LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine sgges ( 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 )

SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:
SGGES 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

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 N = 0, LWORK >= 1, else LWORK >= max(8*N,6*N+16). For good performance , LWORK must generally be larger. 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 SHGEQZ. =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.
Date
November 2011

Definition at line 286 of file sgges.f.

286 *
287 * -- LAPACK driver routine (version 3.4.0) --
288 * -- LAPACK is a software package provided by Univ. of Tennessee, --
289 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
290 * November 2011
291 *
292 * .. Scalar Arguments ..
293  CHARACTER jobvsl, jobvsr, sort
294  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n, sdim
295 * ..
296 * .. Array Arguments ..
297  LOGICAL bwork( * )
298  REAL a( lda, * ), alphai( * ), alphar( * ),
299  \$ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
300  \$ vsr( ldvsr, * ), work( * )
301 * ..
302 * .. Function Arguments ..
303  LOGICAL selctg
304  EXTERNAL selctg
305 * ..
306 *
307 * =====================================================================
308 *
309 * .. Parameters ..
310  REAL zero, one
311  parameter ( zero = 0.0e+0, one = 1.0e+0 )
312 * ..
313 * .. Local Scalars ..
314  LOGICAL cursl, ilascl, ilbscl, ilvsl, ilvsr, lastsl,
315  \$ lquery, lst2sl, wantst
316  INTEGER i, icols, ierr, ihi, ijobvl, ijobvr, ileft,
317  \$ ilo, ip, iright, irows, itau, iwrk, maxwrk,
318  \$ minwrk
319  REAL anrm, anrmto, bignum, bnrm, bnrmto, eps, pvsl,
320  \$ pvsr, safmax, safmin, smlnum
321 * ..
322 * .. Local Arrays ..
323  INTEGER idum( 1 )
324  REAL dif( 2 )
325 * ..
326 * .. External Subroutines ..
327  EXTERNAL sgeqrf, sggbak, sggbal, sgghrd, shgeqz, slabad,
329  \$ xerbla
330 * ..
331 * .. External Functions ..
332  LOGICAL lsame
333  INTEGER ilaenv
334  REAL slamch, slange
335  EXTERNAL lsame, ilaenv, slamch, slange
336 * ..
337 * .. Intrinsic Functions ..
338  INTRINSIC abs, max, sqrt
339 * ..
340 * .. Executable Statements ..
341 *
342 * Decode the input arguments
343 *
344  IF( lsame( jobvsl, 'N' ) ) THEN
345  ijobvl = 1
346  ilvsl = .false.
347  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
348  ijobvl = 2
349  ilvsl = .true.
350  ELSE
351  ijobvl = -1
352  ilvsl = .false.
353  END IF
354 *
355  IF( lsame( jobvsr, 'N' ) ) THEN
356  ijobvr = 1
357  ilvsr = .false.
358  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
359  ijobvr = 2
360  ilvsr = .true.
361  ELSE
362  ijobvr = -1
363  ilvsr = .false.
364  END IF
365 *
366  wantst = lsame( sort, 'S' )
367 *
368 * Test the input arguments
369 *
370  info = 0
371  lquery = ( lwork.EQ.-1 )
372  IF( ijobvl.LE.0 ) THEN
373  info = -1
374  ELSE IF( ijobvr.LE.0 ) THEN
375  info = -2
376  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
377  info = -3
378  ELSE IF( n.LT.0 ) THEN
379  info = -5
380  ELSE IF( lda.LT.max( 1, n ) ) THEN
381  info = -7
382  ELSE IF( ldb.LT.max( 1, n ) ) THEN
383  info = -9
384  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
385  info = -15
386  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
387  info = -17
388  END IF
389 *
390 * Compute workspace
391 * (Note: Comments in the code beginning "Workspace:" describe the
392 * minimal amount of workspace needed at that point in the code,
393 * as well as the preferred amount for good performance.
394 * NB refers to the optimal block size for the immediately
395 * following subroutine, as returned by ILAENV.)
