LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ sgges3()

subroutine sgges3 ( character  JOBVSL,
character  JOBVSR,
character  SORT,
external  SELCTG,
integer  N,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldb, * )  B,
integer  LDB,
integer  SDIM,
real, dimension( * )  ALPHAR,
real, dimension( * )  ALPHAI,
real, dimension( * )  BETA,
real, dimension( ldvsl, * )  VSL,
integer  LDVSL,
real, dimension( ldvsr, * )  VSR,
integer  LDVSR,
real, dimension( * )  WORK,
integer  LWORK,
logical, dimension( * )  BWORK,
integer  INFO 
)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Download SGGES3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
 the generalized eigenvalues, the generalized real Schur form (S,T),
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR). This gives the generalized Schur factorization

          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T.The
 leading columns of VSL and VSR then form an orthonormal basis for the
 corresponding left and right eigenspaces (deflating subspaces).

 (If only the generalized eigenvalues are needed, use the driver
 SGGEV instead, which is faster.)

 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or both being zero.

 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 "standardized" by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]

 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.
Parameters
[in]JOBVSL
          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.
[in]JOBVSR
          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.
[in]SORT
          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);
[in]SELCTG
          SELCTG is a LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.

          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.
[in]N
          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.
[in,out]A
          A is REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.
[in]LDA
          LDA is INTEGER
          The leading dimension of A.  LDA >= max(1,N).
[in,out]B
          B is REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.
[in]LDB
          LDB is INTEGER
          The leading dimension of B.  LDB >= max(1,N).
[out]SDIM
          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)
[out]ALPHAR
          ALPHAR is REAL array, dimension (N)
[out]ALPHAI
          ALPHAI is REAL array, dimension (N)
[out]BETA
          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).
[out]VSL
          VSL is REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.
[in]LDVSL
          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.
[out]VSR
          VSR is REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.
[in]LDVSR
          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.
[out]WORK
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[out]BWORK
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in SLAQZ0.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 279 of file sgges3.f.

