LAPACK  3.9.1
LAPACK: Linear Algebra PACKage

◆ dgges()

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

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

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

Purpose:
 DGGES 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
 DGGEV 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
[out]ALPHAI
          ALPHAI is DOUBLE PRECISION array, dimension (N)
[out]BETA
          BETA is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 >= 8*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 DHGEQZ.
                =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 DTGSEN.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 281 of file dgges.f.

284 *
285 * -- LAPACK driver routine --
286 * -- LAPACK is a software package provided by Univ. of Tennessee, --
287 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288 *
289 * .. Scalar Arguments ..
290  CHARACTER JOBVSL, JOBVSR, SORT
291  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
292 * ..
293 * .. Array Arguments ..
294  LOGICAL BWORK( * )
295  DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
296  $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
297  $ VSR( LDVSR, * ), WORK( * )
298 * ..
299 * .. Function Arguments ..
300  LOGICAL SELCTG
301  EXTERNAL selctg
302 * ..
303 *
304 * =====================================================================
305 *
306 * .. Parameters ..
307  DOUBLE PRECISION ZERO, ONE
308  parameter( zero = 0.0d+0, one = 1.0d+0 )
309 * ..
310 * .. Local Scalars ..
311  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
312  $ LQUERY, LST2SL, WANTST
313  INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
314  $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
315  $ MINWRK
316  DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
317  $ PVSR, SAFMAX, SAFMIN, SMLNUM
318 * ..
319 * .. Local Arrays ..
320  INTEGER IDUM( 1 )
321  DOUBLE PRECISION DIF( 2 )
322 * ..
323 * .. External Subroutines ..
324  EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlabad,
326  $ xerbla
327 * ..
328 * .. External Functions ..
329  LOGICAL LSAME
330  INTEGER ILAENV
331  DOUBLE PRECISION DLAMCH, DLANGE
332  EXTERNAL lsame, ilaenv, dlamch, dlange
333 * ..
334 * .. Intrinsic Functions ..
335  INTRINSIC abs, max, sqrt
336 * ..
337 * .. Executable Statements ..
338 *
339 * Decode the input arguments
340 *
341  IF( lsame( jobvsl, 'N' ) ) THEN
342  ijobvl = 1
343  ilvsl = .false.
344  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
345  ijobvl = 2
346  ilvsl = .true.
347  ELSE
348  ijobvl = -1
349  ilvsl = .false.
350  END IF
351 *
352  IF( lsame( jobvsr, 'N' ) ) THEN
353  ijobvr = 1
354  ilvsr = .false.
355  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
356  ijobvr = 2
357  ilvsr = .true.
358  ELSE
359  ijobvr = -1
360  ilvsr = .false.
361  END IF
362 *
363  wantst = lsame( sort, 'S' )
364 *
365 * Test the input arguments
366 *
367  info = 0
368  lquery = ( lwork.EQ.-1 )
369  IF( ijobvl.LE.0 ) THEN
370  info = -1
371  ELSE IF( ijobvr.LE.0 ) THEN
372  info = -2
373  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
374  info = -3
375  ELSE IF( n.LT.0 ) THEN
376  info = -5
377  ELSE IF( lda.LT.max( 1, n ) ) THEN
378  info = -7
379  ELSE IF( ldb.LT.max( 1, n ) ) THEN
380  info = -9
381  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
382  info = -15
383  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
384  info = -17
385  END IF
386 *
387 * Compute workspace
388 * (Note: Comments in the code beginning "Workspace:" describe the
389 * minimal amount of workspace needed at that point in the code,
390 * as well as the preferred amount for good performance.
391 * NB refers to the optimal block size for the immediately
392 * following subroutine, as returned by ILAENV.)
393 *
394  IF( info.EQ.0 ) THEN
395  IF( n.GT.0 )THEN
396  minwrk = max( 8*n, 6*n + 16 )
397  maxwrk = minwrk - n +
398  $ n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n, 0 )
399  maxwrk = max( maxwrk, minwrk - n +
400  $ n*ilaenv( 1, 'DORMQR', ' ', n, 1, n, -1 ) )
401  IF( ilvsl ) THEN