396 *
397  IF( info.EQ.0 ) THEN
398  IF( n.GT.0 )THEN
399  minwrk = max( 8*n, 6*n + 16 )
400  maxwrk = minwrk - n +
401  \$ n*ilaenv( 1, 'SGEQRF', ' ', n, 1, n, 0 )
402  maxwrk = max( maxwrk, minwrk - n +
403  \$ n*ilaenv( 1, 'SORMQR', ' ', n, 1, n, -1 ) )
404  IF( ilvsl ) THEN
405  maxwrk = max( maxwrk, minwrk - n +
406  \$ n*ilaenv( 1, 'SORGQR', ' ', n, 1, n, -1 ) )
407  END IF
408  ELSE
409  minwrk = 1
410  maxwrk = 1
411  END IF
412  work( 1 ) = maxwrk
413 *
414  IF( lwork.LT.minwrk .AND. .NOT.lquery )
415  \$ info = -19
416  END IF
417 *
418  IF( info.NE.0 ) THEN
419  CALL xerbla( 'SGGES ', -info )
420  RETURN
421  ELSE IF( lquery ) THEN
422  RETURN
423  END IF
424 *
425 * Quick return if possible
426 *
427  IF( n.EQ.0 ) THEN
428  sdim = 0
429  RETURN
430  END IF
431 *
432 * Get machine constants
433 *
434  eps = slamch( 'P' )
435  safmin = slamch( 'S' )
436  safmax = one / safmin
437  CALL slabad( safmin, safmax )
438  smlnum = sqrt( safmin ) / eps
439  bignum = one / smlnum
440 *
441 * Scale A if max element outside range [SMLNUM,BIGNUM]
442 *
443  anrm = slange( 'M', n, n, a, lda, work )
444  ilascl = .false.
445  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
446  anrmto = smlnum
447  ilascl = .true.
448  ELSE IF( anrm.GT.bignum ) THEN
449  anrmto = bignum
450  ilascl = .true.
451  END IF
452  IF( ilascl )
453  \$ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
454 *
455 * Scale B if max element outside range [SMLNUM,BIGNUM]
456 *
457  bnrm = slange( 'M', n, n, b, ldb, work )
458  ilbscl = .false.
459  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
460  bnrmto = smlnum
461  ilbscl = .true.
462  ELSE IF( bnrm.GT.bignum ) THEN
463  bnrmto = bignum
464  ilbscl = .true.
465  END IF
466  IF( ilbscl )
467  \$ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
468 *
469 * Permute the matrix to make it more nearly triangular
470 * (Workspace: need 6*N + 2*N space for storing balancing factors)
471 *
472  ileft = 1
473  iright = n + 1
474  iwrk = iright + n
475  CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
476  \$ work( iright ), work( iwrk ), ierr )
477 *
478 * Reduce B to triangular form (QR decomposition of B)
479 * (Workspace: need N, prefer N*NB)
480 *
481  irows = ihi + 1 - ilo
482  icols = n + 1 - ilo
483  itau = iwrk
484  iwrk = itau + irows
485  CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
486  \$ work( iwrk ), lwork+1-iwrk, ierr )
487 *
488 * Apply the orthogonal transformation to matrix A
489 * (Workspace: need N, prefer N*NB)
490 *
491  CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
492  \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
493  \$ lwork+1-iwrk, ierr )
494 *
495 * Initialize VSL
496 * (Workspace: need N, prefer N*NB)
497 *
498  IF( ilvsl ) THEN
499  CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
500  IF( irows.GT.1 ) THEN
501  CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
502  \$ vsl( ilo+1, ilo ), ldvsl )
503  END IF
504  CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
505  \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
506  END IF
507 *
508 * Initialize VSR
509 *
510  IF( ilvsr )
511  \$ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
512 *
513 * Reduce to generalized Hessenberg form
514 * (Workspace: none needed)
515 *
516  CALL sgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
517  \$ ldvsl, vsr, ldvsr, ierr )
518 *
519 * Perform QZ algorithm, computing Schur vectors if desired
520 * (Workspace: need N)
521 *
522  iwrk = itau
523  CALL shgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
524  \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
525  \$ work( iwrk ), lwork+1-iwrk, ierr )
526  IF( ierr.NE.0 ) THEN
527  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
528  info = ierr
529  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
530  info = ierr - n
531  ELSE
532  info = n + 1
533  END IF
534  GO TO 40
535  END IF
536 *
537 * Sort eigenvalues ALPHA/BETA if desired
538 * (Workspace: need 4*N+16 )
539 *
540  sdim = 0
541  IF( wantst ) THEN
542 *
543 * Undo scaling on eigenvalues before SELCTGing
544 *
545  IF( ilascl ) THEN
546  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
547  \$ ierr )
548  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
549  \$ ierr )
550  END IF
551  IF( ilbscl )
552  \$ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
553 *
554 * Select eigenvalues
555 *
556  DO 10 i = 1, n
557  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
558  10 CONTINUE
559 *
560  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
561  \$ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
562  \$ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
563  \$ ierr )
564  IF( ierr.EQ.1 )
565  \$ info = n + 3
566 *
567  END IF
568 *
569 * Apply back-permutation to VSL and VSR
570 * (Workspace: none needed)
571 *
572  IF( ilvsl )
573  \$ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
574  \$ work( iright ), n, vsl, ldvsl, ierr )
575 *
576  IF( ilvsr )
577  \$ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
578  \$ work( iright ), n, vsr, ldvsr, ierr )
