 LAPACK  3.8.0 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.```
Date
June 2016

Definition at line 289 of file zgeevx.f.

289  implicit none
290 *
291 * -- LAPACK driver routine (version 3.7.0) --
292 * -- LAPACK is a software package provided by Univ. of Tennessee, --
293 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
294 * June 2016
295 *
296 * .. Scalar Arguments ..
297  CHARACTER balanc, jobvl, jobvr, sense
298  INTEGER ihi, ilo, info, lda, ldvl, ldvr, lwork, n
299  DOUBLE PRECISION abnrm
300 * ..
301 * .. Array Arguments ..
302  DOUBLE PRECISION rconde( * ), rcondv( * ), rwork( * ),
303  \$ scale( * )
304  COMPLEX*16 a( lda, * ), vl( ldvl, * ), vr( ldvr, * ),
305  \$ w( * ), work( * )
306 * ..
307 *
308 * =====================================================================
309 *
310 * .. Parameters ..
311  DOUBLE PRECISION zero, one
312  parameter( zero = 0.0d0, one = 1.0d0 )
313 * ..
314 * .. Local Scalars ..
315  LOGICAL lquery, scalea, wantvl, wantvr, wntsnb, wntsne,
316  \$ wntsnn, wntsnv
317  CHARACTER job, side
318  INTEGER hswork, i, icond, ierr, itau, iwrk, k,
319  \$ lwork_trevc, maxwrk, minwrk, nout
320  DOUBLE PRECISION anrm, bignum, cscale, eps, scl, smlnum
321  COMPLEX*16 tmp
322 * ..
323 * .. Local Arrays ..
324  LOGICAL select( 1 )
325  DOUBLE PRECISION dum( 1 )
326 * ..
327 * .. External Subroutines ..
328  EXTERNAL dlabad, dlascl, xerbla, zdscal, zgebak, zgebal,
330  \$ ztrsna, zunghr
331 * ..
332 * .. External Functions ..
333  LOGICAL lsame
334  INTEGER idamax, ilaenv
335  DOUBLE PRECISION dlamch, dznrm2, zlange
336  EXTERNAL lsame, idamax, ilaenv, dlamch, dznrm2, zlange
337 * ..
338 * .. Intrinsic Functions ..
339  INTRINSIC dble, dcmplx, conjg, aimag, max, sqrt
340 * ..
341 * .. Executable Statements ..
342 *
343 * Test the input arguments
344 *
345  info = 0
346  lquery = ( lwork.EQ.-1 )
347  wantvl = lsame( jobvl, 'V' )
348  wantvr = lsame( jobvr, 'V' )
349  wntsnn = lsame( sense, 'N' )
350  wntsne = lsame( sense, 'E' )
351  wntsnv = lsame( sense, 'V' )
352  wntsnb = lsame( sense, 'B' )
353  IF( .NOT.( lsame( balanc, 'N' ) .OR. lsame( balanc, 'S' ) .OR.
354  \$ lsame( balanc, 'P' ) .OR. lsame( balanc, 'B' ) ) ) THEN
355  info = -1
356  ELSE IF( ( .NOT.wantvl ) .AND. ( .NOT.lsame( jobvl, 'N' ) ) ) THEN
357  info = -2
358  ELSE IF( ( .NOT.wantvr ) .AND. ( .NOT.lsame( jobvr, 'N' ) ) ) THEN
359  info = -3
360  ELSE IF( .NOT.( wntsnn .OR. wntsne .OR. wntsnb .OR. wntsnv ) .OR.
361  \$ ( ( wntsne .OR. wntsnb ) .AND. .NOT.( wantvl .AND.
362  \$ wantvr ) ) ) THEN
363  info = -4
364  ELSE IF( n.LT.0 ) THEN
365  info = -5
366  ELSE IF( lda.LT.max( 1, n ) ) THEN
367  info = -7
368  ELSE IF( ldvl.LT.1 .OR. ( wantvl .AND. ldvl.LT.n ) ) THEN
369  info = -10
370  ELSE IF( ldvr.LT.1 .OR. ( wantvr .AND. ldvr.LT.n ) ) THEN
371  info = -12
372  END IF
373 *
374 * Compute workspace
375 * (Note: Comments in the code beginning "Workspace:" describe the
376 * minimal amount of workspace needed at that point in the code,
377 * as well as the preferred amount for good performance.
