1,i) := - tau(i) * V(1:i-1,i:n) * V(i,i:n)'
*
CALL SGEMV( 'No transpose', I-1, N-I+1, -TAU( I ),
$ V( 1, I ), LDV, V( I, I ), LDV, ZERO,
$ T( 1, I ), 1 )
END IF
V( I, I ) = VII
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL STRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
$ LDT, T( 1, I ), 1 )
T( I, I ) = TAU( I )
END IF
20 CONTINUE
ELSE
DO 40 I = K, 1, -1
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO 30 J = I, K
T( J, I ) = ZERO
30 CONTINUE
ELSE
*
* general case
*
IF( I.LT.K ) THEN
IF( LSAME( STOREV, 'C' ) ) THEN
VII = V( N-K+I, I )
V( N-K+I, I ) = ONE
*
* T(i+1:k,i) :=
* - tau(i) * V(1:n-k+i,i+1:k)' * V(1:n-k+i,i)
*
CALL SGEMV( 'Transpose', N-K+I, K-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( 1, I ), 1, ZERO,
$ T( I+1, I ), 1 )
V( N-K+I, I ) = VII
ELSE
VII = V( I, N-K+I )
V( I, N-K+I ) = ONE
*
* T(i+1:k,i) :=
* - tau(i) * V(i+1:k,1:n-k+i) * V(i,1:n-k+i)'
*
CALL SGEMV( 'No transpose', K-I, N-K+I, -TAU( I ),
$ V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO,
$ T( I+1, I ), 1 )
V( I, N-K+I ) = VII
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL STRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
$ T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
END IF
T( I, I ) = TAU( I )
END IF
40 CONTINUE
END IF
RETURN
*
* End of SLARFT
*
END
SUBROUTINE SLARTG( F, G, CS, SN, R )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
REAL CS, F, G, R, SN
* ..
*
* Purpose
* =======
*
* SLARTG generate a plane rotation so that
*
* [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1.
* [ -SN CS ] [ G ] [ 0 ]
*
* This is a slower, more accurate version of the BLAS1 routine SROTG,
* with the following other differences:
* F and G are unchanged on return.
* If G=0, then CS=1 and SN=0.
* If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any
* floating point operations (saves work in SBDSQR when
* there are zeros on the diagonal).
*
* If F exceeds G in magnitude, CS will be positive.
*
* Arguments
* =========
*
* F (input) REAL
* The first component of vector to be rotated.
*
* G (input) REAL
* The second component of vector to be rotated.
*
* CS (output) REAL
* The cosine of the rotation.
*
* SN (output) REAL
* The sine of the rotation.
*
* R (output) REAL
* The nonzero component of the rotated vector.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E0 )
REAL ONE
PARAMETER ( ONE = 1.0E0 )
REAL TWO
PARAMETER ( TWO = 2.0E0 )
* ..
* .. Local Scalars ..
LOGICAL FIRST
INTEGER COUNT, I
REAL EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, SQRT
* ..
* .. Save statement ..
SAVE FIRST, SAFMX2, SAFMIN, SAFMN2
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
SAFMIN = SLAMCH( 'S' )
EPS = SLAMCH( 'E' )
SAFMN2 = SLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( SLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
END IF
IF( G.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
R = F
ELSE IF( F.EQ.ZERO ) THEN
CS = ZERO
SN = ONE
R = G
ELSE
F1 = F
G1 = G
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 ) THEN
COUNT = 0
10 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMN2
G1 = G1*SAFMN2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.GE.SAFMX2 )
$ GO TO 10
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 20 I = 1, COUNT
R = R*SAFMX2
20 CONTINUE
ELSE IF( SCALE.LE.SAFMN2 ) THEN
COUNT = 0
30 CONTINUE
COUNT = COUNT + 1
F1 = F1*SAFMX2
G1 = G1*SAFMX2
SCALE = MAX( ABS( F1 ), ABS( G1 ) )
IF( SCALE.LE.SAFMN2 )
$ GO TO 30
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
DO 40 I = 1, COUNT
R = R*SAFMN2
40 CONTINUE
ELSE
R = SQRT( F1**2+G1**2 )
CS = F1 / R
SN = G1 / R
END IF
IF( ABS( F ).GT.ABS( G ) .AND. CS.LT.ZERO ) THEN
CS = -CS
SN = -SN
R = -R
END IF
END IF
RETURN
*
* End of SLARTG
*
END
SUBROUTINE SLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER TYPE
INTEGER INFO, KL, KU, LDA, M, N
REAL CFROM, CTO
* ..
* .. Array Arguments ..
REAL A( LDA, * )
* ..
*
* Purpose
* =======
*
* SLASCL multiplies the M by N real matrix A by the real scalar
* CTO/CFROM. This is done without over/underflow as long as the final
* result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
* A may be full, upper triangular, lower triangular, upper Hessenberg,
* or banded.
*
* Arguments
* =========
*
* TYPE (input) CHARACTER*1
* TYPE indices the storage type of the input matrix.
* = 'G': A is a full matrix.
* = 'L': A is a lower triangular matrix.
* = 'U': A is an upper triangular matrix.
* = 'H': A is an upper Hessenberg matrix.
* = 'B': A is a symmetric band matrix with lower bandwidth KL
* and upper bandwidth KU and with the only the lower
* half stored.
* = 'Q': A is a symmetric band matrix with lower bandwidth KL
* and upper bandwidth KU and with the only the upper
* half stored.
* = 'Z': A is a band matrix with lower bandwidth KL and upper
* bandwidth KU.
*
* KL (input) INTEGER
* The lower bandwidth of A. Referenced only if TYPE = 'B',
* 'Q' or 'Z'.
*
* KU (input) INTEGER
* The upper bandwidth of A. Referenced only if TYPE = 'B',
* 'Q' or 'Z'.
*
* CFROM (input) REAL
* CTO (input) REAL
* The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
* without over/underflow if the final result CTO*A(I,J)/CFROM
* can be represented without over/underflow. CFROM must be
* nonzero.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,M)
* The matrix to be multiplied by CTO/CFROM. See TYPE for the
* storage type.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* INFO (output) INTEGER
* 0 - successful exit
* <0 - if INFO = -i, the i-th argument had an illegal value.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
INTEGER I, ITYPE, J, K1, K2, K3, K4
REAL BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH
EXTERNAL LSAME, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
*
IF( LSAME( TYPE, 'G' ) ) THEN
ITYPE = 0
ELSE IF( LSAME( TYPE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( TYPE, 'U' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( TYPE, 'H' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( TYPE, 'B' ) ) THEN
ITYPE = 4
ELSE IF( LSAME( TYPE, 'Q' ) ) THEN
ITYPE = 5
ELSE IF( LSAME( TYPE, 'Z' ) ) THEN
ITYPE = 6
ELSE
ITYPE = -1
END IF
*
IF( ITYPE.EQ.-1 ) THEN
INFO = -1
ELSE IF( CFROM.EQ.ZERO ) THEN
INFO = -4
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR.
$ ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN
INFO = -7
ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN
INFO = -9
ELSE IF( ITYPE.GE.4 ) THEN
IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN
INFO = -2
ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR.
$ ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) )
$ THEN
INFO = -3
ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR.
$ ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR.
$ ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN
INFO = -9
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASCL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
* Get machine parameters
*
SMLNUM = SLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
*
CFROMC = CFROM
CTOC = CTO
*
10 CONTINUE
CFROM1 = CFROMC*SMLNUM
CTO1 = CTOC / BIGNUM
IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN
MUL = SMLNUM
DONE = .FALSE.
CFROMC = CFROM1
ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN
MUL = BIGNUM
DONE = .FALSE.
CTOC = CTO1
ELSE
MUL = CTOC / CFROMC
DONE = .TRUE.
END IF
*
IF( ITYPE.EQ.0 ) THEN
*
* Full matrix
*
DO 30 J = 1, N
DO 20 I = 1, M
A( I, J ) = A( I, J )*MUL
20 CONTINUE
30 CONTINUE
*
ELSE IF( ITYPE.EQ.1 ) THEN
*
* Lower triangular matrix
*
DO 50 J = 1, N
DO 40 I = J, M
A( I, J ) = A( I, J )*MUL
40 CONTINUE
50 CONTINUE
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* Upper triangular matrix
*
DO 70 J = 1, N
DO 60 I = 1, MIN( J, M )
A( I, J ) = A( I, J )*MUL
60 CONTINUE
70 CONTINUE
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* Upper Hessenberg matrix
*
DO 90 J = 1, N
DO 80 I = 1, MIN( J+1, M )
A( I, J ) = A( I, J )*MUL
80 CONTINUE
90 CONTINUE
*
ELSE IF( ITYPE.EQ.4 ) THEN
*
* Lower half of a symmetric band matrix
*
K3 = KL + 1
K4 = N + 1
DO 110 J = 1, N
DO 100 I = 1, MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
100 CONTINUE
110 CONTINUE
*
ELSE IF( ITYPE.EQ.5 ) THEN
*
* Upper half of a symmetric band matrix
*
K1 = KU + 2
K3 = KU + 1
DO 130 J = 1, N
DO 120 I = MAX( K1-J, 1 ), K3
A( I, J ) = A( I, J )*MUL
120 CONTINUE
130 CONTINUE
*
ELSE IF( ITYPE.EQ.6 ) THEN
*
* Band matrix
*
K1 = KL + KU + 2
K2 = KL + 1
K3 = 2*KL + KU + 1
K4 = KL + KU + 1 + M
DO 150 J = 1, N
DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J )
A( I, J ) = A( I, J )*MUL
140 CONTINUE
150 CONTINUE
*
END IF
*
IF( .NOT.DONE )
$ GO TO 10
*
RETURN
*
* End of SLASCL
*
END
SUBROUTINE SLASET( UPLO, M, N, ALPHA, BETA, A, LDA )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, M, N
REAL ALPHA, BETA
* ..
* .. Array Arguments ..
REAL A( LDA, * )
* ..
*
* Purpose
* =======
*
* SLASET initializes an m-by-n matrix A to BETA on the diagonal and
* ALPHA on the offdiagonals.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies the part of the matrix A to be set.
* = 'U': Upper triangular part is set; the strictly lower
* triangular part of A is not changed.
* = 'L': Lower triangular part is set; the strictly upper
* triangular part of A is not changed.
* Otherwise: All of the matrix A is set.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* ALPHA (input) REAL
* The constant to which the offdiagonal elements are to be set.
*
* BETA (input) REAL
* The constant to which the diagonal elements are to be set.
*
* A (input/output) REAL array, dimension (LDA,N)
* On exit, the leading m-by-n submatrix of A is set as follows:
*
* if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n,
* if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n,
* otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j,
*
* and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Set the strictly upper triangular or trapezoidal part of the
* array to ALPHA.
*
DO 20 J = 2, N
DO 10 I = 1, MIN( J-1, M )
A( I, J ) = ALPHA
10 CONTINUE
20 CONTINUE
*
ELSE IF( LSAME( UPLO, 'L' ) ) THEN
*
* Set the strictly lower triangular or trapezoidal part of the
* array to ALPHA.
*
DO 40 J = 1, MIN( M, N )
DO 30 I = J + 1, M
A( I, J ) = ALPHA
30 CONTINUE
40 CONTINUE
*
ELSE
*
* Set the leading m-by-n submatrix to ALPHA.
*
DO 60 J = 1, N
DO 50 I = 1, M
A( I, J ) = ALPHA
50 CONTINUE
60 CONTINUE
END IF
*
* Set the first min(M,N) diagonal elements to BETA.
*
DO 70 I = 1, MIN( M, N )
A( I, I ) = BETA
70 CONTINUE
*
RETURN
*
* End of SLASET
*
END
SUBROUTINE SLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER DIRECT, PIVOT, SIDE
INTEGER LDA, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), C( * ), S( * )
* ..
*
* Purpose
* =======
*
* SLASR performs the transformation
*
* A := P*A, when SIDE = 'L' or 'l' ( Left-hand side )
*
* A := A*P', when SIDE = 'R' or 'r' ( Right-hand side )
*
* where A is an m by n real matrix and P is an orthogonal matrix,
* consisting of a sequence of plane rotations determined by the
* parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l'
* and z = n when SIDE = 'R' or 'r' ):
*
* When DIRECT = 'F' or 'f' ( Forward sequence ) then
*
* P = P( z - 1 )*...*P( 2 )*P( 1 ),
*
* and when DIRECT = 'B' or 'b' ( Backward sequence ) then
*
* P = P( 1 )*P( 2 )*...*P( z - 1 ),
*
* where P( k ) is a plane rotation matrix for the following planes:
*
* when PIVOT = 'V' or 'v' ( Variable pivot ),
* the plane ( k, k + 1 )
*
* when PIVOT = 'T' or 't' ( Top pivot ),
* the plane ( 1, k + 1 )
*
* when PIVOT = 'B' or 'b' ( Bottom pivot ),
* the plane ( k, z )
*
* c( k ) and s( k ) must contain the cosine and sine that define the
* matrix P( k ). The two by two plane rotation part of the matrix
* P( k ), R( k ), is assumed to be of the form
*
* R( k ) = ( c( k ) s( k ) ).