402  maxwrk = max( maxwrk, minwrk - n +
403  $ n*ilaenv( 1, 'DORGQR', ' ', n, 1, n, -1 ) )
404  END IF
405  ELSE
406  minwrk = 1
407  maxwrk = 1
408  END IF
409  work( 1 ) = maxwrk
410 *
411  IF( lwork.LT.minwrk .AND. .NOT.lquery )
412  $ info = -19
413  END IF
414 *
415  IF( info.NE.0 ) THEN
416  CALL xerbla( 'DGGES ', -info )
417  RETURN
418  ELSE IF( lquery ) THEN
419  RETURN
420  END IF
421 *
422 * Quick return if possible
423 *
424  IF( n.EQ.0 ) THEN
425  sdim = 0
426  RETURN
427  END IF
428 *
429 * Get machine constants
430 *
431  eps = dlamch( 'P' )
432  safmin = dlamch( 'S' )
433  safmax = one / safmin
434  CALL dlabad( safmin, safmax )
435  smlnum = sqrt( safmin ) / eps
436  bignum = one / smlnum
437 *
438 * Scale A if max element outside range [SMLNUM,BIGNUM]
439 *
440  anrm = dlange( 'M', n, n, a, lda, work )
441  ilascl = .false.
442  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
443  anrmto = smlnum
444  ilascl = .true.
445  ELSE IF( anrm.GT.bignum ) THEN
446  anrmto = bignum
447  ilascl = .true.
448  END IF
449  IF( ilascl )
450  $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
451 *
452 * Scale B if max element outside range [SMLNUM,BIGNUM]
453 *
454  bnrm = dlange( 'M', n, n, b, ldb, work )
455  ilbscl = .false.
456  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
457  bnrmto = smlnum
458  ilbscl = .true.
459  ELSE IF( bnrm.GT.bignum ) THEN
460  bnrmto = bignum
461  ilbscl = .true.
462  END IF
463  IF( ilbscl )
464  $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
465 *
466 * Permute the matrix to make it more nearly triangular
467 * (Workspace: need 6*N + 2*N space for storing balancing factors)
468 *
469  ileft = 1
470  iright = n + 1
471  iwrk = iright + n
472  CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
473  $ work( iright ), work( iwrk ), ierr )
474 *
475 * Reduce B to triangular form (QR decomposition of B)
476 * (Workspace: need N, prefer N*NB)
477 *
478  irows = ihi + 1 - ilo
479  icols = n + 1 - ilo
480  itau = iwrk
481  iwrk = itau + irows
482  CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
483  $ work( iwrk ), lwork+1-iwrk, ierr )
484 *
485 * Apply the orthogonal transformation to matrix A
486 * (Workspace: need N, prefer N*NB)
487 *
488  CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
489  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
490  $ lwork+1-iwrk, ierr )
491 *
492 * Initialize VSL
493 * (Workspace: need N, prefer N*NB)
494 *
495  IF( ilvsl ) THEN
496  CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
497  IF( irows.GT.1 ) THEN
498  CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
499  $ vsl( ilo+1, ilo ), ldvsl )
500  END IF
501  CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
502  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
503  END IF
504 *
505 * Initialize VSR
506 *
507  IF( ilvsr )
508  $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
509 *
510 * Reduce to generalized Hessenberg form
511 * (Workspace: none needed)
512 *
513  CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
514  $ ldvsl, vsr, ldvsr, ierr )
515 *
516 * Perform QZ algorithm, computing Schur vectors if desired
517 * (Workspace: need N)
518 *
519  iwrk = itau
520  CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
521  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
522  $ work( iwrk ), lwork+1-iwrk, ierr )
523  IF( ierr.NE.0 ) THEN
524  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
525  info = ierr
526  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
527  info = ierr - n
528  ELSE
529  info = n + 1
530  END IF
531  GO TO 50
532  END IF
533 *
534 * Sort eigenvalues ALPHA/BETA if desired
535 * (Workspace: need 4*N+16 )
536 *
537  sdim = 0
538  IF( wantst ) THEN
539 *
540 * Undo scaling on eigenvalues before SELCTGing
541 *
542  IF( ilascl ) THEN
543  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
544  $ ierr )
545  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
546  $ ierr )
547  END IF
548  IF( ilbscl )
549  $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
550 *
551 * Select eigenvalues
552 *