378 * CWorkspace refers to complex workspace, and RWorkspace to real
379 * workspace. NB refers to the optimal block size for the
380 * immediately following subroutine, as returned by ILAENV.
381 * HSWORK refers to the workspace preferred by ZHSEQR, as
382 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
383 * the worst case.)
384 *
385  IF( info.EQ.0 ) THEN
386  IF( n.EQ.0 ) THEN
387  minwrk = 1
388  maxwrk = 1
389  ELSE
390  maxwrk = n + n*ilaenv( 1, 'ZGEHRD', ' ', n, 1, n, 0 )
391 *
392  IF( wantvl ) THEN
393  CALL ztrevc3( 'L', 'B', SELECT, n, a, lda,
394  \$ vl, ldvl, vr, ldvr,
395  \$ n, nout, work, -1, rwork, -1, ierr )
396  lwork_trevc = int( work(1) )
397  maxwrk = max( maxwrk, lwork_trevc )
398  CALL zhseqr( 'S', 'V', n, 1, n, a, lda, w, vl, ldvl,
399  \$ work, -1, info )
400  ELSE IF( wantvr ) THEN
401  CALL ztrevc3( 'R', 'B', SELECT, n, a, lda,
402  \$ vl, ldvl, vr, ldvr,
403  \$ n, nout, work, -1, rwork, -1, ierr )
404  lwork_trevc = int( work(1) )
405  maxwrk = max( maxwrk, lwork_trevc )
406  CALL zhseqr( 'S', 'V', n, 1, n, a, lda, w, vr, ldvr,
407  \$ work, -1, info )
408  ELSE
409  IF( wntsnn ) THEN
410  CALL zhseqr( 'E', 'N', n, 1, n, a, lda, w, vr, ldvr,
411  \$ work, -1, info )
412  ELSE
413  CALL zhseqr( 'S', 'N', n, 1, n, a, lda, w, vr, ldvr,
414  \$ work, -1, info )
415  END IF
416  END IF
417  hswork = int( work(1) )
418 *
419  IF( ( .NOT.wantvl ) .AND. ( .NOT.wantvr ) ) THEN
420  minwrk = 2*n
421  IF( .NOT.( wntsnn .OR. wntsne ) )
422  \$ minwrk = max( minwrk, n*n + 2*n )
423  maxwrk = max( maxwrk, hswork )
424  IF( .NOT.( wntsnn .OR. wntsne ) )
425  \$ maxwrk = max( maxwrk, n*n + 2*n )
426  ELSE
427  minwrk = 2*n
428  IF( .NOT.( wntsnn .OR. wntsne ) )
429  \$ minwrk = max( minwrk, n*n + 2*n )
430  maxwrk = max( maxwrk, hswork )
431  maxwrk = max( maxwrk, n + ( n - 1 )*ilaenv( 1, 'ZUNGHR',
432  \$ ' ', n, 1, n, -1 ) )
433  IF( .NOT.( wntsnn .OR. wntsne ) )
434  \$ maxwrk = max( maxwrk, n*n + 2*n )
435  maxwrk = max( maxwrk, 2*n )
436  END IF
437  maxwrk = max( maxwrk, minwrk )
438  END IF
439  work( 1 ) = maxwrk
440 *
441  IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
442  info = -20
443  END IF
444  END IF
445 *
446  IF( info.NE.0 ) THEN
447  CALL xerbla( 'ZGEEVX', -info )
448  RETURN
449  ELSE IF( lquery ) THEN
450  RETURN
451  END IF
452 *
453 * Quick return if possible
454 *
455  IF( n.EQ.0 )
456  \$ RETURN
457 *
458 * Get machine constants
459 *
460  eps = dlamch( 'P' )
461  smlnum = dlamch( 'S' )
462  bignum = one / smlnum
463  CALL dlabad( smlnum, bignum )
464  smlnum = sqrt( smlnum ) / eps
465  bignum = one / smlnum
466 *
467 * Scale A if max element outside range [SMLNUM,BIGNUM]
468 *
469  icond = 0
470  anrm = zlange( 'M', n, n, a, lda, dum )
471  scalea = .false.
