LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ zgeevx()

subroutine zgeevx ( character  BALANC,
character  JOBVL,
character  JOBVR,
character  SENSE,
integer  N,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( * )  W,
complex*16, dimension( ldvl, * )  VL,
integer  LDVL,
complex*16, dimension( ldvr, * )  VR,
integer  LDVR,
integer  ILO,
integer  IHI,
double precision, dimension( * )  SCALE,
double precision  ABNRM,
double precision, dimension( * )  RCONDE,
double precision, dimension( * )  RCONDV,
complex*16, dimension( * )  WORK,
integer  LWORK,
double precision, dimension( * )  RWORK,
integer  INFO 
)

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

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

Purpose:
 ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
 eigenvalues and, optionally, the left and/or right eigenvectors.

 Optionally also, it computes a balancing transformation to improve
 the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
 SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
 (RCONDE), and reciprocal condition numbers for the right
 eigenvectors (RCONDV).

 The right eigenvector v(j) of A satisfies
                  A * v(j) = lambda(j) * v(j)
 where lambda(j) is its eigenvalue.
 The left eigenvector u(j) of A satisfies
               u(j)**H * A = lambda(j) * u(j)**H
 where u(j)**H denotes the conjugate transpose of u(j).

 The computed eigenvectors are normalized to have Euclidean norm
 equal to 1 and largest component real.

 Balancing a matrix means permuting the rows and columns to make it
 more nearly upper triangular, and applying a diagonal similarity
 transformation D * A * D**(-1), where D is a diagonal matrix, to
 make its rows and columns closer in norm and the condition numbers
 of its eigenvalues and eigenvectors smaller.  The computed
 reciprocal condition numbers correspond to the balanced matrix.
 Permuting rows and columns will not change the condition numbers
 (in exact arithmetic) but diagonal scaling will.  For further
 explanation of balancing, see section 4.10.2 of the LAPACK
 Users' Guide.
Parameters
[in]BALANC
          BALANC is CHARACTER*1
          Indicates how the input matrix should be diagonally scaled
          and/or permuted to improve the conditioning of its
          eigenvalues.
          = 'N': Do not diagonally scale or permute;
          = 'P': Perform permutations to make the matrix more nearly
                 upper triangular. Do not diagonally scale;
          = 'S': Diagonally scale the matrix, ie. replace A by
                 D*A*D**(-1), where D is a diagonal matrix chosen
                 to make the rows and columns of A more equal in
                 norm. Do not permute;
          = 'B': Both diagonally scale and permute A.

          Computed reciprocal condition numbers will be for the matrix
          after balancing and/or permuting. Permuting does not change
          condition numbers (in exact arithmetic), but balancing does.
[in]JOBVL
          JOBVL is CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVL must = 'V'.
[in]JOBVR
          JOBVR is CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.
          If SENSE = 'E' or 'B', JOBVR must = 'V'.
[in]SENSE
          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for eigenvalues only;
          = 'V': Computed for right eigenvectors only;
          = 'B': Computed for eigenvalues and right eigenvectors.

