 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ztrevc3()

 subroutine ztrevc3 ( character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer INFO )

ZTREVC3

Download ZTREVC3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` ZTREVC3 computes some or all of the right and/or left eigenvectors of
a complex upper triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a complex general matrix:  A = Q*T*Q**H, as computed by ZHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x,     (y**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal of T.

This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the unitary factor that reduces a matrix A to
Schur form T, then Q*X and Q*Y are the matrices of right and left
eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation.```
Parameters
 [in] SIDE ``` SIDE is CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors.``` [in] HOWMNY ``` HOWMNY is CHARACTER*1 = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, backtransformed using the matrices supplied in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, as indicated by the logical array SELECT.``` [in] SELECT ``` SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed. The eigenvector corresponding to the j-th eigenvalue is computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = 'A' or 'B'.``` [in] N ``` N is INTEGER The order of the matrix T. N >= 0.``` [in,out] T ``` T is COMPLEX*16 array, dimension (LDT,N) The upper triangular matrix T. T is modified, but restored on exit.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N).``` [in,out] VL ``` VL is COMPLEX*16 array, dimension (LDVL,MM) On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. Not referenced if SIDE = 'R'.``` [in] LDVL ``` LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.``` [in,out] VR ``` VR is COMPLEX*16 array, dimension (LDVR,MM) On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. Not referenced if SIDE = 'L'.``` [in] LDVR ``` LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.``` [in] MM ``` MM is INTEGER The number of columns in the arrays VL and/or VR. MM >= M.``` [out] M ``` M is INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected eigenvector occupies one column.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))` [in] LWORK ``` LWORK is INTEGER The dimension of array WORK. LWORK >= max(1,2*N). For optimum performance, LWORK >= N + 2*N*NB, where NB is the optimal blocksize. 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 (LRWORK)` [in] LRWORK ``` LRWORK is INTEGER The dimension of array RWORK. LRWORK >= max(1,N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date
November 2017
Further Details:
```  The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.```

Definition at line 248 of file ztrevc3.f.

248  IMPLICIT NONE
249 *
250 * -- LAPACK computational routine (version 3.8.0) --
251 * -- LAPACK is a software package provided by Univ. of Tennessee, --
252 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
253 * November 2017
254 *
255 * .. Scalar Arguments ..
256  CHARACTER howmny, side
257  INTEGER info, ldt, ldvl, ldvr, lwork, lrwork, m, mm, n
258 * ..
259 * .. Array Arguments ..
260  LOGICAL select( * )
261  DOUBLE PRECISION rwork( * )
262  COMPLEX*16 t( ldt, * ), vl( ldvl, * ), vr( ldvr, * ),
263  \$ work( * )
264 * ..
265 *
266 * =====================================================================
267 *
268 * .. Parameters ..
269  DOUBLE PRECISION zero, one
270  parameter( zero = 0.0d+0, one = 1.0d+0 )
271  COMPLEX*16 czero, cone
272  parameter( czero = ( 0.0d+0, 0.0d+0 ),
273  \$ cone = ( 1.0d+0, 0.0d+0 ) )
274  INTEGER nbmin, nbmax
275  parameter( nbmin = 8, nbmax = 128 )
276 * ..
277 * .. Local Scalars ..
278  LOGICAL allv, bothv, leftv, lquery, over, rightv, somev
279  INTEGER i, ii, is, j, k, ki, iv, maxwrk, nb
280  DOUBLE PRECISION ovfl, remax, scale, smin, smlnum, ulp, unfl
281  COMPLEX*16 cdum
282 * ..
283 * .. External Functions ..
284  LOGICAL lsame
285  INTEGER ilaenv, izamax
286  DOUBLE PRECISION dlamch, dzasum
287  EXTERNAL lsame, ilaenv, izamax, dlamch, dzasum
288 * ..
289 * .. External Subroutines ..
290  EXTERNAL xerbla, zcopy, zdscal, zgemv, zlatrs,
292 * ..
293 * .. Intrinsic Functions ..
294  INTRINSIC abs, dble, dcmplx, conjg, aimag, max
295 * ..
296 * .. Statement Functions ..
297  DOUBLE PRECISION cabs1
298 * ..
299 * .. Statement Function definitions ..