472  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
473  scalea = .true.
474  cscale = smlnum
475  ELSE IF( anrm.GT.bignum ) THEN
476  scalea = .true.
477  cscale = bignum
478  END IF
479  IF( scalea )
480  \$ CALL zlascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
481 *
482 * Balance the matrix and compute ABNRM
483 *
484  CALL zgebal( balanc, n, a, lda, ilo, ihi, scale, ierr )
485  abnrm = zlange( '1', n, n, a, lda, dum )
486  IF( scalea ) THEN
487  dum( 1 ) = abnrm
488  CALL dlascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
489  abnrm = dum( 1 )
490  END IF
491 *
492 * Reduce to upper Hessenberg form
493 * (CWorkspace: need 2*N, prefer N+N*NB)
494 * (RWorkspace: none)
495 *
496  itau = 1
497  iwrk = itau + n
498  CALL zgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
499  \$ lwork-iwrk+1, ierr )
500 *
501  IF( wantvl ) THEN
502 *
503 * Want left eigenvectors
504 * Copy Householder vectors to VL
505 *
506  side = 'L'
507  CALL zlacpy( 'L', n, n, a, lda, vl, ldvl )
508 *
509 * Generate unitary matrix in VL
510 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
511 * (RWorkspace: none)
512 *
513  CALL zunghr( n, ilo, ihi, vl, ldvl, work( itau ), work( iwrk ),
514  \$ lwork-iwrk+1, ierr )
515 *
516 * Perform QR iteration, accumulating Schur vectors in VL
517 * (CWorkspace: need 1, prefer HSWORK (see comments) )
518 * (RWorkspace: none)
519 *
520  iwrk = itau
521  CALL zhseqr( 'S', 'V', n, ilo, ihi, a, lda, w, vl, ldvl,
522  \$ work( iwrk ), lwork-iwrk+1, info )
523 *
524  IF( wantvr ) THEN
525 *
526 * Want left and right eigenvectors
527 * Copy Schur vectors to VR
528 *
529  side = 'B'
530  CALL zlacpy( 'F', n, n, vl, ldvl, vr, ldvr )
531  END IF
532 *
533  ELSE IF( wantvr ) THEN
534 *
535 * Want right eigenvectors
536 * Copy Householder vectors to VR
537 *
538  side = 'R'
539  CALL zlacpy( 'L', n, n, a, lda, vr, ldvr )
540 *
541 * Generate unitary matrix in VR
542 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
543 * (RWorkspace: none)
544 *
545  CALL zunghr( n, ilo, ihi, vr, ldvr, work( itau ), work( iwrk ),
546  \$ lwork-iwrk+1, ierr )
547 *
548 * Perform QR iteration, accumulating Schur vectors in VR
549 * (CWorkspace: need 1, prefer HSWORK (see comments) )
550 * (RWorkspace: none)
551 *
552  iwrk = itau
553  CALL zhseqr( 'S', 'V', n, ilo, ihi, a, lda, w, vr, ldvr,
554  \$ work( iwrk ), lwork-iwrk+1, info )
555 *
556  ELSE
557 *
558 * Compute eigenvalues only
559 * If condition numbers desired, compute Schur form
560 *
561  IF( wntsnn ) THEN
562  job = 'E'
563  ELSE
564  job = 'S'
565  END IF
566 *
567 * (CWorkspace: need 1, prefer HSWORK (see comments) )
568 * (RWorkspace: none)
569 *
570  iwrk = itau
571  CALL zhseqr( job, 'N', n, ilo, ihi, a, lda, w, vr, ldvr,
572  \$ work( iwrk ), lwork-iwrk+1, info )
573  END IF
574 *
575 * If INFO .NE. 0 from ZHSEQR, then quit
576 *
577  IF( info.NE.0 )
578  \$ GO TO 50
579 *
580  IF( wantvl .OR. wantvr ) THEN
581 *
582 * Compute left and/or right eigenvectors
583 * (CWorkspace: need 2*N, prefer N + 2*N*NB)
584 * (RWorkspace: need N)
585 *
586  CALL ztrevc3( side, 'B', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
587  \$ n, nout, work( iwrk ), lwork-iwrk+1,
588  \$ rwork, n, ierr )
589  END IF
590 *
591 * Compute condition numbers if desired
592 * (CWorkspace: need N*N+2*N unless SENSE = 'E')
593 * (RWorkspace: need 2*N unless SENSE = 'E')
