LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dgges3()

subroutine dgges3 ( 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 
)

DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Download DGGES3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DGGES3 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 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.
Date
January 2015

Definition at line 284 of file dgges3.f.

284 *
285 * -- LAPACK driver routine (version 3.6.0) --
286 * -- LAPACK is a software package provided by Univ. of Tennessee, --
287 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288 * January 2015
289 *
290 * .. Scalar Arguments ..
291  CHARACTER jobvsl, jobvsr, sort
292  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n, sdim
293 * ..
294 * .. Array Arguments ..
295  LOGICAL bwork( * )
296  DOUBLE PRECISION a( lda, * ), alphai( * ), alphar( * ),
297  $ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
298  $ vsr( ldvsr, * ), work( * )
299 * ..
300 * .. Function Arguments ..
301  LOGICAL selctg
302  EXTERNAL selctg
303 * ..
304 *
305 * =====================================================================
306 *
307 * .. Parameters ..
308  DOUBLE PRECISION zero, one
309  parameter( zero = 0.0d+0, one = 1.0d+0 )
310 * ..
311 * .. Local Scalars ..
312  LOGICAL cursl, ilascl, ilbscl, ilvsl, ilvsr, lastsl,
313  $ lquery, lst2sl, wantst
314  INTEGER i, icols, ierr, ihi, ijobvl, ijobvr, ileft,
315  $ ilo, ip, iright, irows, itau, iwrk, lwkopt
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, dgghd3, dhgeqz, dlabad,
326  $ xerbla
327 * ..
328 * .. External Functions ..
329  LOGICAL lsame
330  DOUBLE PRECISION dlamch, dlange
331  EXTERNAL lsame, dlamch, dlange
332 * ..
333 * .. Intrinsic Functions ..
334  INTRINSIC abs, max, sqrt
335 * ..
336 * .. Executable Statements ..
337 *
338 * Decode the input arguments
339 *
340  IF( lsame( jobvsl, 'N' ) ) THEN
341  ijobvl = 1
342  ilvsl = .false.
343  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
344  ijobvl = 2
345  ilvsl = .true.
346  ELSE
347  ijobvl = -1
348  ilvsl = .false.
349  END IF
350 *
351  IF( lsame( jobvsr, 'N' ) ) THEN
352  ijobvr = 1
353  ilvsr = .false.
354  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
355  ijobvr = 2
356  ilvsr = .true.
357  ELSE
358  ijobvr = -1
359  ilvsr = .false.
360  END IF
361 *
362  wantst = lsame( sort, 'S' )
363 *
364 * Test the input arguments
365 *
366  info = 0
367  lquery = ( lwork.EQ.-1 )
368  IF( ijobvl.LE.0 ) THEN
369  info = -1
370  ELSE IF( ijobvr.LE.0 ) THEN
371  info = -2
372  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
373  info = -3
374  ELSE IF( n.LT.0 ) THEN
375  info = -5
376  ELSE IF( lda.LT.max( 1, n ) ) THEN
377  info = -7
378  ELSE IF( ldb.LT.max( 1, n ) ) THEN
379  info = -9
380  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
381  info = -15
382  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
383  info = -17
384  ELSE IF( lwork.LT.6*n+16 .AND. .NOT.lquery ) THEN
385  info = -19
386  END IF
387 *
388 * Compute workspace
389 *
390  IF( info.EQ.0 ) THEN
391  CALL dgeqrf( n, n, b, ldb, work, work, -1, ierr )
392  lwkopt = max( 6*n+16, 3*n+int( work( 1 ) ) )
393  CALL dormqr( 'L', 'T', n, n, n, b, ldb, work, a, lda, work,
394  $ -1, ierr )
395  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
396  IF( ilvsl ) THEN
397  CALL dorgqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
398  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
399  END IF
400  CALL dgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
401  $ ldvsl, vsr, ldvsr, work, -1, ierr )
402  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
403  CALL dhgeqz( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
404  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
405  $ work, -1, ierr )
406  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
407  IF( wantst ) THEN
408  CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
409  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
410  $ sdim, pvsl, pvsr, dif, work, -1, idum, 1,
411  $ ierr )
412  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
413  END IF
414  work( 1 ) = lwkopt
415  END IF
416 *
417  IF( info.NE.0 ) THEN
418  CALL xerbla( 'DGGES3 ', -info )
419  RETURN
420  ELSE IF( lquery ) THEN
421  RETURN
422  END IF
423 *
424 * Quick return if possible
425 *
426  IF( n.EQ.0 ) THEN
427  sdim = 0
428  RETURN
429  END IF
430 *
431 * Get machine constants
432 *
433  eps = dlamch( 'P' )
434  safmin = dlamch( 'S' )
435  safmax = one / safmin
436  CALL dlabad( safmin, safmax )
437  smlnum = sqrt( safmin ) / eps
438  bignum = one / smlnum
439 *
440 * Scale A if max element outside range [SMLNUM,BIGNUM]
441 *
442  anrm = dlange( 'M', n, n, a, lda, work )
443  ilascl = .false.
444  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
445  anrmto = smlnum
446  ilascl = .true.
447  ELSE IF( anrm.GT.bignum ) THEN
448  anrmto = bignum
449  ilascl = .true.
450  END IF
451  IF( ilascl )
452  $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
453 *
454 * Scale B if max element outside range [SMLNUM,BIGNUM]
455 *
456  bnrm = dlange( 'M', n, n, b, ldb, work )
457  ilbscl = .false.
458  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
459  bnrmto = smlnum
460  ilbscl = .true.
461  ELSE IF( bnrm.GT.bignum ) THEN
462  bnrmto = bignum
463  ilbscl = .true.
464  END IF
465  IF( ilbscl )
466  $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
467 *
468 * Permute the matrix to make it more nearly triangular
469 *
470  ileft = 1
471  iright = n + 1
472  iwrk = iright + n
473  CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
474  $ work( iright ), work( iwrk ), ierr )
475 *
476 * Reduce B to triangular form (QR decomposition of B)
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 *
487  CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