553  DO 10 i = 1, n
554  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
555  10 CONTINUE
556 *
557  CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
558  $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
559  $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
560  $ ierr )
561  IF( ierr.EQ.1 )
562  $ info = n + 3
563 *
564  END IF
565 *
566 * Apply back-permutation to VSL and VSR
567 * (Workspace: none needed)
568 *
569  IF( ilvsl )
570  $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
571  $ work( iright ), n, vsl, ldvsl, ierr )
572 *
573  IF( ilvsr )
574  $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
575  $ work( iright ), n, vsr, ldvsr, ierr )
576 *
577 * Check if unscaling would cause over/underflow, if so, rescale
578 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
579 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
580 *
581  IF( ilascl ) THEN
582  DO 20 i = 1, n
583  IF( alphai( i ).NE.zero ) THEN
584  IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
585  $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
586  work( 1 ) = abs( a( i, i ) / alphar( i ) )
587  beta( i ) = beta( i )*work( 1 )
588  alphar( i ) = alphar( i )*work( 1 )
589  alphai( i ) = alphai( i )*work( 1 )
590  ELSE IF( ( alphai( i ) / safmax ).GT.
591  $ ( anrmto / anrm ) .OR.
592  $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
593  $ THEN
594  work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
595  beta( i ) = beta( i )*work( 1 )
596  alphar( i ) = alphar( i )*work( 1 )
597  alphai( i ) = alphai( i )*work( 1 )
598  END IF
599  END IF
600  20 CONTINUE
601  END IF
602 *
603  IF( ilbscl ) THEN
604  DO 30 i = 1, n
605  IF( alphai( i ).NE.zero ) THEN
606  IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
607  $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
608  work( 1 ) = abs( b( i, i ) / beta( i ) )
609  beta( i ) = beta( i )*work( 1 )
610  alphar( i ) = alphar( i )*work( 1 )
611  alphai( i ) = alphai( i )*work( 1 )
612  END IF
613  END IF
614  30 CONTINUE
615  END IF
616 *
617 * Undo scaling
618 *
619  IF( ilascl ) THEN
620  CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
621  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
622  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
623  END IF
624 *
625  IF( ilbscl ) THEN
626  CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
627  CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
628  END IF
629 *
630  IF( wantst ) THEN
631 *
632 * Check if reordering is correct
633 *
634  lastsl = .true.
635  lst2sl = .true.
636  sdim = 0
637  ip = 0
638  DO 40 i = 1, n
639  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
640  IF( alphai( i ).EQ.zero ) THEN
641  IF( cursl )
642  $ sdim = sdim + 1
643  ip = 0
644  IF( cursl .AND. .NOT.lastsl )
645  $ info = n + 2
646  ELSE
647  IF( ip.EQ.1 ) THEN
648 *
649 * Last eigenvalue of conjugate pair
650 *
651  cursl = cursl .OR. lastsl
652  lastsl = cursl
653  IF( cursl )
654  $ sdim = sdim + 2
655  ip = -1
656  IF( cursl .AND. .NOT.lst2sl )
657  $ info = n + 2
658  ELSE
659 *
660 * First eigenvalue of conjugate pair
661 *
662  ip = 1
663  END IF
664  END IF
665  lst2sl = lastsl
666  lastsl = cursl
667  40 CONTINUE
668 *
669  END IF
670 *
671  50 CONTINUE
672 *
673  work( 1 ) = maxwrk
674 *
675  RETURN
676 *
677 * End of DGGES
678 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
DGGBAK
Definition: dggbak.f:147
subroutine dggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
DGGBAL
Definition: dggbal.f:177
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dhgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DHGEQZ
Definition: dhgeqz.f:304
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:145
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:128
subroutine dgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
DGGHRD
Definition: dgghrd.f:207
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:167
subroutine dtgsen(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)
DTGSEN
Definition: dtgsen.f:451
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