          If SENSE = 'E' or 'B', both left and right eigenvectors
          must also be computed (JOBVL = 'V' and JOBVR = 'V').
[in]N
          N is INTEGER
          The order of the matrix A. N >= 0.
[in,out]A
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, the N-by-N matrix A.
          On exit, A has been overwritten.  If JOBVL = 'V' or
          JOBVR = 'V', A contains the Schur form of the balanced
          version of the matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
[out]W
          W is COMPLEX*16 array, dimension (N)
          W contains the computed eigenvalues.
[out]VL
          VL is COMPLEX*16 array, dimension (LDVL,N)
          If JOBVL = 'V', the left eigenvectors u(j) are stored one
          after another in the columns of VL, in the same order
          as their eigenvalues.
          If JOBVL = 'N', VL is not referenced.
          u(j) = VL(:,j), the j-th column of VL.
[in]LDVL
          LDVL is INTEGER
          The leading dimension of the array VL.  LDVL >= 1; if
          JOBVL = 'V', LDVL >= N.
[out]VR
          VR is COMPLEX*16 array, dimension (LDVR,N)
          If JOBVR = 'V', the right eigenvectors v(j) are stored one
          after another in the columns of VR, in the same order
          as their eigenvalues.
          If JOBVR = 'N', VR is not referenced.
          v(j) = VR(:,j), the j-th column of VR.
[in]LDVR
          LDVR is INTEGER
          The leading dimension of the array VR.  LDVR >= 1; if
          JOBVR = 'V', LDVR >= N.
[out]ILO
          ILO is INTEGER
[out]IHI
          IHI is INTEGER
          ILO and IHI are integer values determined when A was
          balanced.  The balanced A(i,j) = 0 if I > J and
          J = 1,...,ILO-1 or I = IHI+1,...,N.
[out]SCALE
          SCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          when balancing A.  If P(j) is the index of the row and column
          interchanged with row and column j, and D(j) is the scaling
          factor applied to row and column j, then
          SCALE(J) = P(J),    for J = 1,...,ILO-1
                   = D(J),    for J = ILO,...,IHI
                   = P(J)     for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.
[out]ABNRM
          ABNRM is DOUBLE PRECISION
          The one-norm of the balanced matrix (the maximum
          of the sum of absolute values of elements of any column).
[out]RCONDE
          RCONDE is DOUBLE PRECISION array, dimension (N)
          RCONDE(j) is the reciprocal condition number of the j-th
          eigenvalue.
[out]RCONDV
          RCONDV is DOUBLE PRECISION array, dimension (N)
          RCONDV(j) is the reciprocal condition number of the j-th
          right eigenvector.
[out]WORK
          WORK is COMPLEX*16 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 SENSE = 'N' or 'E',
          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
          LWORK >= N*N+2*N.
          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]RWORK
          RWORK is DOUBLE PRECISION array, dimension (2*N)
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the QR algorithm failed to compute all the
                eigenvalues, and no eigenvectors or condition numbers
                have been computed; elements 1:ILO-1 and i+1:N of W
                contain eigenvalues which have converged.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 285 of file zgeevx.f.