* ( -s( k ) c( k ) )
*
* This version vectorises across rows of the array A when SIDE = 'L'.
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* Specifies whether the plane rotation matrix P is applied to
* A on the left or the right.
* = 'L': Left, compute A := P*A
* = 'R': Right, compute A:= A*P'
*
* DIRECT (input) CHARACTER*1
* Specifies whether P is a forward or backward sequence of
* plane rotations.
* = 'F': Forward, P = P( z - 1 )*...*P( 2 )*P( 1 )
* = 'B': Backward, P = P( 1 )*P( 2 )*...*P( z - 1 )
*
* PIVOT (input) CHARACTER*1
* Specifies the plane for which P(k) is a plane rotation
* matrix.
* = 'V': Variable pivot, the plane (k,k+1)
* = 'T': Top pivot, the plane (1,k+1)
* = 'B': Bottom pivot, the plane (k,z)
*
* M (input) INTEGER
* The number of rows of the matrix A. If m <= 1, an immediate
* return is effected.
*
* N (input) INTEGER
* The number of columns of the matrix A. If n <= 1, an
* immediate return is effected.
*
* C, S (input) REAL arrays, dimension
* (M-1) if SIDE = 'L'
* (N-1) if SIDE = 'R'
* c(k) and s(k) contain the cosine and sine that define the
* matrix P(k). The two by two plane rotation part of the
* matrix P(k), R(k), is assumed to be of the form
* R( k ) = ( c( k ) s( k ) ).
* ( -s( k ) c( k ) )
*
* A (input/output) REAL array, dimension (LDA,N)
* The m by n matrix A. On exit, A is overwritten by P*A if
* SIDE = 'R' or by A*P' if SIDE = 'L'.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J
REAL CTEMP, STEMP, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( .NOT.( LSAME( SIDE, 'L' ) .OR. LSAME( SIDE, 'R' ) ) ) THEN
INFO = 1
ELSE IF( .NOT.( LSAME( PIVOT, 'V' ) .OR. LSAME( PIVOT,
$ 'T' ) .OR. LSAME( PIVOT, 'B' ) ) ) THEN
INFO = 2
ELSE IF( .NOT.( LSAME( DIRECT, 'F' ) .OR. LSAME( DIRECT, 'B' ) ) )
$ THEN
INFO = 3
ELSE IF( M.LT.0 ) THEN
INFO = 4
ELSE IF( N.LT.0 ) THEN
INFO = 5
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = 9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASR ', INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )
$ RETURN
IF( LSAME( SIDE, 'L' ) ) THEN
*
* Form P * A
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 20 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 10 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
10 CONTINUE
END IF
20 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 40 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 30 I = 1, N
TEMP = A( J+1, I )
A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I )
A( J, I ) = STEMP*TEMP + CTEMP*A( J, I )
30 CONTINUE
END IF
40 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 60 J = 2, M
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 50 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
50 CONTINUE
END IF
60 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 80 J = M, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 70 I = 1, N
TEMP = A( J, I )
A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I )
A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I )
70 CONTINUE
END IF
80 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 100 J = 1, M - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 90 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
90 CONTINUE
END IF
100 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 120 J = M - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 110 I = 1, N
TEMP = A( J, I )
A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP
A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP
110 CONTINUE
END IF
120 CONTINUE
END IF
END IF
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
*
* Form A * P'
*
IF( LSAME( PIVOT, 'V' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 140 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 130 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
130 CONTINUE
END IF
140 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 160 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 150 I = 1, M
TEMP = A( I, J+1 )
A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J )
A( I, J ) = STEMP*TEMP + CTEMP*A( I, J )
150 CONTINUE
END IF
160 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'T' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 180 J = 2, N
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 170 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
170 CONTINUE
END IF
180 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 200 J = N, 2, -1
CTEMP = C( J-1 )
STEMP = S( J-1 )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 190 I = 1, M
TEMP = A( I, J )
A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 )
A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 )
190 CONTINUE
END IF
200 CONTINUE
END IF
ELSE IF( LSAME( PIVOT, 'B' ) ) THEN
IF( LSAME( DIRECT, 'F' ) ) THEN
DO 220 J = 1, N - 1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 210 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
210 CONTINUE
END IF
220 CONTINUE
ELSE IF( LSAME( DIRECT, 'B' ) ) THEN
DO 240 J = N - 1, 1, -1
CTEMP = C( J )
STEMP = S( J )
IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN
DO 230 I = 1, M
TEMP = A( I, J )
A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP
A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP
230 CONTINUE
END IF
240 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of SLASR
*
END
SUBROUTINE SLASRT( ID, N, D, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER ID
INTEGER INFO, N
* ..
* .. Array Arguments ..
REAL D( * )
* ..
*
* Purpose
* =======
*
* Sort the numbers in D in increasing order (if ID = 'I') or
* in decreasing order (if ID = 'D' ).
*
* Use Quick Sort, reverting to Insertion sort on arrays of
* size <= 20. Dimension of STACK limits N to about 2**32.
*
* Arguments
* =========
*
* ID (input) CHARACTER*1
* = 'I': sort D in increasing order;
* = 'D': sort D in decreasing order.
*
* N (input) INTEGER
* The length of the array D.
*
* D (input/output) REAL array, dimension (N)
* On entry, the array to be sorted.
* On exit, D has been sorted into increasing order
* (D(1) <= ... <= D(N) ) or into decreasing order
* (D(1) >= ... >= D(N) ), depending on ID.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
INTEGER SELECT
PARAMETER ( SELECT = 20 )
* ..
* .. Local Scalars ..
INTEGER DIR, ENDD, I, J, START, STKPNT
REAL D1, D2, D3, DMNMX, TMP
* ..
* .. Local Arrays ..
INTEGER STACK( 2, 32 )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
* Test the input paramters.
*
INFO = 0
DIR = -1
IF( LSAME( ID, 'D' ) ) THEN
DIR = 0
ELSE IF( LSAME( ID, 'I' ) ) THEN
DIR = 1
END IF
IF( DIR.EQ.-1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASRT', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
STKPNT = 1
STACK( 1, 1 ) = 1
STACK( 2, 1 ) = N
10 CONTINUE
START = STACK( 1, STKPNT )
ENDD = STACK( 2, STKPNT )
STKPNT = STKPNT - 1
IF( ENDD-START.LE.SELECT .AND. ENDD-START.GT.0 ) THEN
*
* Do Insertion sort on D( START:ENDD )
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
DO 30 I = START + 1, ENDD
DO 20 J = I, START + 1, -1
IF( D( J ).GT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
*
ELSE
*
* Sort into increasing order
*
DO 50 I = START + 1, ENDD
DO 40 J = I, START + 1, -1
IF( D( J ).LT.D( J-1 ) ) THEN
DMNMX = D( J )
D( J ) = D( J-1 )
D( J-1 ) = DMNMX
ELSE
GO TO 50
END IF
40 CONTINUE
50 CONTINUE
*
END IF
*
ELSE IF( ENDD-START.GT.SELECT ) THEN
*
* Partition D( START:ENDD ) and stack parts, largest one first
*
* Choose partition entry as median of 3
*
D1 = D( START )
D2 = D( ENDD )
I = ( START+ENDD ) / 2
D3 = D( I )
IF( D1.LT.D2 ) THEN
IF( D3.LT.D1 ) THEN
DMNMX = D1
ELSE IF( D3.LT.D2 ) THEN
DMNMX = D3
ELSE
DMNMX = D2
END IF
ELSE
IF( D3.LT.D2 ) THEN
DMNMX = D2
ELSE IF( D3.LT.D1 ) THEN
DMNMX = D3
ELSE
DMNMX = D1
END IF
END IF
*
IF( DIR.EQ.0 ) THEN
*
* Sort into decreasing order
*
I = START - 1
J = ENDD + 1
60 CONTINUE
70 CONTINUE
J = J - 1
IF( D( J ).LT.DMNMX )
$ GO TO 70
80 CONTINUE
I = I + 1
IF( D( I ).GT.DMNMX )
$ GO TO 80
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 60
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
ELSE
*
* Sort into increasing order
*
I = START - 1
J = ENDD + 1
90 CONTINUE
100 CONTINUE
J = J - 1
IF( D( J ).GT.DMNMX )
$ GO TO 100
110 CONTINUE
I = I + 1
IF( D( I ).LT.DMNMX )
$ GO TO 110
IF( I.LT.J ) THEN
TMP = D( I )
D( I ) = D( J )
D( J ) = TMP
GO TO 90
END IF
IF( J-START.GT.ENDD-J-1 ) THEN
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
ELSE
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = J + 1
STACK( 2, STKPNT ) = ENDD
STKPNT = STKPNT + 1
STACK( 1, STKPNT ) = START
STACK( 2, STKPNT ) = J
END IF
END IF
END IF
IF( STKPNT.GT.0 )
$ GO TO 10
RETURN
*
* End of SLASRT
*
END
SUBROUTINE SLASSQ( N, X, INCX, SCALE, SUMSQ )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
INTEGER INCX, N
REAL SCALE, SUMSQ
* ..
* .. Array Arguments ..
REAL X( * )
* ..
*
* Purpose
* =======
*
* SLASSQ returns the values scl and smsq such that
*
* ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,
*
* where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is
* assumed to be non-negative and scl returns the value
*
* scl = max( scale, abs( x( i ) ) ).
*
* scale and sumsq must be supplied in SCALE and SUMSQ and
* scl and smsq are overwritten on SCALE and SUMSQ respectively.
*
* The routine makes only one pass through the vector x.
*
* Arguments
* =========
*
* N (input) INTEGER
* The number of elements to be used from the vector X.
*
* X (input) REAL
* The vector for which a scaled sum of squares is computed.
* x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.
*
* INCX (input) INTEGER
* The increment between successive values of the vector X.
* INCX > 0.
*
* SCALE (input/output) REAL
* On entry, the value scale in the equation above.
* On exit, SCALE is overwritten with scl , the scaling factor
* for the sum of squares.
*
* SUMSQ (input/output) REAL
* On entry, the value sumsq in the equation above.
* On exit, SUMSQ is overwritten with smsq , the basic sum of
* squares from which scl has been factored out.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER IX
REAL ABSXI
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
IF( N.GT.0 ) THEN
DO 10 IX = 1, 1 + ( N-1 )*INCX, INCX
IF( X( IX ).NE.ZERO ) THEN
ABSXI = ABS( X( IX ) )
IF( SCALE.LT.ABSXI ) THEN
SUMSQ = 1 + SUMSQ*( SCALE / ABSXI )**2
SCALE = ABSXI
ELSE
SUMSQ = SUMSQ + ( ABSXI / SCALE )**2
END IF
END IF
10 CONTINUE
END IF
RETURN
*
* End of SLASSQ
*
END
SUBROUTINE SLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
REAL A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
* Purpose
* =======
*
* SLATRD reduces NB rows and columns of a real symmetric matrix A to
* symmetric tridiagonal form by an orthogonal similarity
* transformation Q' * A * Q, and returns the matrices V and W which are
* needed to apply the transformation to the unreduced part of A.
*
* If UPLO = 'U', SLATRD reduces the last NB rows and columns of a
* matrix, of which the upper triangle is supplied;
* if UPLO = 'L', SLATRD reduces the first NB rows and columns of a
* matrix, of which the lower triangle is supplied.
*
* This is an auxiliary routine called by SSYTRD.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER
* Specifies whether the upper or lower triangular part of the
* symmetric matrix A is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The order of the matrix A.
*
* NB (input) INTEGER
* The number of rows and columns to be reduced.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* n-by-n upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading n-by-n lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
* On exit:
* if UPLO = 'U', the last NB columns have been reduced to
* tridiagonal form, with the diagonal elements overwriting
* the diagonal elements of A; the elements above the diagonal
* with the array TAU, represent the orthogonal matrix Q as a
* product of elementary reflectors;
* if UPLO = 'L', the first NB columns have been reduced to
* tridiagonal form, with the diagonal elements overwriting
* the diagonal elements of A; the elements below the diagonal
* with the array TAU, represent the orthogonal matrix Q as a
* product of elementary reflectors.
* See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= (1,N).
*
* E (output) REAL array, dimension (N-1)
* If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
* elements of the last NB columns of the reduced matrix;
* if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
* the first NB columns of the reduced matrix.
*
* TAU (output) REAL array, dimension (N-1)
* The scalar factors of the elementary reflectors, stored in
* TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
* See Further Details.
*
* W (output) REAL array, dimension (LDW,NB)
* The n-by-nb matrix W required to update the unreduced part
* of A.
*
* LDW (input) INTEGER
* The leading dimension of the array W. LDW >= max(1,N).
*
* Further Details
* ===============
*
* If UPLO = 'U', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(n) H(n-1) . . . H(n-nb+1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
* and tau in TAU(i-1).
*
* If UPLO = 'L', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(1) H(2) . . . H(nb).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
* and tau in TAU(i).