300  cabs1( cdum ) = abs( dble( cdum ) ) + abs( aimag( cdum ) )
301 * ..
302 * .. Executable Statements ..
303 *
304 * Decode and test the input parameters
305 *
306  bothv = lsame( side, 'B' )
307  rightv = lsame( side, 'R' ) .OR. bothv
308  leftv = lsame( side, 'L' ) .OR. bothv
309 *
310  allv = lsame( howmny, 'A' )
311  over = lsame( howmny, 'B' )
312  somev = lsame( howmny, 'S' )
313 *
314 * Set M to the number of columns required to store the selected
315 * eigenvectors.
316 *
317  IF( somev ) THEN
318  m = 0
319  DO 10 j = 1, n
320  IF( SELECT( j ) )
321  \$ m = m + 1
322  10 CONTINUE
323  ELSE
324  m = n
325  END IF
326 *
327  info = 0
328  nb = ilaenv( 1, 'ZTREVC', side // howmny, n, -1, -1, -1 )
329  maxwrk = n + 2*n*nb
330  work(1) = maxwrk
331  rwork(1) = n
332  lquery = ( lwork.EQ.-1 .OR. lrwork.EQ.-1 )
333  IF( .NOT.rightv .AND. .NOT.leftv ) THEN
334  info = -1
335  ELSE IF( .NOT.allv .AND. .NOT.over .AND. .NOT.somev ) THEN
336  info = -2
337  ELSE IF( n.LT.0 ) THEN
338  info = -4
339  ELSE IF( ldt.LT.max( 1, n ) ) THEN
340  info = -6
341  ELSE IF( ldvl.LT.1 .OR. ( leftv .AND. ldvl.LT.n ) ) THEN
342  info = -8
343  ELSE IF( ldvr.LT.1 .OR. ( rightv .AND. ldvr.LT.n ) ) THEN
344  info = -10
345  ELSE IF( mm.LT.m ) THEN
346  info = -11
347  ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
348  info = -14
349  ELSE IF ( lrwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
350  info = -16
351  END IF
352  IF( info.NE.0 ) THEN
353  CALL xerbla( 'ZTREVC3', -info )
354  RETURN
355  ELSE IF( lquery ) THEN
356  RETURN
357  END IF
358 *
359 * Quick return if possible.
360 *
361  IF( n.EQ.0 )
362  \$ RETURN
363 *
364 * Use blocked version of back-transformation if sufficient workspace.
365 * Zero-out the workspace to avoid potential NaN propagation.
366 *
367  IF( over .AND. lwork .GE. n + 2*n*nbmin ) THEN
368  nb = (lwork - n) / (2*n)
369  nb = min( nb, nbmax )
370  CALL zlaset( 'F', n, 1+2*nb, czero, czero, work, n )
371  ELSE
372  nb = 1
373  END IF
374 *
375 * Set the constants to control overflow.
376 *
377  unfl = dlamch( 'Safe minimum' )
378  ovfl = one / unfl
379  CALL dlabad( unfl, ovfl )
380  ulp = dlamch( 'Precision' )
381  smlnum = unfl*( n / ulp )
382 *
383 * Store the diagonal elements of T in working array WORK.
384 *
385  DO 20 i = 1, n
386  work( i ) = t( i, i )
387  20 CONTINUE
388 *
389 * Compute 1-norm of each column of strictly upper triangular
390 * part of T to control overflow in triangular solver.
391 *
392  rwork( 1 ) = zero
393  DO 30 j = 2, n
394  rwork( j ) = dzasum( j-1, t( 1, j ), 1 )
395  30 CONTINUE
396 *
397  IF( rightv ) THEN
398 *
399 * ============================================================
400 * Compute right eigenvectors.
401 *
402 * IV is index of column in current block.
403 * Non-blocked version always uses IV=NB=1;
404 * blocked version starts with IV=NB, goes down to 1.
405 * (Note the "0-th" column is used to store the original diagonal.)
406  iv = nb
407  is = m
408  DO 80 ki = n, 1, -1
409  IF( somev ) THEN
410  IF( .NOT.SELECT( ki ) )
411  \$ GO TO 80
412  END IF
413  smin = max( ulp*( cabs1( t( ki, ki ) ) ), smlnum )
414 *
415 * --------------------------------------------------------
416 * Complex right eigenvector
417 *
418  work( ki + iv*n ) = cone
419 *
420 * Form right-hand side.