282 *
283 * -- LAPACK driver routine --
284 * -- LAPACK is a software package provided by Univ. of Tennessee, --
285 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
286 *
287 * .. Scalar Arguments ..
288  CHARACTER JOBVSL, JOBVSR, SORT
289  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
290 * ..
291 * .. Array Arguments ..
292  LOGICAL BWORK( * )
293  REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
294  $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
295  $ VSR( LDVSR, * ), WORK( * )
296 * ..
297 * .. Function Arguments ..
298  LOGICAL SELCTG
299  EXTERNAL selctg
300 * ..
301 *
302 * =====================================================================
303 *
304 * .. Parameters ..
305  REAL ZERO, ONE
306  parameter( zero = 0.0e+0, one = 1.0e+0 )
307 * ..
308 * .. Local Scalars ..
309  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
310  $ LQUERY, LST2SL, WANTST
311  INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
312  $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, LWKOPT
313  REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
314  $ PVSR, SAFMAX, SAFMIN, SMLNUM
315 * ..
316 * .. Local Arrays ..
317  INTEGER IDUM( 1 )
318  REAL DIF( 2 )
319 * ..
320 * .. External Subroutines ..
321  EXTERNAL sgeqrf, sggbak, sggbal, sgghd3, slaqz0, slabad,
323  $ xerbla
324 * ..
325 * .. External Functions ..
326  LOGICAL LSAME
327  REAL SLAMCH, SLANGE
328  EXTERNAL lsame, slamch, slange
329 * ..
330 * .. Intrinsic Functions ..
331  INTRINSIC abs, max, sqrt
332 * ..
333 * .. Executable Statements ..
334 *
335 * Decode the input arguments
336 *
337  IF( lsame( jobvsl, 'N' ) ) THEN
338  ijobvl = 1
339  ilvsl = .false.
340  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
341  ijobvl = 2
342  ilvsl = .true.
343  ELSE
344  ijobvl = -1
345  ilvsl = .false.
346  END IF
347 *
348  IF( lsame( jobvsr, 'N' ) ) THEN
349  ijobvr = 1
350  ilvsr = .false.
351  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
352  ijobvr = 2
353  ilvsr = .true.
354  ELSE
355  ijobvr = -1
356  ilvsr = .false.
357  END IF
358 *
359  wantst = lsame( sort, 'S' )
360 *
361 * Test the input arguments
362 *
363  info = 0
364  lquery = ( lwork.EQ.-1 )
365  IF( ijobvl.LE.0 ) THEN
366  info = -1
367  ELSE IF( ijobvr.LE.0 ) THEN
368  info = -2
369  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
370  info = -3
371  ELSE IF( n.LT.0 ) THEN
372  info = -5
373  ELSE IF( lda.LT.max( 1, n ) ) THEN
374  info = -7
375  ELSE IF( ldb.LT.max( 1, n ) ) THEN
376  info = -9
377  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
378  info = -15
379  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
380  info = -17
381  ELSE IF( lwork.LT.6*n+16 .AND. .NOT.lquery ) THEN
382  info = -19
383  END IF
384 *
385 * Compute workspace
386 *
387  IF( info.EQ.0 ) THEN
388  CALL sgeqrf( n, n, b, ldb, work, work, -1, ierr )
389  lwkopt = max( 6*n+16, 3*n+int( work( 1 ) ) )
390  CALL sormqr( 'L', 'T', n, n, n, b, ldb, work, a, lda, work,
391  $ -1, ierr )
392  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
393  IF( ilvsl ) THEN
394  CALL sorgqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
395  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
396  END IF
397  CALL sgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
398  $ ldvsl, vsr, ldvsr, work, -1, ierr )
399  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
400  CALL slaqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
401  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
402  $ work, -1, 0, ierr )
403  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
404  IF( wantst ) THEN
405  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
406  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
407  $ sdim, pvsl, pvsr, dif, work, -1, idum, 1,
408  $ ierr )
409  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
410  END IF
411  work( 1 ) = lwkopt
412  END IF
413 *
414  IF( info.NE.0 ) THEN
415  CALL xerbla( 'SGGES3 ', -info )
416  RETURN
417  ELSE IF( lquery ) THEN
418  RETURN
419  END IF
420 *
421 * Quick return if possible
422 *
423  IF( n.EQ.0 ) THEN
424  sdim = 0
425  RETURN
426  END IF
427 *
428 * Get machine constants
429 *
430  eps = slamch( 'P' )
431  safmin = slamch( 'S' )
432  safmax = one / safmin
433  CALL slabad( safmin, safmax )
434  smlnum = sqrt( safmin ) / eps
435  bignum = one / smlnum
436 *
437 * Scale A if max element outside range [SMLNUM,BIGNUM]
438 *
439  anrm = slange( 'M', n, n, a, lda, work )
440  ilascl = .false.
441  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
442  anrmto = smlnum
443  ilascl = .true.
444  ELSE IF( anrm.GT.bignum ) THEN
445  anrmto = bignum
446  ilascl = .true.
447  END IF
448  IF( ilascl )
449  $ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
450 *
451 * Scale B if max element outside range [SMLNUM,BIGNUM]
452 *
453  bnrm = slange( 'M', n, n, b, ldb, work )
454  ilbscl = .false.
455  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
456  bnrmto = smlnum
457  ilbscl = .true.
458  ELSE IF( bnrm.GT.bignum ) THEN
459  bnrmto = bignum
460  ilbscl = .true.
461  END IF
462  IF( ilbscl )
463  $ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
464 *
465 * Permute the matrix to make it more nearly triangular
466 *
467  ileft = 1
468  iright = n + 1
469  iwrk = iright + n
470  CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
471  $ work( iright ), work( iwrk ), ierr )
472 *
473 * Reduce B to triangular form (QR decomposition of B)
474 *
475  irows = ihi + 1 - ilo
476  icols = n + 1 - ilo
477  itau = iwrk
478  iwrk = itau + irows
479  CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
480  $ work( iwrk ), lwork+1-iwrk, ierr )
481 *
482 * Apply the orthogonal transformation to matrix A
483 *
484  CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