488  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
489  $ lwork+1-iwrk, ierr )
490 *
491 * Initialize VSL
492 *
493  IF( ilvsl ) THEN
494  CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
495  IF( irows.GT.1 ) THEN
496  CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
497  $ vsl( ilo+1, ilo ), ldvsl )
498  END IF
499  CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
500  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
501  END IF
502 *
503 * Initialize VSR
504 *
505  IF( ilvsr )
506  $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
507 *
508 * Reduce to generalized Hessenberg form
509 *
510  CALL dgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
511  $ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk,
512  $ ierr )
513 *
514 * Perform QZ algorithm, computing Schur vectors if desired
515 *
516  iwrk = itau
517  CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
518  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
519  $ work( iwrk ), lwork+1-iwrk, ierr )
520  IF( ierr.NE.0 ) THEN
521  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
522  info = ierr
523  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
524  info = ierr - n
525  ELSE
526  info = n + 1
527  END IF
528  GO TO 50
529  END IF
530 *
531 * Sort eigenvalues ALPHA/BETA if desired
532 *
533  sdim = 0
534  IF( wantst ) THEN
535 *
536 * Undo scaling on eigenvalues before SELCTGing
537 *
538  IF( ilascl ) THEN
539  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
540  $ ierr )
541  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
542  $ ierr )
543  END IF
544  IF( ilbscl )
545  $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
546 *
547 * Select eigenvalues
548 *
549  DO 10 i = 1, n
550  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
551  10 CONTINUE
552 *
553  CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
554  $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
555  $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
556  $ ierr )
557  IF( ierr.EQ.1 )
558  $ info = n + 3
559 *
560  END IF
561 *
562 * Apply back-permutation to VSL and VSR
563 *
564  IF( ilvsl )
565  $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
566  $ work( iright ), n, vsl, ldvsl, ierr )
567 *
568  IF( ilvsr )
569  $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
570  $ work( iright ), n, vsr, ldvsr, ierr )
571 *
572 * Check if unscaling would cause over/underflow, if so, rescale
573 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
574 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
575 *
576  IF( ilascl ) THEN
577  DO 20 i = 1, n
578  IF( alphai( i ).NE.zero ) THEN
579  IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
580  $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
581  work( 1 ) = abs( a( i, i ) / alphar( i ) )
582  beta( i ) = beta( i )*work( 1 )
583  alphar( i ) = alphar( i )*work( 1 )
584  alphai( i ) = alphai( i )*work( 1 )
585  ELSE IF( ( alphai( i ) / safmax ).GT.
586  $ ( anrmto / anrm ) .OR.
587  $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
588  $ THEN
589  work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
590  beta( i ) = beta( i )*work( 1 )
591  alphar( i ) = alphar( i )*work( 1 )
592  alphai( i ) = alphai( i )*work( 1 )
593  END IF
594  END IF
595  20 CONTINUE
596  END IF
597 *
598  IF( ilbscl ) THEN
599  DO 30 i = 1, n
600  IF( alphai( i ).NE.zero ) THEN
601  IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
602  $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
603  work( 1 ) = abs( b( i, i ) / beta( i ) )
604  beta( i ) = beta( i )*work( 1 )
605  alphar( i ) = alphar( i )*work( 1 )
606  alphai( i ) = alphai( i )*work( 1 )
607  END IF
608  END IF
609  30 CONTINUE
610  END IF
611 *
612 * Undo scaling
613 *
614  IF( ilascl ) THEN
615  CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
616  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
617  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
618  END IF
619 *
620  IF( ilbscl ) THEN
621  CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
622  CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
623  END IF
624 *
625  IF( wantst ) THEN
626 *
627 * Check if reordering is correct
628 *
629  lastsl = .true.
630  lst2sl = .true.
631  sdim = 0
632  ip = 0
633  DO 40 i = 1, n
634  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
635  IF( alphai( i ).EQ.zero ) THEN
636  IF( cursl )
637  $ sdim = sdim + 1
638  ip = 0
639  IF( cursl .AND. .NOT.lastsl )
640  $ info = n + 2
641  ELSE
642  IF( ip.EQ.1 ) THEN
643 *
644 * Last eigenvalue of conjugate pair
645 *
646  cursl = cursl .OR. lastsl
647  lastsl = cursl
648  IF( cursl )
649  $ sdim = sdim + 2
650  ip = -1
651  IF( cursl .AND. .NOT.lst2sl )
652  $ info = n + 2
653  ELSE
654 *
655 * First eigenvalue of conjugate pair
656 *
657  ip = 1
658  END IF
659  END IF
660  lst2sl = lastsl
661  lastsl = cursl
662  40 CONTINUE
663 *
664  END IF
665 *
666  50 CONTINUE
667 *
668  work( 1 ) = lwkopt
669 *
670  RETURN
671 *
672 * End of DGGES3
673 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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:453
subroutine dggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
DGGBAK
Definition: dggbak.f:149
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:145
subroutine dggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
DGGBAL
Definition: dggbal.f:179
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:112
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:169
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:130
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:116
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:306
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:138
subroutine dgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DGGHD3
Definition: dgghd3.f:232
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