288  implicit none
289 *
290 * -- LAPACK driver routine --
291 * -- LAPACK is a software package provided by Univ. of Tennessee, --
292 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
293 *
294 * .. Scalar Arguments ..
295  CHARACTER BALANC, JOBVL, JOBVR, SENSE
296  INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
297  DOUBLE PRECISION ABNRM
298 * ..
299 * .. Array Arguments ..
300  DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ),
301  $ SCALE( * )
302  COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
303  $ W( * ), WORK( * )
304 * ..
305 *
306 * =====================================================================
307 *
308 * .. Parameters ..
309  DOUBLE PRECISION ZERO, ONE
310  parameter( zero = 0.0d0, one = 1.0d0 )
311 * ..
312 * .. Local Scalars ..
313  LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
314  $ WNTSNN, WNTSNV
315  CHARACTER JOB, SIDE
316  INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K,
317  $ LWORK_TREVC, MAXWRK, MINWRK, NOUT
318  DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
319  COMPLEX*16 TMP
320 * ..
321 * .. Local Arrays ..
322  LOGICAL SELECT( 1 )
323  DOUBLE PRECISION DUM( 1 )
324 * ..
325 * .. External Subroutines ..
326  EXTERNAL dlabad, dlascl, xerbla, zdscal, zgebak, zgebal,
328  $ ztrsna, zunghr
329 * ..
330 * .. External Functions ..
331  LOGICAL LSAME
332  INTEGER IDAMAX, ILAENV
333  DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE
334  EXTERNAL lsame, idamax, ilaenv, dlamch, dznrm2, zlange
335 * ..
336 * .. Intrinsic Functions ..
337  INTRINSIC dble, dcmplx, conjg, aimag, max, sqrt
338 * ..
339 * .. Executable Statements ..
340 *
341 * Test the input arguments
342 *
343  info = 0
344  lquery = ( lwork.EQ.-1 )
345  wantvl = lsame( jobvl, 'V' )
346  wantvr = lsame( jobvr, 'V' )
347  wntsnn = lsame( sense, 'N' )
348  wntsne = lsame( sense, 'E' )
349  wntsnv = lsame( sense, 'V' )
350  wntsnb = lsame( sense, 'B' )
351  IF( .NOT.( lsame( balanc, 'N' ) .OR. lsame( balanc, 'S' ) .OR.
352  $ lsame( balanc, 'P' ) .OR. lsame( balanc, 'B' ) ) ) THEN
353  info = -1
354  ELSE IF( ( .NOT.wantvl ) .AND. ( .NOT.lsame( jobvl, 'N' ) ) ) THEN
355  info = -2
356  ELSE IF( ( .NOT.wantvr ) .AND. ( .NOT.lsame( jobvr, 'N' ) ) ) THEN
357  info = -3
358  ELSE IF( .NOT.( wntsnn .OR. wntsne .OR. wntsnb .OR. wntsnv ) .OR.
359  $ ( ( wntsne .OR. wntsnb ) .AND. .NOT.( wantvl .AND.
360  $ wantvr ) ) ) THEN
361  info = -4
362  ELSE IF( n.LT.0 ) THEN
363  info = -5
364  ELSE IF( lda.LT.max( 1, n ) ) THEN
365  info = -7
366  ELSE IF( ldvl.LT.1 .OR. ( wantvl .AND. ldvl.LT.n ) ) THEN
367  info = -10
368  ELSE IF( ldvr.LT.1 .OR. ( wantvr .AND. ldvr.LT.n ) ) THEN
369  info = -12
370  END IF
371 *
372 * Compute workspace
373 * (Note: Comments in the code beginning "Workspace:" describe the
374 * minimal amount of workspace needed at that point in the code,
375 * as well as the preferred amount for good performance.
376 * CWorkspace refers to complex workspace, and RWorkspace to real
377 * workspace. NB refers to the optimal block size for the
378 * immediately following subroutine, as returned by ILAENV.
379 * HSWORK refers to the workspace preferred by ZHSEQR, as
380 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
381 * the worst case.)
382 *
383  IF( info.EQ.0 ) THEN
384  IF( n.EQ.0 ) THEN
385  minwrk = 1
386  maxwrk = 1
387  ELSE
388  maxwrk = n + n*ilaenv( 1, 'ZGEHRD', ' ', n, 1, n, 0 )
389 *
390  IF( wantvl ) THEN
391  CALL ztrevc3( 'L', 'B', SELECT, n, a, lda,
392  $ vl, ldvl, vr, ldvr,
393  $ n, nout, work, -1, rwork, -1, ierr )
394  lwork_trevc = int( work(1) )
395  maxwrk = max( maxwrk, lwork_trevc )
396  CALL zhseqr( 'S', 'V', n, 1, n, a, lda, w, vl, ldvl,
397  $ work, -1, info )
398  ELSE IF( wantvr ) THEN
399  CALL ztrevc3( 'R', 'B', SELECT, n, a, lda,
400  $ vl, ldvl, vr, ldvr,
401  $ n, nout, work, -1, rwork, -1, ierr )
402  lwork_trevc = int( work(1) )
403  maxwrk = max( maxwrk, lwork_trevc )
404  CALL zhseqr( 'S', 'V', n, 1, n, a, lda, w, vr, ldvr,
405  $ work, -1, info )
406  ELSE