594 *
595  IF( .NOT.wntsnn ) THEN
596  CALL ztrsna( sense, 'A', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
597  \$ rconde, rcondv, n, nout, work( iwrk ), n, rwork,
598  \$ icond )
599  END IF
600 *
601  IF( wantvl ) THEN
602 *
603 * Undo balancing of left eigenvectors
604 *
605  CALL zgebak( balanc, 'L', n, ilo, ihi, scale, n, vl, ldvl,
606  \$ ierr )
607 *
608 * Normalize left eigenvectors and make largest component real
609 *
610  DO 20 i = 1, n
611  scl = one / dznrm2( n, vl( 1, i ), 1 )
612  CALL zdscal( n, scl, vl( 1, i ), 1 )
613  DO 10 k = 1, n
614  rwork( k ) = dble( vl( k, i ) )**2 +
615  \$ aimag( vl( k, i ) )**2
616  10 CONTINUE
617  k = idamax( n, rwork, 1 )
618  tmp = conjg( vl( k, i ) ) / sqrt( rwork( k ) )
619  CALL zscal( n, tmp, vl( 1, i ), 1 )
620  vl( k, i ) = dcmplx( dble( vl( k, i ) ), zero )
621  20 CONTINUE
622  END IF
623 *
624  IF( wantvr ) THEN
625 *
626 * Undo balancing of right eigenvectors
627 *
628  CALL zgebak( balanc, 'R', n, ilo, ihi, scale, n, vr, ldvr,
629  \$ ierr )
630 *
631 * Normalize right eigenvectors and make largest component real
632 *
633  DO 40 i = 1, n
634  scl = one / dznrm2( n, vr( 1, i ), 1 )
635  CALL zdscal( n, scl, vr( 1, i ), 1 )
636  DO 30 k = 1, n
637  rwork( k ) = dble( vr( k, i ) )**2 +
638  \$ aimag( vr( k, i ) )**2
639  30 CONTINUE
640  k = idamax( n, rwork, 1 )
641  tmp = conjg( vr( k, i ) ) / sqrt( rwork( k ) )
642  CALL zscal( n, tmp, vr( 1, i ), 1 )
643  vr( k, i ) = dcmplx( dble( vr( k, i ) ), zero )
644  40 CONTINUE
645  END IF
646 *
647 * Undo scaling if necessary
648 *
649  50 CONTINUE
650  IF( scalea ) THEN
651  CALL zlascl( 'G', 0, 0, cscale, anrm, n-info, 1, w( info+1 ),
652  \$ max( n-info, 1 ), ierr )
653  IF( info.EQ.0 ) THEN
654  IF( ( wntsnv .OR. wntsnb ) .AND. icond.EQ.0 )
655  \$ CALL dlascl( 'G', 0, 0, cscale, anrm, n, 1, rcondv, n,
656  \$ ierr )
657  ELSE
658  CALL zlascl( 'G', 0, 0, cscale, anrm, ilo-1, 1, w, n, ierr )
659  END IF
660  END IF
661 *
662  work( 1 ) = maxwrk
663  RETURN
664 *
665 * End of ZGEEVX
666 *
subroutine zunghr(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGHR
Definition: zunghr.f:128
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:117
subroutine ztrsna(JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO)
ZTRSNA
Definition: ztrsna.f:251
subroutine zgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZGEHRD
Definition: zgehrd.f:169
subroutine zgebak(JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
ZGEBAK
Definition: zgebak.f:133
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
double precision function dznrm2(N, X, INCX)
DZNRM2
Definition: dznrm2.f:77
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zhseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
ZHSEQR
Definition: zhseqr.f:301
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:80
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:73
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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 zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine ztrevc3(SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, RWORK, LRWORK, INFO)
ZTREVC3
Definition: ztrevc3.f:248
subroutine zgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
ZGEBAL
Definition: zgebal.f:164
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:145
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