*
* The elements of the vectors v together form the n-by-nb matrix V
* which is needed, with W, to apply the transformation to the unreduced
* part of the matrix, using a symmetric rank-2k update of the form:
* A := A - V*W' - W*V'.
*
* The contents of A on exit are illustrated by the following examples
* with n = 5 and nb = 2:
*
* if UPLO = 'U': if UPLO = 'L':
*
* ( a a a v4 v5 ) ( d )
* ( a a v4 v5 ) ( 1 d )
* ( a 1 v5 ) ( v1 1 a )
* ( d 1 ) ( v1 v2 a a )
* ( d ) ( v1 v2 a a a )
*
* where d denotes a diagonal element of the reduced matrix, a denotes
* an element of the original matrix that is unchanged, and vi denotes
* an element of the vector defining H(i).
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, HALF
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, HALF = 0.5E+0 )
* ..
* .. Local Scalars ..
INTEGER I, IW
REAL ALPHA
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SGEMV, SLARFG, SSCAL, SSYMV
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SDOT
EXTERNAL LSAME, SDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Reduce last NB columns of upper triangle
*
DO 10 I = N, N - NB + 1, -1
IW = I - N + NB
IF( I.LT.N ) THEN
*
* Update A(1:i,i)
*
CALL SGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
$ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
CALL SGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
$ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
END IF
IF( I.GT.1 ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(1:i-2,i)
*
CALL SLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
E( I-1 ) = A( I-1, I )
A( I-1, I ) = ONE
*
* Compute W(1:i-1,i)
*
CALL SSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
$ ZERO, W( 1, IW ), 1 )
IF( I.LT.N ) THEN
CALL SGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
$ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL SGEMV( 'No transpose', I-1, N-I, -ONE,
$ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
CALL SGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
$ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL SGEMV( 'No transpose', I-1, N-I, -ONE,
$ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
END IF
CALL SSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
ALPHA = -HALF*TAU( I-1 )*SDOT( I-1, W( 1, IW ), 1,
$ A( 1, I ), 1 )
CALL SAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
END IF
*
10 CONTINUE
ELSE
*
* Reduce first NB columns of lower triangle
*
DO 20 I = 1, NB
*
* Update A(i:n,i)
*
CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
$ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
IF( I.LT.N ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:n,i)
*
CALL SLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAU( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute W(i+1:n,i)
*
CALL SSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
CALL SGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL SGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL SGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL SGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
$ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL SSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
ALPHA = -HALF*TAU( I )*SDOT( N-I, W( I+1, I ), 1,
$ A( I+1, I ), 1 )
CALL SAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
END IF
*
20 CONTINUE
END IF
*
RETURN
*
* End of SLATRD
*
END
REAL FUNCTION SNRM2 ( N, X, INCX )
* .. Scalar Arguments ..
INTEGER INCX, N
* .. Array Arguments ..
REAL X( * )
* ..
*
* SNRM2 returns the euclidean norm of a vector via the function
* name, so that
*
* SNRM2 := sqrt( x'*x )
*
*
*
* -- This version written on 25-October-1982.
* Modified on 14-October-1993 to inline the call to SLASSQ.
* Sven Hammarling, Nag Ltd.
*
*
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* .. Local Scalars ..
INTEGER IX
REAL ABSXI, NORM, SCALE, SSQ
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
IF( N.LT.1 .OR. INCX.LT.1 )THEN
NORM = ZERO
ELSE IF( N.EQ.1 )THEN
NORM = ABS( X( 1 ) )
ELSE
SCALE = ZERO
SSQ = ONE
* The following loop is equivalent to this call to the LAPACK
* auxiliary routine:
* CALL SLASSQ( N, X, INCX, SCALE, SSQ )
*
DO 10, IX = 1, 1 + ( N - 1 )*INCX, INCX
IF( X( IX ).NE.ZERO )THEN
ABSXI = ABS( X( IX ) )
IF( SCALE.LT.ABSXI )THEN
SSQ = ONE + SSQ*( SCALE/ABSXI )**2
SCALE = ABSXI
ELSE
SSQ = SSQ + ( ABSXI/SCALE )**2
END IF
END IF
10 CONTINUE
NORM = SCALE * SQRT( SSQ )
END IF
*
SNRM2 = NORM
RETURN
*
* End of SNRM2.
*
END
SUBROUTINE SORG2L( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SORG2L generates an m by n real matrix Q with orthonormal columns,
* which is defined as the last n columns of a product of k elementary
* reflectors of order m
*
* Q = H(k) . . . H(2) H(1)
*
* as returned by SGEQLF.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix Q. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix Q. M >= N >= 0.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines the
* matrix Q. N >= K >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the (n-k+i)-th column must contain the vector which
* defines the elementary reflector H(i), for i = 1,2,...,k, as
* returned by SGEQLF in the last k columns of its array
* argument A.
* On exit, the m by n matrix Q.
*
* LDA (input) INTEGER
* The first dimension of the array A. LDA >= max(1,M).
*
* TAU (input) REAL array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by SGEQLF.
*
* WORK (workspace) REAL array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument has an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, II, J, L
* ..
* .. External Subroutines ..
EXTERNAL SLARF, SSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORG2L', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Initialise columns 1:n-k to columns of the unit matrix
*
DO 20 J = 1, N - K
DO 10 L = 1, M
A( L, J ) = ZERO
10 CONTINUE
A( M-N+J, J ) = ONE
20 CONTINUE
*
DO 40 I = 1, K
II = N - K + I
*
* Apply H(i) to A(1:m-k+i,1:n-k+i) from the left
*
A( M-N+II, II ) = ONE
CALL SLARF( 'Left', M-N+II, II-1, A( 1, II ), 1, TAU( I ), A,
$ LDA, WORK )
CALL SSCAL( M-N+II-1, -TAU( I ), A( 1, II ), 1 )
A( M-N+II, II ) = ONE - TAU( I )
*
* Set A(m-k+i+1:m,n-k+i) to zero
*
DO 30 L = M - N + II + 1, M
A( L, II ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of SORG2L
*
END
SUBROUTINE SORG2R( M, N, K, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SORG2R generates an m by n real matrix Q with orthonormal columns,
* which is defined as the first n columns of a product of k elementary
* reflectors of order m
*
* Q = H(1) H(2) . . . H(k)
*
* as returned by SGEQRF.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix Q. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix Q. M >= N >= 0.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines the
* matrix Q. N >= K >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the i-th column must contain the vector which
* defines the elementary reflector H(i), for i = 1,2,...,k, as
* returned by SGEQRF in the first k columns of its array
* argument A.
* On exit, the m-by-n matrix Q.
*
* LDA (input) INTEGER
* The first dimension of the array A. LDA >= max(1,M).
*
* TAU (input) REAL array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by SGEQRF.
*
* WORK (workspace) REAL array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument has an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, L
* ..
* .. External Subroutines ..
EXTERNAL SLARF, SSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORG2R', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Initialise columns k+1:n to columns of the unit matrix
*
DO 20 J = K + 1, N
DO 10 L = 1, M
A( L, J ) = ZERO
10 CONTINUE
A( J, J ) = ONE
20 CONTINUE
*
DO 40 I = K, 1, -1
*
* Apply H(i) to A(i:m,i:n) from the left
*
IF( I.LT.N ) THEN
A( I, I ) = ONE
CALL SLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ),
$ A( I, I+1 ), LDA, WORK )
END IF
IF( I.LT.M )
$ CALL SSCAL( M-I, -TAU( I ), A( I+1, I ), 1 )
A( I, I ) = ONE - TAU( I )
*
* Set A(1:i-1,i) to zero
*
DO 30 L = 1, I - 1
A( L, I ) = ZERO
30 CONTINUE
40 CONTINUE
RETURN
*
* End of SORG2R
*
END
SUBROUTINE SORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), TAU( * ), WORK( LWORK )
* ..
*
* Purpose
* =======
*
* SORGQL generates an M-by-N real matrix Q with orthonormal columns,
* which is defined as the last N columns of a product of K elementary
* reflectors of order M
*
* Q = H(k) . . . H(2) H(1)
*
* as returned by SGEQLF.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix Q. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix Q. M >= N >= 0.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines the
* matrix Q. N >= K >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the (n-k+i)-th column must contain the vector which
* defines the elementary reflector H(i), for i = 1,2,...,k, as
* returned by SGEQLF in the last k columns of its array
* argument A.
* On exit, the M-by-N matrix Q.
*
* LDA (input) INTEGER
* The first dimension of the array A. LDA >= max(1,M).
*
* TAU (input) REAL array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by SGEQLF.
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument has an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, IB, IINFO, IWS, J, KK, L, LDWORK, NB, NBMIN,
$ NX
* ..
* .. External Subroutines ..
EXTERNAL SLARFB, SLARFT, SORG2L, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORGQL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size.
*
NB = ILAENV( 1, 'SORGQL', ' ', M, N, K, -1 )
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'SORGQL', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'SORGQL', ' ', M, N, K, -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the first block.
* The last kk columns are handled by the block method.
*
KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB )
*
* Set A(m-kk+1:m,1:n-kk) to zero.
*
DO 20 J = 1, N - KK
DO 10 I = M - KK + 1, M
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the first or only block.
*
CALL SORG2L( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = K - KK + 1, K, NB
IB = MIN( NB, K-I+1 )
IF( N-K+I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL SLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB,
$ A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
*
CALL SLARFB( 'Left', 'No transpose', 'Backward',
$ 'Columnwise', M-K+I+IB-1, N-K+I-1, IB,
$ A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA,
$ WORK( IB+1 ), LDWORK )
END IF
*
* Apply H to rows 1:m-k+i+ib-1 of current block
*
CALL SORG2L( M-K+I+IB-1, IB, IB, A( 1, N-K+I ), LDA,
$ TAU( I ), WORK, IINFO )
*
* Set rows m-k+i+ib:m of current block to zero
*
DO 40 J = N - K + I, N - K + I + IB - 1
DO 30 L = M - K + I + IB, M
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of SORGQL
*
END
SUBROUTINE SORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), TAU( * ), WORK( LWORK )
* ..
*
* Purpose
* =======
*
* SORGQR generates an M-by-N real matrix Q with orthonormal columns,
* which is defined as the first N columns of a product of K elementary
* reflectors of order M
*
* Q = H(1) H(2) . . . H(k)
*
* as returned by SGEQRF.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix Q. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix Q. M >= N >= 0.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines the
* matrix Q. N >= K >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the i-th column must contain the vector which
* defines the elementary reflector H(i), for i = 1,2,...,k, as
* returned by SGEQRF in the first k columns of its array
* argument A.
* On exit, the M-by-N matrix Q.
*
* LDA (input) INTEGER
* The first dimension of the array A. LDA >= max(1,M).
*
* TAU (input) REAL array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by SGEQRF.
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument has an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL SLARFB, SLARFT, SORG2R, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORGQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size.
*
NB = ILAENV( 1, 'SORGQR', ' ', M, N, K, -1 )
NBMIN = 2
NX = 0
IWS = N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'SORGQR', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'SORGQR', ' ', M, N, K, -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the last block.
* The first kk columns are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
* Set A(1:kk,kk+1:n) to zero.
*
DO 20 J = KK + 1, N
DO 10 I = 1, KK
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the last or only block.
*
IF( KK.LT.N )
$ CALL SORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA,
$ TAU( KK+1 ), WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF( I+IB.LE.N ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i) H(i+1) . . . H(i+ib-1)
*
CALL SLARFT( 'Forward', 'Columnwise', M-I+1, IB,
$ A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(i:m,i+ib:n) from the left
*
CALL SLARFB( 'Left', 'No transpose', 'Forward',
$ 'Columnwise', M-I+1, N-I-IB+1, IB,
$ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
$ LDA, WORK( IB+1 ), LDWORK )
END IF
*
* Apply H to rows i:m of current block
*
CALL SORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ), WORK,
$ IINFO )
*
* Set rows 1:i-1 of current block to zero
*
DO 40 J = I, I + IB - 1
DO 30 L = 1, I - 1
A( L, J ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of SORGQR
*
END
SUBROUTINE SORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), TAU( * ), WORK( LWORK )
* ..
*
* Purpose
* =======
*
* SORGTR generates a real orthogonal matrix Q which is defined as the
* product of n-1 elementary reflectors of order N, as returned by
* SSYTRD:
*
* if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
*
* if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A contains elementary reflectors
* from SSYTRD;
* = 'L': Lower triangle of A contains elementary reflectors
* from SSYTRD.
*
* N (input) INTEGER
* The order of the matrix Q. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the vectors which define the elementary reflectors,
* as returned by SSYTRD.
* On exit, the N-by-N orthogonal matrix Q.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* TAU (input) REAL array, dimension (N-1)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by SSYTRD.
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N-1).