407  IF( wntsnn ) THEN
408  CALL zhseqr( 'E', 'N', n, 1, n, a, lda, w, vr, ldvr,
409  $ work, -1, info )
410  ELSE
411  CALL zhseqr( 'S', 'N', n, 1, n, a, lda, w, vr, ldvr,
412  $ work, -1, info )
413  END IF
414  END IF
415  hswork = int( work(1) )
416 *
417  IF( ( .NOT.wantvl ) .AND. ( .NOT.wantvr ) ) THEN
418  minwrk = 2*n
419  IF( .NOT.( wntsnn .OR. wntsne ) )
420  $ minwrk = max( minwrk, n*n + 2*n )
421  maxwrk = max( maxwrk, hswork )
422  IF( .NOT.( wntsnn .OR. wntsne ) )
423  $ maxwrk = max( maxwrk, n*n + 2*n )
424  ELSE
425  minwrk = 2*n
426  IF( .NOT.( wntsnn .OR. wntsne ) )
427  $ minwrk = max( minwrk, n*n + 2*n )
428  maxwrk = max( maxwrk, hswork )
429  maxwrk = max( maxwrk, n + ( n - 1 )*ilaenv( 1, 'ZUNGHR',
430  $ ' ', n, 1, n, -1 ) )
431  IF( .NOT.( wntsnn .OR. wntsne ) )
432  $ maxwrk = max( maxwrk, n*n + 2*n )
433  maxwrk = max( maxwrk, 2*n )
434  END IF
435  maxwrk = max( maxwrk, minwrk )
436  END IF
437  work( 1 ) = maxwrk
438 *
439  IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
440  info = -20
441  END IF
442  END IF
443 *
444  IF( info.NE.0 ) THEN
445  CALL xerbla( 'ZGEEVX', -info )
446  RETURN
447  ELSE IF( lquery ) THEN
448  RETURN
449  END IF
450 *
451 * Quick return if possible
452 *
453  IF( n.EQ.0 )
454  $ RETURN
455 *
456 * Get machine constants
457 *
458  eps = dlamch( 'P' )
459  smlnum = dlamch( 'S' )
460  bignum = one / smlnum
461  CALL dlabad( smlnum, bignum )
462  smlnum = sqrt( smlnum ) / eps
463  bignum = one / smlnum
464 *
465 * Scale A if max element outside range [SMLNUM,BIGNUM]
466 *
467  icond = 0
468  anrm = zlange( 'M', n, n, a, lda, dum )
469  scalea = .false.
470  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
471  scalea = .true.
472  cscale = smlnum
473  ELSE IF( anrm.GT.bignum ) THEN
474  scalea = .true.
475  cscale = bignum
476  END IF
477  IF( scalea )
478  $ CALL zlascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
479 *
480 * Balance the matrix and compute ABNRM
481 *
482  CALL zgebal( balanc, n, a, lda, ilo, ihi, scale, ierr )
483  abnrm = zlange( '1', n, n, a, lda, dum )
484  IF( scalea ) THEN
485  dum( 1 ) = abnrm
486  CALL dlascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
487  abnrm = dum( 1 )
488  END IF
489 *
490 * Reduce to upper Hessenberg form
491 * (CWorkspace: need 2*N, prefer N+N*NB)
492 * (RWorkspace: none)
493 *
494  itau = 1
495  iwrk = itau + n
496  CALL zgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
497  $ lwork-iwrk+1, ierr )
498 *
499  IF( wantvl ) THEN
500 *
501 * Want left eigenvectors
502 * Copy Householder vectors to VL
503 *
504  side = 'L'
505  CALL zlacpy( 'L', n, n, a, lda, vl, ldvl )
506 *
507 * Generate unitary matrix in VL
508 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
509 * (RWorkspace: none)
510 *
511  CALL zunghr( n, ilo, ihi, vl, ldvl, work( itau ), work( iwrk ),
512  $ lwork-iwrk+1, ierr )
513 *
514 * Perform QR iteration, accumulating Schur vectors in VL
515 * (CWorkspace: need 1, prefer HSWORK (see comments) )
516 * (RWorkspace: none)
517 *
518  iwrk = itau
519  CALL zhseqr( 'S', 'V', n, ilo, ihi, a, lda, w, vl, ldvl,
520  $ work( iwrk ), lwork-iwrk+1, info )
521 *
522  IF( wantvr ) THEN
523 *
524 * Want left and right eigenvectors
525 * Copy Schur vectors to VR
526 *
527  side = 'B'
528  CALL zlacpy( 'F', n, n, vl, ldvl, vr, ldvr )
529  END IF
530 *
531  ELSE IF( wantvr ) THEN
532 *
533 * Want right eigenvectors
534 * Copy Householder vectors to VR
535 *
536  side = 'R'
537  CALL zlacpy( 'L', n, n, a, lda, vr, ldvr )
538 *
539 * Generate unitary matrix in VR
540 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
541 * (RWorkspace: none)
542 *
543  CALL zunghr( n, ilo, ihi, vr, ldvr, work( itau ), work( iwrk ),
544  $ lwork-iwrk+1, ierr )
545 *
546 * Perform QR iteration, accumulating Schur vectors in VR
547 * (CWorkspace: need 1, prefer HSWORK (see comments) )
548 * (RWorkspace: none)
549 *
550  iwrk = itau
551  CALL zhseqr( 'S', 'V', n, ilo, ihi, a, lda, w, vr, ldvr,
552  $ work( iwrk ), lwork-iwrk+1, info )
553 *
554  ELSE