485  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
486  $ lwork+1-iwrk, ierr )
487 *
488 * Initialize VSL
489 *
490  IF( ilvsl ) THEN
491  CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
492  IF( irows.GT.1 ) THEN
493  CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
494  $ vsl( ilo+1, ilo ), ldvsl )
495  END IF
496  CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
497  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
498  END IF
499 *
500 * Initialize VSR
501 *
502  IF( ilvsr )
503  $ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
504 *
505 * Reduce to generalized Hessenberg form
506 *
507  CALL sgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
508  $ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
509 *
510 * Perform QZ algorithm, computing Schur vectors if desired
511 *
512  iwrk = itau
513  CALL slaqz0( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
514  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
515  $ work( iwrk ), lwork+1-iwrk, 0, ierr )
516  IF( ierr.NE.0 ) THEN
517  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
518  info = ierr
519  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
520  info = ierr - n
521  ELSE
522  info = n + 1
523  END IF
524  GO TO 40
525  END IF
526 *
527 * Sort eigenvalues ALPHA/BETA if desired
528 *
529  sdim = 0
530  IF( wantst ) THEN
531 *
532 * Undo scaling on eigenvalues before SELCTGing
533 *
534  IF( ilascl ) THEN
535  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
536  $ ierr )
537  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
538  $ ierr )
539  END IF
540  IF( ilbscl )
541  $ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
542 *
543 * Select eigenvalues
544 *
545  DO 10 i = 1, n
546  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
547  10 CONTINUE
548 *
549  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
550  $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
551  $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
552  $ ierr )
553  IF( ierr.EQ.1 )
554  $ info = n + 3
555 *
556  END IF
557 *
558 * Apply back-permutation to VSL and VSR
559 *
560  IF( ilvsl )
561  $ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
562  $ work( iright ), n, vsl, ldvsl, ierr )
563 *
564  IF( ilvsr )
565  $ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
566  $ work( iright ), n, vsr, ldvsr, ierr )
567 *
568 * Check if unscaling would cause over/underflow, if so, rescale
569 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
570 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
571 *
572  IF( ilascl )THEN
573  DO 50 i = 1, n
574  IF( alphai( i ).NE.zero ) THEN
575  IF( ( alphar( i )/safmax ).GT.( anrmto/anrm ) .OR.
576  $ ( safmin/alphar( i ) ).GT.( anrm/anrmto ) ) THEN
577  work( 1 ) = abs( a( i, i )/alphar( i ) )
578  beta( i ) = beta( i )*work( 1 )
579  alphar( i ) = alphar( i )*work( 1 )
580  alphai( i ) = alphai( i )*work( 1 )
581  ELSE IF( ( alphai( i )/safmax ).GT.( anrmto/anrm ) .OR.
582  $ ( safmin/alphai( i ) ).GT.( anrm/anrmto ) ) THEN
583  work( 1 ) = abs( a( i, i+1 )/alphai( i ) )
584  beta( i ) = beta( i )*work( 1 )
585  alphar( i ) = alphar( i )*work( 1 )
586  alphai( i ) = alphai( i )*work( 1 )
587  END IF
588  END IF
589  50 CONTINUE
590  END IF
591 *
592  IF( ilbscl )THEN
593  DO 60 i = 1, n
594  IF( alphai( i ).NE.zero ) THEN
595  IF( ( beta( i )/safmax ).GT.( bnrmto/bnrm ) .OR.
596  $ ( safmin/beta( i ) ).GT.( bnrm/bnrmto ) ) THEN
597  work( 1 ) = abs(b( i, i )/beta( i ))
598  beta( i ) = beta( i )*work( 1 )
599  alphar( i ) = alphar( i )*work( 1 )
600  alphai( i ) = alphai( i )*work( 1 )
601  END IF
602  END IF
603  60 CONTINUE
604  END IF
605 *
606 * Undo scaling
607 *
608  IF( ilascl ) THEN
609  CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
610  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
611  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
612  END IF
613 *
614  IF( ilbscl ) THEN
615  CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
616  CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
617  END IF
618 *
619  IF( wantst ) THEN
620 *
621 * Check if reordering is correct
622 *
623  lastsl = .true.
624  lst2sl = .true.
625  sdim = 0
626  ip = 0
627  DO 30 i = 1, n
628  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
629  IF( alphai( i ).EQ.zero ) THEN
630  IF( cursl )
631  $ sdim = sdim + 1
632  ip = 0
633  IF( cursl .AND. .NOT.lastsl )
634  $ info = n + 2
635  ELSE
636  IF( ip.EQ.1 ) THEN
637 *
638 * Last eigenvalue of conjugate pair
639 *
640  cursl = cursl .OR. lastsl
641  lastsl = cursl
642  IF( cursl )
643  $ sdim = sdim + 2
644  ip = -1
645  IF( cursl .AND. .NOT.lst2sl )
646  $ info = n + 2
647  ELSE
648 *
649 * First eigenvalue of conjugate pair
650 *
651  ip = 1
652  END IF
653  END IF
654  lst2sl = lastsl
655  lastsl = cursl
656  30 CONTINUE
657 *
658  END IF
659 *
660  40 CONTINUE
661 *
662  work( 1 ) = lwkopt
663 *
664  RETURN
665 *
666 * End of SGGES3
667 *
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:143
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
recursive subroutine slaqz0(WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
SLAQZ0
Definition: slaqz0.f:304
subroutine sggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
SGGBAK
Definition: sggbak.f:147
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:177
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:145
subroutine stgsen(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
STGSEN
Definition: stgsen.f:451
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine sgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SGGHD3
Definition: sgghd3.f:230
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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