* For optimum performance LWORK >= (N-1)*NB, where NB is
* the optimal blocksize.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IINFO, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SORGQL, SORGQR, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MAX( 1, N-1 ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORGTR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( UPPER ) THEN
*
* Q was determined by a call to SSYTRD with UPLO = 'U'
*
* Shift the vectors which define the elementary reflectors one
* column to the left, and set the last row and column of Q to
* those of the unit matrix
*
DO 20 J = 1, N - 1
DO 10 I = 1, J - 1
A( I, J ) = A( I, J+1 )
10 CONTINUE
A( N, J ) = ZERO
20 CONTINUE
DO 30 I = 1, N - 1
A( I, N ) = ZERO
30 CONTINUE
A( N, N ) = ONE
*
* Generate Q(1:n-1,1:n-1)
*
CALL SORGQL( N-1, N-1, N-1, A, LDA, TAU, WORK, LWORK, IINFO )
*
ELSE
*
* Q was determined by a call to SSYTRD with UPLO = 'L'.
*
* Shift the vectors which define the elementary reflectors one
* column to the right, and set the first row and column of Q to
* those of the unit matrix
*
DO 50 J = N, 2, -1
A( 1, J ) = ZERO
DO 40 I = J + 1, N
A( I, J ) = A( I, J-1 )
40 CONTINUE
50 CONTINUE
A( 1, 1 ) = ONE
DO 60 I = 2, N
A( I, 1 ) = ZERO
60 CONTINUE
IF( N.GT.1 ) THEN
*
* Generate Q(2:n,2:n)
*
CALL SORGQR( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK,
$ LWORK, IINFO )
END IF
END IF
RETURN
*
* End of SORGTR
*
END
subroutine sscal(n,sa,sx,incx)
c
c scales a vector by a constant.
c uses unrolled loops for increment equal to 1.
c jack dongarra, linpack, 3/11/78.
c modified 3/93 to return if incx .le. 0.
c modified 12/3/93, array(1) declarations changed to array(*)
c
real sa,sx(*)
integer i,incx,m,mp1,n,nincx
c
if( n.le.0 .or. incx.le.0 )return
if(incx.eq.1)go to 20
c
c code for increment not equal to 1
c
nincx = n*incx
do 10 i = 1,nincx,incx
sx(i) = sa*sx(i)
10 continue
return
c
c code for increment equal to 1
c
c
c clean-up loop
c
20 m = mod(n,5)
if( m .eq. 0 ) go to 40
do 30 i = 1,m
sx(i) = sa*sx(i)
30 continue
if( n .lt. 5 ) return
40 mp1 = m + 1
do 50 i = mp1,n,5
sx(i) = sa*sx(i)
sx(i + 1) = sa*sx(i + 1)
sx(i + 2) = sa*sx(i + 2)
sx(i + 3) = sa*sx(i + 3)
sx(i + 4) = sa*sx(i + 4)
50 continue
return
end
SUBROUTINE SSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, LDZ, N
* ..
* .. Array Arguments ..
REAL D( * ), E( * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SSTEQR computes all eigenvalues and, optionally, eigenvectors of a
* symmetric tridiagonal matrix using the implicit QL or QR method.
* The eigenvectors of a full or band symmetric matrix can also be found
* if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to
* tridiagonal form.
*
* Arguments
* =========
*
* COMPZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only.
* = 'V': Compute eigenvalues and eigenvectors of the original
* symmetric matrix. On entry, Z must contain the
* orthogonal matrix used to reduce the original matrix
* to tridiagonal form.
* = 'I': Compute eigenvalues and eigenvectors of the
* tridiagonal matrix. Z is initialized to the identity
* matrix.
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the diagonal elements of the tridiagonal matrix.
* On exit, if INFO = 0, the eigenvalues in ascending order.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix.
* On exit, E has been destroyed.
*
* Z (input/output) REAL array, dimension (LDZ, N)
* On entry, if COMPZ = 'V', then Z contains the orthogonal
* matrix used in the reduction to tridiagonal form.
* On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
* orthonormal eigenvectors of the original symmetric matrix,
* and if COMPZ = 'I', Z contains the orthonormal eigenvectors
* of the symmetric tridiagonal matrix.
* If COMPZ = 'N', then Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* eigenvectors are desired, then LDZ >= max(1,N).
*
* WORK (workspace) REAL array, dimension (max(1,2*N-2))
* If COMPZ = 'N', then WORK is not referenced.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: the algorithm has failed to find all the eigenvalues in
* a total of 30*N iterations; if INFO = i, then i
* elements of E have not converged to zero; on exit, D
* and E contain the elements of a symmetric tridiagonal
* matrix which is orthogonally similar to the original
* matrix.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0,
$ THREE = 3.0E0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
$ LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
$ NM1, NMAXIT
REAL ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
$ S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANST, SLAPY2
EXTERNAL LSAME, SLAMCH, SLANST, SLAPY2
* ..
* .. External Subroutines ..
EXTERNAL SLAE2, SLAEV2, SLARTG, SLASCL, SLASET, SLASR,
$ SLASRT, SSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ICOMPZ = 0
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ICOMPZ = 2
ELSE
ICOMPZ = -1
END IF
IF( ICOMPZ.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
$ N ) ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSTEQR', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ICOMPZ.EQ.2 )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Determine the unit roundoff and over/underflow thresholds.
*
EPS = SLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = SLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
*
* Compute the eigenvalues and eigenvectors of the tridiagonal
* matrix.
*
IF( ICOMPZ.EQ.2 )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
NMAXIT = N*MAXIT
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
NM1 = N - 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 160
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
IF( L1.LE.NM1 ) THEN
DO 20 M = L1, NM1
TST = ABS( E( M ) )
IF( TST.EQ.ZERO )
$ GO TO 30
IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
END IF
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = SLANST( 'I', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.EQ.ZERO )
$ GO TO 10
IF( ANORM.GT.SSFMAX ) THEN
ISCALE = 1
CALL SLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL SLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL SLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL SLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GT.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
40 CONTINUE
IF( L.NE.LEND ) THEN
LENDM1 = LEND - 1
DO 50 M = L, LENDM1
TST = ABS( E( M ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+
$ SAFMIN )GO TO 60
50 CONTINUE
END IF
*
M = LEND
*
60 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 80
*
* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L+1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL SLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C, S )
WORK( L ) = C
WORK( N-1+L ) = S
CALL SLASR( 'R', 'V', 'B', N, 2, WORK( L ),
$ WORK( N-1+L ), Z( 1, L ), LDZ )
ELSE
CALL SLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 )
END IF
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L+1 )-P ) / ( TWO*E( L ) )
R = SLAPY2( G, ONE )
G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
MM1 = M - 1
DO 70 I = MM1, L, -1
F = S*E( I )
B = C*E( I )
CALL SLARTG( G, F, C, S, R )
IF( I.NE.M-1 )
$ E( I+1 ) = R
G = D( I+1 ) - P
R = ( D( I )-G )*S + TWO*C*B
P = S*R
D( I+1 ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = -S
END IF
*
70 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = M - L + 1
CALL SLASR( 'R', 'V', 'B', N, MM, WORK( L ), WORK( N-1+L ),
$ Z( 1, L ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( L ) = G
GO TO 40
*
* Eigenvalue found.
*
80 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 40
GO TO 140
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
90 CONTINUE
IF( L.NE.LEND ) THEN
LENDP1 = LEND + 1
DO 100 M = L, LENDP1, -1
TST = ABS( E( M-1 ) )**2
IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+
$ SAFMIN )GO TO 110
100 CONTINUE
END IF
*
M = LEND
*
110 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 130
*
* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
* to compute its eigensystem.
*
IF( M.EQ.L-1 ) THEN
IF( ICOMPZ.GT.0 ) THEN
CALL SLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C, S )
WORK( M ) = C
WORK( N-1+M ) = S
CALL SLASR( 'R', 'V', 'F', N, 2, WORK( M ),
$ WORK( N-1+M ), Z( 1, L-1 ), LDZ )
ELSE
CALL SLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
END IF
D( L-1 ) = RT1
D( L ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 140
JTOT = JTOT + 1
*
* Form shift.
*
G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
R = SLAPY2( G, ONE )
G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
*
S = ONE
C = ONE
P = ZERO
*
* Inner loop
*
LM1 = L - 1
DO 120 I = M, LM1
F = S*E( I )
B = C*E( I )
CALL SLARTG( G, F, C, S, R )
IF( I.NE.M )
$ E( I-1 ) = R
G = D( I ) - P
R = ( D( I+1 )-G )*S + TWO*C*B
P = S*R
D( I ) = G + P
G = C*R - B
*
* If eigenvectors are desired, then save rotations.
*
IF( ICOMPZ.GT.0 ) THEN
WORK( I ) = C
WORK( N-1+I ) = S
END IF
*
120 CONTINUE
*
* If eigenvectors are desired, then apply saved rotations.
*
IF( ICOMPZ.GT.0 ) THEN
MM = L - M + 1
CALL SLASR( 'R', 'V', 'F', N, MM, WORK( M ), WORK( N-1+M ),
$ Z( 1, M ), LDZ )
END IF
*
D( L ) = D( L ) - P
E( LM1 ) = G
GO TO 90
*
* Eigenvalue found.
*
130 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 90
GO TO 140
*
END IF
*
* Undo scaling if necessary
*
140 CONTINUE
IF( ISCALE.EQ.1 ) THEN
CALL SLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL SLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1, E( LSV ),
$ N, INFO )
ELSE IF( ISCALE.EQ.2 ) THEN
CALL SLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
CALL SLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1, E( LSV ),
$ N, INFO )
END IF
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.LT.NMAXIT )
$ GO TO 10
DO 150 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
150 CONTINUE
GO TO 190
*
* Order eigenvalues and eigenvectors.
*
160 CONTINUE
IF( ICOMPZ.EQ.0 ) THEN
*
* Use Quick Sort
*
CALL SLASRT( 'I', N, D, INFO )
*
ELSE
*
* Use Selection Sort to minimize swaps of eigenvectors
*
DO 180 II = 2, N
I = II - 1
K = I
P = D( I )
DO 170 J = II, N
IF( D( J ).LT.P ) THEN
K = J
P = D( J )
END IF
170 CONTINUE
IF( K.NE.I ) THEN
D( K ) = D( I )
D( I ) = P
CALL SSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
END IF
180 CONTINUE
END IF
*
190 CONTINUE
RETURN
*
* End of SSTEQR
*
END
SUBROUTINE SSTERF( N, D, E, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER INFO, N
* ..
* .. Array Arguments ..
REAL D( * ), E( * )
* ..
*
* Purpose
* =======
*
* SSTERF computes all eigenvalues of a symmetric tridiagonal matrix
* using the Pal-Walker-Kahan variant of the QL or QR algorithm.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal matrix.
* On exit, if INFO = 0, the eigenvalues in ascending order.
*
* E (input/output) REAL array, dimension (N-1)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix.
* On exit, E has been destroyed.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: the algorithm failed to find all of the eigenvalues in
* a total of 30*N iterations; if INFO = i, then i
* elements of E have not converged to zero.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO, THREE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0,
$ THREE = 3.0E0 )
INTEGER MAXIT
PARAMETER ( MAXIT = 30 )
* ..
* .. Local Scalars ..
INTEGER I, ISCALE, JTOT, L, L1, LEND, LENDM1, LENDP1,
$ LENDSV, LM1, LSV, M, MM1, NM1, NMAXIT
REAL ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC,
$ OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN,
$ SIGMA, SSFMAX, SSFMIN, TST
* ..
* .. External Functions ..
REAL SLAMCH, SLANST, SLAPY2
EXTERNAL SLAMCH, SLANST, SLAPY2
* ..
* .. External Subroutines ..
EXTERNAL SLAE2, SLASCL, SLASRT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
* Quick return if possible
*
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'SSTERF', -INFO )
RETURN
END IF
IF( N.LE.1 )
$ RETURN
*
* Determine the unit roundoff for this environment.
*
EPS = SLAMCH( 'E' )
EPS2 = EPS**2
SAFMIN = SLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
SSFMAX = SQRT( SAFMAX ) / THREE
SSFMIN = SQRT( SAFMIN ) / EPS2
*
* Compute the eigenvalues of the tridiagonal matrix.
*
NMAXIT = N*MAXIT
SIGMA = ZERO
JTOT = 0
*
* Determine where the matrix splits and choose QL or QR iteration
* for each block, according to whether top or bottom diagonal
* element is smaller.
*
L1 = 1
NM1 = N - 1
*
10 CONTINUE
IF( L1.GT.N )
$ GO TO 170
IF( L1.GT.1 )
$ E( L1-1 ) = ZERO
IF( L1.LE.NM1 ) THEN
DO 20 M = L1, NM1
TST = ABS( E( M ) )
IF( TST.EQ.ZERO )
$ GO TO 30
IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+
$ 1 ) ) ) )*EPS ) THEN
E( M ) = ZERO
GO TO 30
END IF
20 CONTINUE
END IF
M = N
*
30 CONTINUE
L = L1
LSV = L
LEND = M
LENDSV = LEND
L1 = M + 1
IF( LEND.EQ.L )
$ GO TO 10
*
* Scale submatrix in rows and columns L to LEND
*
ANORM = SLANST( 'I', LEND-L+1, D( L ), E( L ) )
ISCALE = 0
IF( ANORM.GT.SSFMAX ) THEN
ISCALE = 1
CALL SLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL SLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N,
$ INFO )
ELSE IF( ANORM.LT.SSFMIN ) THEN
ISCALE = 2
CALL SLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N,
$ INFO )
CALL SLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N,
$ INFO )
END IF
*
DO 40 I = L, LEND - 1
E( I ) = E( I )**2
40 CONTINUE
*
* Choose between QL and QR iteration
*
IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN
LEND = LSV
L = LENDSV
END IF
*
IF( LEND.GE.L ) THEN
*
* QL Iteration
*
* Look for small subdiagonal element.
