LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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:146
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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