555 *
556 * Compute eigenvalues only
557 * If condition numbers desired, compute Schur form
558 *
559  IF( wntsnn ) THEN
560  job = 'E'
561  ELSE
562  job = 'S'
563  END IF
564 *
565 * (CWorkspace: need 1, prefer HSWORK (see comments) )
566 * (RWorkspace: none)
567 *
568  iwrk = itau
569  CALL zhseqr( job, 'N', n, ilo, ihi, a, lda, w, vr, ldvr,
570  $ work( iwrk ), lwork-iwrk+1, info )
571  END IF
572 *
573 * If INFO .NE. 0 from ZHSEQR, then quit
574 *
575  IF( info.NE.0 )
576  $ GO TO 50
577 *
578  IF( wantvl .OR. wantvr ) THEN
579 *
580 * Compute left and/or right eigenvectors
581 * (CWorkspace: need 2*N, prefer N + 2*N*NB)
582 * (RWorkspace: need N)
583 *
584  CALL ztrevc3( side, 'B', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
585  $ n, nout, work( iwrk ), lwork-iwrk+1,
586  $ rwork, n, ierr )
587  END IF
588 *
589 * Compute condition numbers if desired
590 * (CWorkspace: need N*N+2*N unless SENSE = 'E')
591 * (RWorkspace: need 2*N unless SENSE = 'E')
592 *
593  IF( .NOT.wntsnn ) THEN
594  CALL ztrsna( sense, 'A', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
595  $ rconde, rcondv, n, nout, work( iwrk ), n, rwork,
596  $ icond )
597  END IF
598 *
599  IF( wantvl ) THEN
600 *
601 * Undo balancing of left eigenvectors
602 *
603  CALL zgebak( balanc, 'L', n, ilo, ihi, scale, n, vl, ldvl,
604  $ ierr )
605 *
606 * Normalize left eigenvectors and make largest component real
607 *
608  DO 20 i = 1, n
609  scl = one / dznrm2( n, vl( 1, i ), 1 )
610  CALL zdscal( n, scl, vl( 1, i ), 1 )
611  DO 10 k = 1, n
612  rwork( k ) = dble( vl( k, i ) )**2 +
613  $ aimag( vl( k, i ) )**2
614  10 CONTINUE
615  k = idamax( n, rwork, 1 )
616  tmp = conjg( vl( k, i ) ) / sqrt( rwork( k ) )
617  CALL zscal( n, tmp, vl( 1, i ), 1 )
618  vl( k, i ) = dcmplx( dble( vl( k, i ) ), zero )
619  20 CONTINUE
620  END IF
621 *
622  IF( wantvr ) THEN
623 *
624 * Undo balancing of right eigenvectors
625 *
626  CALL zgebak( balanc, 'R', n, ilo, ihi, scale, n, vr, ldvr,
627  $ ierr )
628 *
629 * Normalize right eigenvectors and make largest component real
630 *
631  DO 40 i = 1, n
632  scl = one / dznrm2( n, vr( 1, i ), 1 )
633  CALL zdscal( n, scl, vr( 1, i ), 1 )
634  DO 30 k = 1, n
635  rwork( k ) = dble( vr( k, i ) )**2 +
636  $ aimag( vr( k, i ) )**2
637  30 CONTINUE
638  k = idamax( n, rwork, 1 )
639  tmp = conjg( vr( k, i ) ) / sqrt( rwork( k ) )
640  CALL zscal( n, tmp, vr( 1, i ), 1 )
641  vr( k, i ) = dcmplx( dble( vr( k, i ) ), zero )
642  40 CONTINUE
643  END IF
644 *
645 * Undo scaling if necessary
646 *
647  50 CONTINUE
648  IF( scalea ) THEN
649  CALL zlascl( 'G', 0, 0, cscale, anrm, n-info, 1, w( info+1 ),
650  $ max( n-info, 1 ), ierr )
651  IF( info.EQ.0 ) THEN
652  IF( ( wntsnv .OR. wntsnb ) .AND. icond.EQ.0 )
653  $ CALL dlascl( 'G', 0, 0, cscale, anrm, n, 1, rcondv, n,
654  $ ierr )
655  ELSE
656  CALL zlascl( 'G', 0, 0, cscale, anrm, ilo-1, 1, w, n, ierr )
657  END IF
658  END IF
659 *
660  work( 1 ) = maxwrk
661  RETURN
662 *
663 * End of ZGEEVX
664 *
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
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
subroutine zgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
ZGEBAL
Definition: zgebal.f:162
subroutine zgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZGEHRD
Definition: zgehrd.f:167
subroutine zgebak(JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
ZGEBAK
Definition: zgebak.f:131
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zhseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
ZHSEQR
Definition: zhseqr.f:299
subroutine ztrsna(JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO)
ZTRSNA
Definition: ztrsna.f:249
subroutine ztrevc3(SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, RWORK, LRWORK, INFO)
ZTREVC3
Definition: ztrevc3.f:244
subroutine zunghr(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGHR
Definition: zunghr.f:126
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition: dznrm2.f90:90
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