*
50 CONTINUE
IF( L.NE.LEND ) THEN
LENDM1 = LEND - 1
DO 60 M = L, LENDM1
TST = ABS( E( M ) )
IF( TST.LE. EPS2*ABS( D( M ) * D( M+1 ) ) )GO TO 70
60 CONTINUE
END IF
*
M = LEND
*
70 CONTINUE
IF( M.LT.LEND )
$ E( M ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 90
*
* If remaining matrix is 2 by 2, use SLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L+1 ) THEN
RTE = SQRT( E( L ) )
CALL SLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 )
D( L ) = RT1
D( L+1 ) = RT2
E( L ) = ZERO
L = L + 2
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L ) )
SIGMA = ( D( L+1 )-P ) / ( TWO*RTE )
R = SLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
MM1 = M - 1
DO 80 I = MM1, L, -1
BB = E( I )
R = P + BB
IF( I.NE.M-1 )
$ E( I+1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I+1 ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
80 CONTINUE
*
E( L ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 50
*
* Eigenvalue found.
*
90 CONTINUE
D( L ) = P
*
L = L + 1
IF( L.LE.LEND )
$ GO TO 50
GO TO 150
*
ELSE
*
* QR Iteration
*
* Look for small superdiagonal element.
*
100 CONTINUE
IF( L.NE.LEND ) THEN
LENDP1 = LEND + 1
DO 110 M = L, LENDP1, -1
TST = ABS( E( M-1 ) )
IF( TST.LE. EPS2*ABS( D( M ) * D( M-1 ) ) )GO TO 120
110 CONTINUE
END IF
*
M = LEND
*
120 CONTINUE
IF( M.GT.LEND )
$ E( M-1 ) = ZERO
P = D( L )
IF( M.EQ.L )
$ GO TO 140
*
* If remaining matrix is 2 by 2, use SLAE2 to compute its
* eigenvalues.
*
IF( M.EQ.L-1 ) THEN
RTE = SQRT( E( L-1 ) )
CALL SLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 )
D( L ) = RT1
D( L-1 ) = RT2
E( L-1 ) = ZERO
L = L - 2
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
END IF
*
IF( JTOT.EQ.NMAXIT )
$ GO TO 150
JTOT = JTOT + 1
*
* Form shift.
*
RTE = SQRT( E( L-1 ) )
SIGMA = ( D( L-1 )-P ) / ( TWO*RTE )
R = SLAPY2( SIGMA, ONE )
SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) )
*
C = ONE
S = ZERO
GAMMA = D( M ) - SIGMA
P = GAMMA*GAMMA
*
* Inner loop
*
LM1 = L - 1
DO 130 I = M, LM1
BB = E( I )
R = P + BB
IF( I.NE.M )
$ E( I-1 ) = S*R
OLDC = C
C = P / R
S = BB / R
OLDGAM = GAMMA
ALPHA = D( I+1 )
GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM
D( I ) = OLDGAM + ( ALPHA-GAMMA )
IF( C.NE.ZERO ) THEN
P = ( GAMMA*GAMMA ) / C
ELSE
P = OLDC*BB
END IF
130 CONTINUE
*
E( LM1 ) = S*P
D( L ) = SIGMA + GAMMA
GO TO 100
*
* Eigenvalue found.
*
140 CONTINUE
D( L ) = P
*
L = L - 1
IF( L.GE.LEND )
$ GO TO 100
GO TO 150
*
END IF
*
* Undo scaling if necessary
*
150 CONTINUE
IF( ISCALE.EQ.1 )
$ CALL SLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
IF( ISCALE.EQ.2 )
$ CALL SLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
$ D( LSV ), N, INFO )
*
* Check for no convergence to an eigenvalue after a total
* of N*MAXIT iterations.
*
IF( JTOT.EQ.NMAXIT ) THEN
DO 160 I = 1, N - 1
IF( E( I ).NE.ZERO )
$ INFO = INFO + 1
160 CONTINUE
RETURN
END IF
GO TO 10
*
* Sort eigenvalues in increasing order.
*
170 CONTINUE
CALL SLASRT( 'I', N, D, INFO )
*
RETURN
*
* End of SSTERF
*
END
subroutine sswap (n,sx,incx,sy,incy)
c
c interchanges two vectors.
c uses unrolled loops for increments equal to 1.
c jack dongarra, linpack, 3/11/78.
c modified 12/3/93, array(1) declarations changed to array(*)
c
real sx(*),sy(*),stemp
integer i,incx,incy,ix,iy,m,mp1,n
c
if(n.le.0)return
if(incx.eq.1.and.incy.eq.1)go to 20
c
c code for unequal increments or equal increments not equal
c to 1
c
ix = 1
iy = 1
if(incx.lt.0)ix = (-n+1)*incx + 1
if(incy.lt.0)iy = (-n+1)*incy + 1
do 10 i = 1,n
stemp = sx(ix)
sx(ix) = sy(iy)
sy(iy) = stemp
ix = ix + incx
iy = iy + incy
10 continue
return
c
c code for both increments equal to 1
c
c
c clean-up loop
c
20 m = mod(n,3)
if( m .eq. 0 ) go to 40
do 30 i = 1,m
stemp = sx(i)
sx(i) = sy(i)
sy(i) = stemp
30 continue
if( n .lt. 3 ) return
40 mp1 = m + 1
do 50 i = mp1,n,3
stemp = sx(i)
sx(i) = sy(i)
sy(i) = stemp
stemp = sx(i + 1)
sx(i + 1) = sy(i + 1)
sy(i + 1) = stemp
stemp = sx(i + 2)
sx(i + 2) = sy(i + 2)
sy(i + 2) = stemp
50 continue
return
end
SUBROUTINE SSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
*
* -- LAPACK driver routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), W( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SSYEV computes all eigenvalues and, optionally, eigenvectors of a
* real symmetric matrix A.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only;
* = 'V': Compute eigenvalues and eigenvectors.
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA, N)
* On entry, the symmetric matrix A. If UPLO = 'U', the
* leading N-by-N upper triangular part of A contains the
* upper triangular part of the matrix A. If UPLO = 'L',
* the leading N-by-N lower triangular part of A contains
* the lower triangular part of the matrix A.
* On exit, if JOBZ = 'V', then if INFO = 0, A contains the
* orthonormal eigenvectors of the matrix A.
* If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
* or the upper triangle (if UPLO='U') of A, including the
* diagonal, is destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* W (output) REAL array, dimension (N)
* If INFO = 0, the eigenvalues in ascending order.
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The length of the array WORK. LWORK >= max(1,3*N-1).
* For optimal efficiency, LWORK >= (NB+2)*N,
* where NB is the blocksize for SSYTRD returned by ILAENV.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the algorithm failed to converge; i
* off-diagonal elements of an intermediate tridiagonal
* form did not converge to zero.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, WANTZ
INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
$ LLWORK, LOPT
REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANSY
EXTERNAL LSAME, SLAMCH, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SLASCL, SORGTR, SSCAL, SSTEQR, SSTERF, SSYTRD,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LWORK.LT.MAX( 1, 3*N-1 ) ) THEN
INFO = -8
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSYEV ', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( N.EQ.1 ) THEN
W( 1 ) = A( 1, 1 )
WORK( 1 ) = 3
IF( WANTZ )
$ A( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = SLANSY( 'M', UPLO, N, A, LDA, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 )
$ CALL SLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
* Call SSYTRD to reduce symmetric matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
CALL SSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
LOPT = 2*N + WORK( INDWRK )
*
* For eigenvalues only, call SSTERF. For eigenvectors, first call
* SORGTR to generate the orthogonal matrix, then call SSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL SSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL SORGTR( UPLO, N, A, LDA, WORK( INDTAU ), WORK( INDWRK ),
$ LLWORK, IINFO )
CALL SSTEQR( JOBZ, N, W, WORK( INDE ), A, LDA, WORK( INDTAU ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = MAX( 3*N-1, LOPT )
*
RETURN
*
* End of SSYEV
*
END
SUBROUTINE SSYMV ( UPLO, N, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
* .. Scalar Arguments ..
REAL ALPHA, BETA
INTEGER INCX, INCY, LDA, N
CHARACTER*1 UPLO
* .. Array Arguments ..
REAL A( LDA, * ), X( * ), Y( * )
* ..
*
* Purpose
* =======
*
* SSYMV performs the matrix-vector operation
*
* y := alpha*A*x + beta*y,
*
* where alpha and beta are scalars, x and y are n element vectors and
* A is an n by n symmetric matrix.
*
* Parameters
* ==========
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the upper or lower
* triangular part of the array A is to be referenced as
* follows:
*
* UPLO = 'U' or 'u' Only the upper triangular part of A
* is to be referenced.
*
* UPLO = 'L' or 'l' Only the lower triangular part of A
* is to be referenced.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, n ).
* Before entry with UPLO = 'U' or 'u', the leading n by n
* upper triangular part of the array A must contain the upper
* triangular part of the symmetric matrix and the strictly
* lower triangular part of A is not referenced.
* Before entry with UPLO = 'L' or 'l', the leading n by n
* lower triangular part of the array A must contain the lower
* triangular part of the symmetric matrix and the strictly
* upper triangular part of A is not referenced.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, n ).
* Unchanged on exit.
*
* X - REAL array of dimension at least
* ( 1 + ( n - 1 )*abs( INCX ) ).
* Before entry, the incremented array X must contain the n
* element vector x.
* Unchanged on exit.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
* BETA - REAL .
* On entry, BETA specifies the scalar beta. When BETA is
* supplied as zero then Y need not be set on input.
* Unchanged on exit.
*
* Y - REAL array of dimension at least
* ( 1 + ( n - 1 )*abs( INCY ) ).
* Before entry, the incremented array Y must contain the n
* element vector y. On exit, Y is overwritten by the updated
* vector y.
*
* INCY - INTEGER.
* On entry, INCY specifies the increment for the elements of
* Y. INCY must not be zero.
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* .. Local Scalars ..
REAL TEMP1, TEMP2
INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF ( .NOT.LSAME( UPLO, 'U' ).AND.
$ .NOT.LSAME( UPLO, 'L' ) )THEN
INFO = 1
ELSE IF( N.LT.0 )THEN
INFO = 2
ELSE IF( LDA.LT.MAX( 1, N ) )THEN
INFO = 5
ELSE IF( INCX.EQ.0 )THEN
INFO = 7
ELSE IF( INCY.EQ.0 )THEN
INFO = 10
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SSYMV ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( N.EQ.0 ).OR.( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) )
$ RETURN
*
* Set up the start points in X and Y.
*
IF( INCX.GT.0 )THEN
KX = 1
ELSE
KX = 1 - ( N - 1 )*INCX
END IF
IF( INCY.GT.0 )THEN
KY = 1
ELSE
KY = 1 - ( N - 1 )*INCY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through the triangular part
* of A.
*
* First form y := beta*y.
*
IF( BETA.NE.ONE )THEN
IF( INCY.EQ.1 )THEN
IF( BETA.EQ.ZERO )THEN
DO 10, I = 1, N
Y( I ) = ZERO
10 CONTINUE
ELSE
DO 20, I = 1, N
Y( I ) = BETA*Y( I )
20 CONTINUE
END IF
ELSE
IY = KY
IF( BETA.EQ.ZERO )THEN
DO 30, I = 1, N
Y( IY ) = ZERO
IY = IY + INCY
30 CONTINUE
ELSE
DO 40, I = 1, N
Y( IY ) = BETA*Y( IY )
IY = IY + INCY
40 CONTINUE
END IF
END IF
END IF
IF( ALPHA.EQ.ZERO )
$ RETURN
IF( LSAME( UPLO, 'U' ) )THEN
*
* Form y when A is stored in upper triangle.
*
IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN
DO 60, J = 1, N
TEMP1 = ALPHA*X( J )
TEMP2 = ZERO
DO 50, I = 1, J - 1
Y( I ) = Y( I ) + TEMP1*A( I, J )
TEMP2 = TEMP2 + A( I, J )*X( I )
50 CONTINUE
Y( J ) = Y( J ) + TEMP1*A( J, J ) + ALPHA*TEMP2
60 CONTINUE
ELSE
JX = KX
JY = KY
DO 80, J = 1, N
TEMP1 = ALPHA*X( JX )
TEMP2 = ZERO
IX = KX
IY = KY
DO 70, I = 1, J - 1
Y( IY ) = Y( IY ) + TEMP1*A( I, J )
TEMP2 = TEMP2 + A( I, J )*X( IX )
IX = IX + INCX
IY = IY + INCY
70 CONTINUE
Y( JY ) = Y( JY ) + TEMP1*A( J, J ) + ALPHA*TEMP2
JX = JX + INCX
JY = JY + INCY
80 CONTINUE
END IF
ELSE
*
* Form y when A is stored in lower triangle.