579 *
580 * Check if unscaling would cause over/underflow, if so, rescale
581 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
582 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
583 *
584  IF( ilascl )THEN
585  DO 50 i = 1, n
586  IF( alphai( i ).NE.zero ) THEN
587  IF( ( alphar( i )/safmax ).GT.( anrmto/anrm ) .OR.
588  \$ ( safmin/alphar( i ) ).GT.( anrm/anrmto ) ) THEN
589  work( 1 ) = abs( a( i, i )/alphar( i ) )
590  beta( i ) = beta( i )*work( 1 )
591  alphar( i ) = alphar( i )*work( 1 )
592  alphai( i ) = alphai( i )*work( 1 )
593  ELSE IF( ( alphai( i )/safmax ).GT.( anrmto/anrm ) .OR.
594  \$ ( safmin/alphai( i ) ).GT.( anrm/anrmto ) ) THEN
595  work( 1 ) = abs( a( i, i+1 )/alphai( i ) )
596  beta( i ) = beta( i )*work( 1 )
597  alphar( i ) = alphar( i )*work( 1 )
598  alphai( i ) = alphai( i )*work( 1 )
599  END IF
600  END IF
601  50 CONTINUE
602  END IF
603 *
604  IF( ilbscl )THEN
605  DO 60 i = 1, n
606  IF( alphai( i ).NE.zero ) THEN
607  IF( ( beta( i )/safmax ).GT.( bnrmto/bnrm ) .OR.
608  \$ ( safmin/beta( i ) ).GT.( bnrm/bnrmto ) ) THEN
609  work( 1 ) = abs(b( i, i )/beta( i ))
610  beta( i ) = beta( i )*work( 1 )
611  alphar( i ) = alphar( i )*work( 1 )
612  alphai( i ) = alphai( i )*work( 1 )
613  END IF
614  END IF
615  60 CONTINUE
616  END IF
617 *
618 * Undo scaling
619 *
620  IF( ilascl ) THEN
621  CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
622  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
623  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
624  END IF
625 *
626  IF( ilbscl ) THEN
627  CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
628  CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
629  END IF
630 *
631  IF( wantst ) THEN
632 *
633 * Check if reordering is correct
634 *
635  lastsl = .true.
636  lst2sl = .true.
637  sdim = 0
638  ip = 0
639  DO 30 i = 1, n
640  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
641  IF( alphai( i ).EQ.zero ) THEN
642  IF( cursl )
643  \$ sdim = sdim + 1
644  ip = 0
645  IF( cursl .AND. .NOT.lastsl )
646  \$ info = n + 2
647  ELSE
648  IF( ip.EQ.1 ) THEN
649 *
650 * Last eigenvalue of conjugate pair
651 *
652  cursl = cursl .OR. lastsl
653  lastsl = cursl
654  IF( cursl )
655  \$ sdim = sdim + 2
656  ip = -1
657  IF( cursl .AND. .NOT.lst2sl )
658  \$ info = n + 2
659  ELSE
660 *
661 * First eigenvalue of conjugate pair
662 *
663  ip = 1
664  END IF
665  END IF
666  lst2sl = lastsl
667  lastsl = cursl
668  30 CONTINUE
669 *
670  END IF
671 *
672  40 CONTINUE
673 *
674  work( 1 ) = maxwrk
675 *
676  RETURN
677 *
678 * End of SGGES
679 *
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:179
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:453
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:170
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:145
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:138
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine shgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ
Definition: shgeqz.f:306
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:112
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:116
subroutine sggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
SGGBAK
Definition: sggbak.f:149
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
subroutine sgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
SGGHRD
Definition: sgghrd.f:209
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:130
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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