*
IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN
DO 100, J = 1, N
TEMP1 = ALPHA*X( J )
TEMP2 = ZERO
Y( J ) = Y( J ) + TEMP1*A( J, J )
DO 90, I = J + 1, N
Y( I ) = Y( I ) + TEMP1*A( I, J )
TEMP2 = TEMP2 + A( I, J )*X( I )
90 CONTINUE
Y( J ) = Y( J ) + ALPHA*TEMP2
100 CONTINUE
ELSE
JX = KX
JY = KY
DO 120, J = 1, N
TEMP1 = ALPHA*X( JX )
TEMP2 = ZERO
Y( JY ) = Y( JY ) + TEMP1*A( J, J )
IX = JX
IY = JY
DO 110, I = J + 1, N
IX = IX + INCX
IY = IY + INCY
Y( IY ) = Y( IY ) + TEMP1*A( I, J )
TEMP2 = TEMP2 + A( I, J )*X( IX )
110 CONTINUE
Y( JY ) = Y( JY ) + ALPHA*TEMP2
JX = JX + INCX
JY = JY + INCY
120 CONTINUE
END IF
END IF
*
RETURN
*
* End of SSYMV .
*
END
SUBROUTINE SSYR2 ( UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA )
* .. Scalar Arguments ..
REAL ALPHA
INTEGER INCX, INCY, LDA, N
CHARACTER*1 UPLO
* .. Array Arguments ..
REAL A( LDA, * ), X( * ), Y( * )
* ..
*
* Purpose
* =======
*
* SSYR2 performs the symmetric rank 2 operation
*
* A := alpha*x*y' + alpha*y*x' + A,
*
* where alpha is a scalar, x and y are n element vectors and A is an n
* by n symmetric matrix.
*
* Parameters
* ==========
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the upper or lower
* triangular part of the array A is to be referenced as
* follows:
*
* UPLO = 'U' or 'u' Only the upper triangular part of A
* is to be referenced.
*
* UPLO = 'L' or 'l' Only the lower triangular part of A
* is to be referenced.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* X - REAL array of dimension at least
* ( 1 + ( n - 1 )*abs( INCX ) ).
* Before entry, the incremented array X must contain the n
* element vector x.
* Unchanged on exit.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
* Y - REAL array of dimension at least
* ( 1 + ( n - 1 )*abs( INCY ) ).
* Before entry, the incremented array Y must contain the n
* element vector y.
* Unchanged on exit.
*
* INCY - INTEGER.
* On entry, INCY specifies the increment for the elements of
* Y. INCY must not be zero.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, n ).
* Before entry with UPLO = 'U' or 'u', the leading n by n
* upper triangular part of the array A must contain the upper
* triangular part of the symmetric matrix and the strictly
* lower triangular part of A is not referenced. On exit, the
* upper triangular part of the array A is overwritten by the
* upper triangular part of the updated matrix.
* Before entry with UPLO = 'L' or 'l', the leading n by n
* lower triangular part of the array A must contain the lower
* triangular part of the symmetric matrix and the strictly
* upper triangular part of A is not referenced. On exit, the
* lower triangular part of the array A is overwritten by the
* lower triangular part of the updated matrix.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, n ).
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* .. Local Scalars ..
REAL TEMP1, TEMP2
INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF ( .NOT.LSAME( UPLO, 'U' ).AND.
$ .NOT.LSAME( UPLO, 'L' ) )THEN
INFO = 1
ELSE IF( N.LT.0 )THEN
INFO = 2
ELSE IF( INCX.EQ.0 )THEN
INFO = 5
ELSE IF( INCY.EQ.0 )THEN
INFO = 7
ELSE IF( LDA.LT.MAX( 1, N ) )THEN
INFO = 9
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SSYR2 ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) )
$ RETURN
*
* Set up the start points in X and Y if the increments are not both
* unity.
*
IF( ( INCX.NE.1 ).OR.( INCY.NE.1 ) )THEN
IF( INCX.GT.0 )THEN
KX = 1
ELSE
KX = 1 - ( N - 1 )*INCX
END IF
IF( INCY.GT.0 )THEN
KY = 1
ELSE
KY = 1 - ( N - 1 )*INCY
END IF
JX = KX
JY = KY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through the triangular part
* of A.
*
IF( LSAME( UPLO, 'U' ) )THEN
*
* Form A when A is stored in the upper triangle.
*
IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN
DO 20, J = 1, N
IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN
TEMP1 = ALPHA*Y( J )
TEMP2 = ALPHA*X( J )
DO 10, I = 1, J
A( I, J ) = A( I, J ) + X( I )*TEMP1 + Y( I )*TEMP2
10 CONTINUE
END IF
20 CONTINUE
ELSE
DO 40, J = 1, N
IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN
TEMP1 = ALPHA*Y( JY )
TEMP2 = ALPHA*X( JX )
IX = KX
IY = KY
DO 30, I = 1, J
A( I, J ) = A( I, J ) + X( IX )*TEMP1
$ + Y( IY )*TEMP2
IX = IX + INCX
IY = IY + INCY
30 CONTINUE
END IF
JX = JX + INCX
JY = JY + INCY
40 CONTINUE
END IF
ELSE
*
* Form A when A is stored in the lower triangle.
*
IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN
DO 60, J = 1, N
IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN
TEMP1 = ALPHA*Y( J )
TEMP2 = ALPHA*X( J )
DO 50, I = J, N
A( I, J ) = A( I, J ) + X( I )*TEMP1 + Y( I )*TEMP2
50 CONTINUE
END IF
60 CONTINUE
ELSE
DO 80, J = 1, N
IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN
TEMP1 = ALPHA*Y( JY )
TEMP2 = ALPHA*X( JX )
IX = JX
IY = JY
DO 70, I = J, N
A( I, J ) = A( I, J ) + X( IX )*TEMP1
$ + Y( IY )*TEMP2
IX = IX + INCX
IY = IY + INCY
70 CONTINUE
END IF
JX = JX + INCX
JY = JY + INCY
80 CONTINUE
END IF
END IF
*
RETURN
*
* End of SSYR2 .
*
END
SUBROUTINE SSYR2K( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB,
$ BETA, C, LDC )
* .. Scalar Arguments ..
CHARACTER*1 UPLO, TRANS
INTEGER N, K, LDA, LDB, LDC
REAL ALPHA, BETA
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * ), C( LDC, * )
* ..
*
* Purpose
* =======
*
* SSYR2K performs one of the symmetric rank 2k operations
*
* C := alpha*A*B' + alpha*B*A' + beta*C,
*
* or
*
* C := alpha*A'*B + alpha*B'*A + beta*C,
*
* where alpha and beta are scalars, C is an n by n symmetric matrix
* and A and B are n by k matrices in the first case and k by n
* matrices in the second case.
*
* Parameters
* ==========
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the upper or lower
* triangular part of the array C is to be referenced as
* follows:
*
* UPLO = 'U' or 'u' Only the upper triangular part of C
* is to be referenced.
*
* UPLO = 'L' or 'l' Only the lower triangular part of C
* is to be referenced.
*
* Unchanged on exit.
*
* TRANS - CHARACTER*1.
* On entry, TRANS specifies the operation to be performed as
* follows:
*
* TRANS = 'N' or 'n' C := alpha*A*B' + alpha*B*A' +
* beta*C.
*
* TRANS = 'T' or 't' C := alpha*A'*B + alpha*B'*A +
* beta*C.
*
* TRANS = 'C' or 'c' C := alpha*A'*B + alpha*B'*A +
* beta*C.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix C. N must be
* at least zero.
* Unchanged on exit.
*
* K - INTEGER.
* On entry with TRANS = 'N' or 'n', K specifies the number
* of columns of the matrices A and B, and on entry with
* TRANS = 'T' or 't' or 'C' or 'c', K specifies the number
* of rows of the matrices A and B. K must be at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, ka ), where ka is
* k when TRANS = 'N' or 'n', and is n otherwise.
* Before entry with TRANS = 'N' or 'n', the leading n by k
* part of the array A must contain the matrix A, otherwise
* the leading k by n part of the array A must contain the
* matrix A.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When TRANS = 'N' or 'n'
* then LDA must be at least max( 1, n ), otherwise LDA must
* be at least max( 1, k ).
* Unchanged on exit.
*
* B - REAL array of DIMENSION ( LDB, kb ), where kb is
* k when TRANS = 'N' or 'n', and is n otherwise.
* Before entry with TRANS = 'N' or 'n', the leading n by k
* part of the array B must contain the matrix B, otherwise
* the leading k by n part of the array B must contain the
* matrix B.
* Unchanged on exit.
*
* LDB - INTEGER.
* On entry, LDB specifies the first dimension of B as declared
* in the calling (sub) program. When TRANS = 'N' or 'n'
* then LDB must be at least max( 1, n ), otherwise LDB must
* be at least max( 1, k ).
* Unchanged on exit.
*
* BETA - REAL .
* On entry, BETA specifies the scalar beta.
* Unchanged on exit.
*
* C - REAL array of DIMENSION ( LDC, n ).
* Before entry with UPLO = 'U' or 'u', the leading n by n
* upper triangular part of the array C must contain the upper
* triangular part of the symmetric matrix and the strictly
* lower triangular part of C is not referenced. On exit, the
* upper triangular part of the array C is overwritten by the
* upper triangular part of the updated matrix.
* Before entry with UPLO = 'L' or 'l', the leading n by n
* lower triangular part of the array C must contain the lower
* triangular part of the symmetric matrix and the strictly
* upper triangular part of C is not referenced. On exit, the
* lower triangular part of the array C is overwritten by the
* lower triangular part of the updated matrix.
*
* LDC - INTEGER.
* On entry, LDC specifies the first dimension of C as declared
* in the calling (sub) program. LDC must be at least
* max( 1, n ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, INFO, J, L, NROWA
REAL TEMP1, TEMP2
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
IF( LSAME( TRANS, 'N' ) )THEN
NROWA = N
ELSE
NROWA = K
END IF
UPPER = LSAME( UPLO, 'U' )
*
INFO = 0
IF( ( .NOT.UPPER ).AND.
$ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN
INFO = 1
ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND.
$ ( .NOT.LSAME( TRANS, 'T' ) ).AND.
$ ( .NOT.LSAME( TRANS, 'C' ) ) )THEN
INFO = 2
ELSE IF( N .LT.0 )THEN
INFO = 3
ELSE IF( K .LT.0 )THEN
INFO = 4
ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
INFO = 7
ELSE IF( LDB.LT.MAX( 1, NROWA ) )THEN
INFO = 9
ELSE IF( LDC.LT.MAX( 1, N ) )THEN
INFO = 12
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'SSYR2K', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( ( N.EQ.0 ).OR.
$ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
$ RETURN
*
* And when alpha.eq.zero.
*
IF( ALPHA.EQ.ZERO )THEN
IF( UPPER )THEN
IF( BETA.EQ.ZERO )THEN
DO 20, J = 1, N
DO 10, I = 1, J
C( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40, J = 1, N
DO 30, I = 1, J
C( I, J ) = BETA*C( I, J )
30 CONTINUE
40 CONTINUE
END IF
ELSE
IF( BETA.EQ.ZERO )THEN
DO 60, J = 1, N
DO 50, I = J, N
C( I, J ) = ZERO
50 CONTINUE
60 CONTINUE
ELSE
DO 80, J = 1, N
DO 70, I = J, N
C( I, J ) = BETA*C( I, J )
70 CONTINUE
80 CONTINUE
END IF
END IF
RETURN
END IF
*
* Start the operations.
*
IF( LSAME( TRANS, 'N' ) )THEN
*
* Form C := alpha*A*B' + alpha*B*A' + C.
*
IF( UPPER )THEN
DO 130, J = 1, N
IF( BETA.EQ.ZERO )THEN
DO 90, I = 1, J
C( I, J ) = ZERO
90 CONTINUE
ELSE IF( BETA.NE.ONE )THEN
DO 100, I = 1, J
C( I, J ) = BETA*C( I, J )
100 CONTINUE
END IF
DO 120, L = 1, K
IF( ( A( J, L ).NE.ZERO ).OR.
$ ( B( J, L ).NE.ZERO ) )THEN
TEMP1 = ALPHA*B( J, L )
TEMP2 = ALPHA*A( J, L )
DO 110, I = 1, J
C( I, J ) = C( I, J ) +
$ A( I, L )*TEMP1 + B( I, L )*TEMP2
110 CONTINUE
END IF
120 CONTINUE
130 CONTINUE
ELSE
DO 180, J = 1, N
IF( BETA.EQ.ZERO )THEN
DO 140, I = J, N
C( I, J ) = ZERO
140 CONTINUE
ELSE IF( BETA.NE.ONE )THEN
DO 150, I = J, N
C( I, J ) = BETA*C( I, J )
150 CONTINUE
END IF
DO 170, L = 1, K
IF( ( A( J, L ).NE.ZERO ).OR.
$ ( B( J, L ).NE.ZERO ) )THEN
TEMP1 = ALPHA*B( J, L )
TEMP2 = ALPHA*A( J, L )
DO 160, I = J, N
C( I, J ) = C( I, J ) +
$ A( I, L )*TEMP1 + B( I, L )*TEMP2
160 CONTINUE
END IF
170 CONTINUE
180 CONTINUE
END IF
ELSE
*
* Form C := alpha*A'*B + alpha*B'*A + C.
*
IF( UPPER )THEN
DO 210, J = 1, N
DO 200, I = 1, J
TEMP1 = ZERO
TEMP2 = ZERO
DO 190, L = 1, K
TEMP1 = TEMP1 + A( L, I )*B( L, J )
TEMP2 = TEMP2 + B( L, I )*A( L, J )
190 CONTINUE
IF( BETA.EQ.ZERO )THEN
C( I, J ) = ALPHA*TEMP1 + ALPHA*TEMP2
ELSE
C( I, J ) = BETA *C( I, J ) +
$ ALPHA*TEMP1 + ALPHA*TEMP2
END IF
200 CONTINUE
210 CONTINUE
ELSE
DO 240, J = 1, N
DO 230, I = J, N
TEMP1 = ZERO
TEMP2 = ZERO
DO 220, L = 1, K
TEMP1 = TEMP1 + A( L, I )*B( L, J )
TEMP2 = TEMP2 + B( L, I )*A( L, J )
220 CONTINUE
IF( BETA.EQ.ZERO )THEN
C( I, J ) = ALPHA*TEMP1 + ALPHA*TEMP2
ELSE
C( I, J ) = BETA *C( I, J ) +
$ ALPHA*TEMP1 + ALPHA*TEMP2
END IF
230 CONTINUE
240 CONTINUE
END IF
END IF
*
RETURN
*
* End of SSYR2K.
*
END
SUBROUTINE SSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), TAU( * )
* ..
*
* Purpose
* =======
*
* SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
* form T by an orthogonal similarity transformation: Q' * A * Q = T.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* symmetric matrix A is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* n-by-n upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading n-by-n lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
* On exit, if UPLO = 'U', the diagonal and first superdiagonal
* of A are overwritten by the corresponding elements of the
* tridiagonal matrix T, and the elements above the first
* superdiagonal, with the array TAU, represent the orthogonal
* matrix Q as a product of elementary reflectors; if UPLO
* = 'L', the diagonal and first subdiagonal of A are over-
* written by the corresponding elements of the tridiagonal
* matrix T, and the elements below the first subdiagonal, with
* the array TAU, represent the orthogonal matrix Q as a product
* of elementary reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* D (output) REAL array, dimension (N)
* The diagonal elements of the tridiagonal matrix T:
* D(i) = A(i,i).
*
* E (output) REAL array, dimension (N-1)
* The off-diagonal elements of the tridiagonal matrix T:
* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*
* TAU (output) REAL array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* If UPLO = 'U', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(n-1) . . . H(2) H(1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
* A(1:i-1,i+1), and tau in TAU(i).
*
* If UPLO = 'L', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(1) H(2) . . . H(n-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
* and tau in TAU(i).
*
* The contents of A on exit are illustrated by the following examples
* with n = 5:
*
* if UPLO = 'U': if UPLO = 'L':
*
* ( d e v2 v3 v4 ) ( d )
* ( d e v3 v4 ) ( e d )
* ( d e v4 ) ( v1 e d )
* ( d e ) ( v1 v2 e d )
* ( d ) ( v1 v2 v3 e d )
*
* where d and e denote diagonal and off-diagonal elements of T, and vi
* denotes an element of the vector defining H(i).
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO, HALF
PARAMETER ( ONE = 1.0, ZERO = 0.0, HALF = 1.0 / 2.0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I
REAL ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SLARFG, SSYMV, SSYR2, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SDOT
EXTERNAL LSAME, SDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSYTD2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A
*
DO 10 I = N - 1, 1, -1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(1:i-1,i+1)
*
CALL SLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
E( I ) = A( I, I+1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(1:i,1:i)
*
A( I, I+1 ) = ONE
*
* Compute x := tau * A * v storing x in TAU(1:i)
*
CALL SSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
$ TAU, 1 )
*
* Compute w := x - 1/2 * tau * (x'*v) * v
*
ALPHA = -HALF*TAUI*SDOT( I, TAU, 1, A( 1, I+1 ), 1 )
CALL SAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL SSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
$ LDA )
*
A( I, I+1 ) = E( I )
END IF
D( I+1 ) = A( I+1, I+1 )
TAU( I ) = TAUI
10 CONTINUE
D( 1 ) = A( 1, 1 )
ELSE
*
* Reduce the lower triangle of A
*
DO 20 I = 1, N - 1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(i+2:n,i)
*
CALL SLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAUI )
E( I ) = A( I+1, I )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(i+1:n,i+1:n)
*
A( I+1, I ) = ONE
*
* Compute x := tau * A * v storing y in TAU(i:n-1)
*
CALL SSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
*
* Compute w := x - 1/2 * tau * (x'*v) * v
*
ALPHA = -HALF*TAUI*SDOT( N-I, TAU( I ), 1, A( I+1, I ),
$ 1 )
CALL SAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL SSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
$ A( I+1, I+1 ), LDA )
*
A( I+1, I ) = E( I )
END IF
D( I ) = A( I, I )
TAU( I ) = TAUI
20 CONTINUE
D( N ) = A( N, N )
END IF
*
RETURN
*
* End of SSYTD2
*
END
SUBROUTINE SSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), TAU( * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* SSYTRD reduces a real symmetric matrix A to real symmetric
* tridiagonal form T by an orthogonal similarity transformation:
* Q**T * A * Q = T.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A. If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
* On exit, if UPLO = 'U', the diagonal and first superdiagonal
* of A are overwritten by the corresponding elements of the
* tridiagonal matrix T, and the elements above the first
* superdiagonal, with the array TAU, represent the orthogonal
* matrix Q as a product of elementary reflectors; if UPLO
* = 'L', the diagonal and first subdiagonal of A are over-
* written by the corresponding elements of the tridiagonal
* matrix T, and the elements below the first subdiagonal, with
* the array TAU, represent the orthogonal matrix Q as a product
* of elementary reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* D (output) REAL array, dimension (N)
* The diagonal elements of the tridiagonal matrix T:
* D(i) = A(i,i).
*
* E (output) REAL array, dimension (N-1)
* The off-diagonal elements of the tridiagonal matrix T:
* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*
* TAU (output) REAL array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) REAL array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 1.
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* If UPLO = 'U', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(n-1) . . . H(2) H(1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
* A(1:i-1,i+1), and tau in TAU(i).
*
* If UPLO = 'L', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(1) H(2) . . . H(n-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
* and tau in TAU(i).
*
* The contents of A on exit are illustrated by the following examples
* with n = 5:
*
* if UPLO = 'U': if UPLO = 'L':
*
* ( d e v2 v3 v4 ) ( d )
* ( d e v3 v4 ) ( e d )
* ( d e v4 ) ( v1 e d )
* ( d e ) ( v1 v2 e d )
* ( d ) ( v1 v2 v3 e d )
*
* where d and e denote diagonal and off-diagonal elements of T, and vi
* denotes an element of the vector defining H(i).
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IINFO, IWS, J, KK, LDWORK, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL SLATRD, SSYR2K, SSYTD2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSYTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
* Determine the block size.
*
NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
NX = N
IWS = 1
IF( NB.GT.1 .AND. NB.LT.N ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code).
*
NX = MAX( NB, ILAENV( 3, 'SSYTRD', UPLO, N, -1, -1, -1 ) )
IF( NX.LT.N ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code by setting NX = N.
*
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = ILAENV( 2, 'SSYTRD', UPLO, N, -1, -1, -1 )
IF( NB.LT.NBMIN )
$ NX = N
END IF
ELSE
NX = N
END IF
ELSE
NB = 1
END IF
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* Columns 1:kk are handled by the unblocked method.
*
KK = N - ( ( N-NX+NB-1 ) / NB )*NB
DO 20 I = N - NB + 1, KK + 1, -NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL SLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
$ LDWORK )
*
* Update the unreduced submatrix A(1:i-1,1:i-1), using an
* update of the form: A := A - V*W' - W*V'
*
CALL SSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
$ LDA, WORK, LDWORK, ONE, A, LDA )
*
* Copy superdiagonal elements back into A, and diagonal
* elements into D
*
DO 10 J = I, I + NB - 1
A( J-1, J ) = E( J-1 )
D( J ) = A( J, J )
10 CONTINUE
20 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL SSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
ELSE
*
* Reduce the lower triangle of A
*
DO 40 I = 1, N - NX, NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL SLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
$ TAU( I ), WORK, LDWORK )
*
* Update the unreduced submatrix A(i+ib:n,i+ib:n), using
* an update of the form: A := A - V*W' - W*V'
*
CALL SSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
$ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
$ A( I+NB, I+NB ), LDA )
*
* Copy subdiagonal elements back into A, and diagonal
* elements into D
*
DO 30 J = I, I + NB - 1
A( J+1, J ) = E( J )
D( J ) = A( J, J )
30 CONTINUE
40 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL SSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAU( I ), IINFO )
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of SSYTRD
*
END
SUBROUTINE STRMM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA,
$ B, LDB )
* .. Scalar Arguments ..
CHARACTER*1 SIDE, UPLO, TRANSA, DIAG
INTEGER M, N, LDA, LDB
REAL ALPHA
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* STRMM performs one of the matrix-matrix operations
*
* B := alpha*op( A )*B, or B := alpha*B*op( A ),
*
* where alpha is a scalar, B is an m by n matrix, A is a unit, or
* non-unit, upper or lower triangular matrix and op( A ) is one of
*
* op( A ) = A or op( A ) = A'.
*
* Parameters
* ==========
*
* SIDE - CHARACTER*1.
* On entry, SIDE specifies whether op( A ) multiplies B from
* the left or right as follows:
*
* SIDE = 'L' or 'l' B := alpha*op( A )*B.
*
* SIDE = 'R' or 'r' B := alpha*B*op( A ).
*
* Unchanged on exit.
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the matrix A is an upper or
* lower triangular matrix as follows:
*
* UPLO = 'U' or 'u' A is an upper triangular matrix.
*
* UPLO = 'L' or 'l' A is a lower triangular matrix.
*
* Unchanged on exit.
*
* TRANSA - CHARACTER*1.
* On entry, TRANSA specifies the form of op( A ) to be used in
* the matrix multiplication as follows:
*
* TRANSA = 'N' or 'n' op( A ) = A.
*
* TRANSA = 'T' or 't' op( A ) = A'.
*
* TRANSA = 'C' or 'c' op( A ) = A'.
*
* Unchanged on exit.
*
* DIAG - CHARACTER*1.
* On entry, DIAG specifies whether or not A is unit triangular
* as follows:
*
* DIAG = 'U' or 'u' A is assumed to be unit triangular.
*
* DIAG = 'N' or 'n' A is not assumed to be unit
* triangular.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of B. M must be at
* least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of B. N must be
* at least zero.
* Unchanged on exit.
*
* ALPHA - REAL .
* On entry, ALPHA specifies the scalar alpha. When alpha is
* zero then A is not referenced and B need not be set before
* entry.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, k ), where k is m
* when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
* Before entry with UPLO = 'U' or 'u', the leading k by k
* upper triangular part of the array A must contain the upper
* triangular matrix and the strictly lower triangular part of
* A is not referenced.
* Before entry with UPLO = 'L' or 'l', the leading k by k
* lower triangular part of the array A must contain the lower
* triangular matrix and the strictly upper triangular part of
* A is not referenced.
* Note that when DIAG = 'U' or 'u', the diagonal elements of
* A are not referenced either, but are assumed to be unity.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When SIDE = 'L' or 'l' then
* LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
* then LDA must be at least max( 1, n ).
* Unchanged on exit.
*
* B - REAL array of DIMENSION ( LDB, n ).
* Before entry, the leading m by n part of the array B must
* contain the matrix B, and on exit is overwritten by the
* transformed matrix.
*
* LDB - INTEGER.
* On entry, LDB specifies the first dimension of B as declared
* in the calling (sub) program. LDB must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Local Scalars ..
LOGICAL LSIDE, NOUNIT, UPPER
INTEGER I, INFO, J, K, NROWA
REAL TEMP
* .. Parameters ..
REAL ONE , ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
LSIDE = LSAME( SIDE , 'L' )
IF( LSIDE )THEN
NROWA = M
ELSE
NROWA = N
END IF
NOUNIT = LSAME( DIAG , 'N' )
UPPER = LSAME( UPLO , 'U' )
*
INFO = 0
IF( ( .NOT.LSIDE ).AND.
$ ( .NOT.LSAME( SIDE , 'R' ) ) )THEN
INFO = 1
ELSE IF( ( .NOT.UPPER ).AND.
$ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN
INFO = 2
ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND.
$ ( .NOT.LSAME( TRANSA, 'T' ) ).AND.
$ ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN
INFO = 3
ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND.
$ ( .NOT.LSAME( DIAG , 'N' ) ) )THEN
INFO = 4
ELSE IF( M .LT.0 )THEN
INFO = 5
ELSE IF( N .LT.0 )THEN
INFO = 6
ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
INFO = 9
ELSE IF( LDB.LT.MAX( 1, M ) )THEN
INFO = 11
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'STRMM ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* And when alpha.eq.zero.
*
IF( ALPHA.EQ.ZERO )THEN
DO 20, J = 1, N
DO 10, I = 1, M
B( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
RETURN
END IF
*
* Start the operations.
*
IF( LSIDE )THEN
IF( LSAME( TRANSA, 'N' ) )THEN
*
* Form B := alpha*A*B.
*
IF( UPPER )THEN
DO 50, J = 1, N
DO 40, K = 1, M
IF( B( K, J ).NE.ZERO )THEN
TEMP = ALPHA*B( K, J )
DO 30, I = 1, K - 1
B( I, J ) = B( I, J ) + TEMP*A( I, K )
30 CONTINUE
IF( NOUNIT )
$ TEMP = TEMP*A( K, K )
B( K, J ) = TEMP
END IF
40 CONTINUE
50 CONTINUE
ELSE
DO 80, J = 1, N
DO 70 K = M, 1, -1
IF( B( K, J ).NE.ZERO )THEN
TEMP = ALPHA*B( K, J )
B( K, J ) = TEMP
IF( NOUNIT )
$ B( K, J ) = B( K, J )*A( K, K )
DO 60, I = K + 1, M
B( I, J ) = B( I, J ) + TEMP*A( I, K )
60 CONTINUE
END IF
70 CONTINUE
80 CONTINUE
END IF
ELSE
*
* Form B := alpha*B*A'.
*
IF( UPPER )THEN
DO 110, J = 1, N
DO 100, I = M, 1, -1
TEMP = B( I, J )
IF( NOUNIT )
$ TEMP = TEMP*A( I, I )
DO 90, K = 1, I - 1
TEMP = TEMP + A( K, I )*B( K, J )
90 CONTINUE
B( I, J ) = ALPHA*TEMP
100 CONTINUE
110 CONTINUE
ELSE
DO 140, J = 1, N
DO 130, I = 1, M
TEMP = B( I, J )
IF( NOUNIT )
$ TEMP = TEMP*A( I, I )
DO 120, K = I + 1, M
TEMP = TEMP + A( K, I )*B( K, J )
120 CONTINUE
B( I, J ) = ALPHA*TEMP
130 CONTINUE
140 CONTINUE
END IF
END IF
ELSE
IF( LSAME( TRANSA, 'N' ) )THEN
*
* Form B := alpha*B*A.
*
IF( UPPER )THEN
DO 180, J = N, 1, -1
TEMP = ALPHA
IF( NOUNIT )
$ TEMP = TEMP*A( J, J )
DO 150, I = 1, M
B( I, J ) = TEMP*B( I, J )
150 CONTINUE
DO 170, K = 1, J - 1
IF( A( K, J ).NE.ZERO )THEN
TEMP = ALPHA*A( K, J )
DO 160, I = 1, M
B( I, J ) = B( I, J ) + TEMP*B( I, K )
160 CONTINUE
END IF
170 CONTINUE
180 CONTINUE
ELSE
DO 220, J = 1, N
TEMP = ALPHA
IF( NOUNIT )
$ TEMP = TEMP*A( J, J )
DO 190, I = 1, M
B( I, J ) = TEMP*B( I, J )
190 CONTINUE
DO 210, K = J + 1, N
IF( A( K, J ).NE.ZERO )THEN
TEMP = ALPHA*A( K, J )
DO 200, I = 1, M
B( I, J ) = B( I, J ) + TEMP*B( I, K )
200 CONTINUE
END IF
210 CONTINUE
220 CONTINUE
END IF
ELSE
*
* Form B := alpha*B*A'.
*
IF( UPPER )THEN
DO 260, K = 1, N
DO 240, J = 1, K - 1
IF( A( J, K ).NE.ZERO )THEN
TEMP = ALPHA*A( J, K )
DO 230, I = 1, M
B( I, J ) = B( I, J ) + TEMP*B( I, K )
230 CONTINUE
END IF
240 CONTINUE
TEMP = ALPHA
IF( NOUNIT )
$ TEMP = TEMP*A( K, K )
IF( TEMP.NE.ONE )THEN
DO 250, I = 1, M
B( I, K ) = TEMP*B( I, K )
250 CONTINUE
END IF
260 CONTINUE
ELSE
DO 300, K = N, 1, -1
DO 280, J = K + 1, N
IF( A( J, K ).NE.ZERO )THEN
TEMP = ALPHA*A( J, K )
DO 270, I = 1, M
B( I, J ) = B( I, J ) + TEMP*B( I, K )
270 CONTINUE
END IF
280 CONTINUE
TEMP = ALPHA
IF( NOUNIT )
$ TEMP = TEMP*A( K, K )
IF( TEMP.NE.ONE )THEN
DO 290, I = 1, M
B( I, K ) = TEMP*B( I, K )
290 CONTINUE
END IF
300 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of STRMM .
*
END
SUBROUTINE STRMV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
* .. Scalar Arguments ..
INTEGER INCX, LDA, N
CHARACTER*1 DIAG, TRANS, UPLO
* .. Array Arguments ..
REAL A( LDA, * ), X( * )
* ..
*
* Purpose
* =======
*
* STRMV performs one of the matrix-vector operations
*
* x := A*x, or x := A'*x,
*
* where x is an n element vector and A is an n by n unit, or non-unit,
* upper or lower triangular matrix.
*
* Parameters
* ==========
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the matrix is an upper or
* lower triangular matrix as follows:
*
* UPLO = 'U' or 'u' A is an upper triangular matrix.
*
* UPLO = 'L' or 'l' A is a lower triangular matrix.
*
* Unchanged on exit.
*
* TRANS - CHARACTER*1.
* On entry, TRANS specifies the operation to be performed as
* follows:
*
* TRANS = 'N' or 'n' x := A*x.
*
* TRANS = 'T' or 't' x := A'*x.
*
* TRANS = 'C' or 'c' x := A'*x.
*
* Unchanged on exit.
*
* DIAG - CHARACTER*1.
* On entry, DIAG specifies whether or not A is unit
* triangular as follows:
*
* DIAG = 'U' or 'u' A is assumed to be unit triangular.
*
* DIAG = 'N' or 'n' A is not assumed to be unit
* triangular.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* A - REAL array of DIMENSION ( LDA, n ).
* Before entry with UPLO = 'U' or 'u', the leading n by n
* upper triangular part of the array A must contain the upper
* triangular matrix and the strictly lower triangular part of
* A is not referenced.
* Before entry with UPLO = 'L' or 'l', the leading n by n
* lower triangular part of the array A must contain the lower
* triangular matrix and the strictly upper triangular part of
* A is not referenced.
* Note that when DIAG = 'U' or 'u', the diagonal elements of
* A are not referenced either, but are assumed to be unity.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, n ).
* Unchanged on exit.
*
* X - REAL array of dimension at least
* ( 1 + ( n - 1 )*abs( INCX ) ).
* Before entry, the incremented array X must contain the n
* element vector x. On exit, X is overwritten with the
* tranformed vector x.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* .. Local Scalars ..
REAL TEMP
INTEGER I, INFO, IX, J, JX, KX
LOGICAL NOUNIT
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL XERBLA
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF ( .NOT.LSAME( UPLO , 'U' ).AND.
$ .NOT.LSAME( UPLO , 'L' ) )THEN
INFO = 1
ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND.
$ .NOT.LSAME( TRANS, 'T' ).AND.
$ .NOT.LSAME( TRANS, 'C' ) )THEN
INFO = 2
ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND.
$ .NOT.LSAME( DIAG , 'N' ) )THEN
INFO = 3
ELSE IF( N.LT.0 )THEN
INFO = 4
ELSE IF( LDA.LT.MAX( 1, N ) )THEN
INFO = 6
ELSE IF( INCX.EQ.0 )THEN
INFO = 8
END IF
IF( INFO.NE.0 )THEN
CALL XERBLA( 'STRMV ', INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
NOUNIT = LSAME( DIAG, 'N' )
*
* Set up the start point in X if the increment is not unity. This
* will be ( N - 1 )*INCX too small for descending loops.
*
IF( INCX.LE.0 )THEN
KX = 1 - ( N - 1 )*INCX
ELSE IF( INCX.NE.1 )THEN
KX = 1
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through A.
*
IF( LSAME( TRANS, 'N' ) )THEN
*
* Form x := A*x.
*
IF( LSAME( UPLO, 'U' ) )THEN
IF( INCX.EQ.1 )THEN
DO 20, J = 1, N
IF( X( J ).NE.ZERO )THEN
TEMP = X( J )
DO 10, I = 1, J - 1
X( I ) = X( I ) + TEMP*A( I, J )
10 CONTINUE
IF( NOUNIT )
$ X( J ) = X( J )*A( J, J )
END IF
20 CONTINUE
ELSE
JX = KX
DO 40, J = 1, N
IF( X( JX ).NE.ZERO )THEN
TEMP = X( JX )
IX = KX
DO 30, I = 1, J - 1
X( IX ) = X( IX ) + TEMP*A( I, J )
IX = IX + INCX
30 CONTINUE
IF( NOUNIT )
$ X( JX ) = X( JX )*A( J, J )
END IF
JX = JX + INCX
40 CONTINUE
END IF
ELSE
IF( INCX.EQ.1 )THEN
DO 60, J = N, 1, -1
IF( X( J ).NE.ZERO )THEN
TEMP = X( J )
DO 50, I = N, J + 1, -1
X( I ) = X( I ) + TEMP*A( I, J )
50 CONTINUE
IF( NOUNIT )
$ X( J ) = X( J )*A( J, J )
END IF
60 CONTINUE
ELSE
KX = KX + ( N - 1 )*INCX
JX = KX
DO 80, J = N, 1, -1
IF( X( JX ).NE.ZERO )THEN
TEMP = X( JX )
IX = KX
DO 70, I = N, J + 1, -1
X( IX ) = X( IX ) + TEMP*A( I, J )
IX = IX - INCX
70 CONTINUE
IF( NOUNIT )
$ X( JX ) = X( JX )*A( J, J )
END IF
JX = JX - INCX
80 CONTINUE
END IF
END IF
ELSE
*
* Form x := A'*x.
*
IF( LSAME( UPLO, 'U' ) )THEN
IF( INCX.EQ.1 )THEN
DO 100, J = N, 1, -1
TEMP = X( J )
IF( NOUNIT )
$ TEMP = TEMP*A( J, J )
DO 90, I = J - 1, 1, -1
TEMP = TEMP + A( I, J )*X( I )
90 CONTINUE
X( J ) = TEMP
100 CONTINUE
ELSE
JX = KX + ( N - 1 )*INCX
DO 120, J = N, 1, -1
TEMP = X( JX )
IX = JX
IF( NOUNIT )
$ TEMP = TEMP*A( J, J )
DO 110, I = J - 1, 1, -1
IX = IX - INCX
TEMP = TEMP + A( I, J )*X( IX )
110 CONTINUE
X( JX ) = TEMP
JX = JX - INCX
120 CONTINUE
END IF
ELSE
IF( INCX.EQ.1 )THEN
DO 140, J = 1, N
TEMP = X( J )
IF( NOUNIT )
$ TEMP = TEMP*A( J, J )
DO 130, I = J + 1, N
TEMP = TEMP + A( I, J )*X( I )
130 CONTINUE
X( J ) = TEMP
140 CONTINUE
ELSE
JX = KX
DO 160, J = 1, N
TEMP = X( JX )
IX = JX
IF( NOUNIT )
$ TEMP = TEMP*A( J, J )
DO 150, I = J + 1, N
IX = IX + INCX
TEMP = TEMP + A( I, J )*X( IX )
150 CONTINUE
X( JX ) = TEMP
JX = JX + INCX
160 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of STRMV .
*
END
SUBROUTINE XERBLA( SRNAME, INFO )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER*6 SRNAME
INTEGER INFO
* ..
*
* Purpose
* =======
*
* XERBLA is an error handler for the LAPACK routines.
* It is called by an LAPACK routine if an input parameter has an
* invalid value. A message is printed and execution stops.
*
* Installers may consider modifying the STOP statement in order to
* call system-specific exception-handling facilities.
*
* Arguments
* =========
*
* SRNAME (input) CHARACTER*6
* The name of the routine which called XERBLA.
*
* INFO (input) INTEGER
* The position of the invalid parameter in the parameter list
* of the calling routine.
*
* =====================================================================
*
* .. Executable Statements ..
*
WRITE( *, FMT = 9999 )SRNAME, INFO
*
STOP
*
9999 FORMAT( ' ** On entry to ', A6, ' parameter number ', I2, ' had ',
$ 'an illegal value' )
*
* End of XERBLA
*
END