421 *
422  DO 40 k = 1, ki - 1
423  work( k + iv*n ) = -t( k, ki )
424  40 CONTINUE
425 *
426 * Solve upper triangular system:
427 * [ T(1:KI-1,1:KI-1) - T(KI,KI) ]*X = SCALE*WORK.
428 *
429  DO 50 k = 1, ki - 1
430  t( k, k ) = t( k, k ) - t( ki, ki )
431  IF( cabs1( t( k, k ) ).LT.smin )
432  \$ t( k, k ) = smin
433  50 CONTINUE
434 *
435  IF( ki.GT.1 ) THEN
436  CALL zlatrs( 'Upper', 'No transpose', 'Non-unit', 'Y',
437  \$ ki-1, t, ldt, work( 1 + iv*n ), scale,
438  \$ rwork, info )
439  work( ki + iv*n ) = scale
440  END IF
441 *
442 * Copy the vector x or Q*x to VR and normalize.
443 *
444  IF( .NOT.over ) THEN
445 * ------------------------------
446 * no back-transform: copy x to VR and normalize.
447  CALL zcopy( ki, work( 1 + iv*n ), 1, vr( 1, is ), 1 )
448 *
449  ii = izamax( ki, vr( 1, is ), 1 )
450  remax = one / cabs1( vr( ii, is ) )
451  CALL zdscal( ki, remax, vr( 1, is ), 1 )
452 *
453  DO 60 k = ki + 1, n
454  vr( k, is ) = czero
455  60 CONTINUE
456 *
457  ELSE IF( nb.EQ.1 ) THEN
458 * ------------------------------
459 * version 1: back-transform each vector with GEMV, Q*x.
460  IF( ki.GT.1 )
461  \$ CALL zgemv( 'N', n, ki-1, cone, vr, ldvr,
462  \$ work( 1 + iv*n ), 1, dcmplx( scale ),
463  \$ vr( 1, ki ), 1 )
464 *
465  ii = izamax( n, vr( 1, ki ), 1 )
466  remax = one / cabs1( vr( ii, ki ) )
467  CALL zdscal( n, remax, vr( 1, ki ), 1 )
468 *
469  ELSE
470 * ------------------------------
471 * version 2: back-transform block of vectors with GEMM
472 * zero out below vector
473  DO k = ki + 1, n
474  work( k + iv*n ) = czero
475  END DO
476 *
477 * Columns IV:NB of work are valid vectors.
478 * When the number of vectors stored reaches NB,
479 * or if this was last vector, do the GEMM
480  IF( (iv.EQ.1) .OR. (ki.EQ.1) ) THEN
481  CALL zgemm( 'N', 'N', n, nb-iv+1, ki+nb-iv, cone,
482  \$ vr, ldvr,
483  \$ work( 1 + (iv)*n ), n,
484  \$ czero,
485  \$ work( 1 + (nb+iv)*n ), n )
486 * normalize vectors
487  DO k = iv, nb
488  ii = izamax( n, work( 1 + (nb+k)*n ), 1 )
489  remax = one / cabs1( work( ii + (nb+k)*n ) )
490  CALL zdscal( n, remax, work( 1 + (nb+k)*n ), 1 )
491  END DO
492  CALL zlacpy( 'F', n, nb-iv+1,
493  \$ work( 1 + (nb+iv)*n ), n,
494  \$ vr( 1, ki ), ldvr )
495  iv = nb
496  ELSE
497  iv = iv - 1
498  END IF
499  END IF
500 *
501 * Restore the original diagonal elements of T.
502 *
503  DO 70 k = 1, ki - 1
504  t( k, k ) = work( k )
505  70 CONTINUE
506 *
507  is = is - 1
508  80 CONTINUE
509  END IF
510 *
511  IF( leftv ) THEN
512 *
513 * ============================================================
514 * Compute left eigenvectors.
515 *
516 * IV is index of column in current block.
517 * Non-blocked version always uses IV=1;
518 * blocked version starts with IV=1, goes up to NB.
519 * (Note the "0-th" column is used to store the original diagonal.)
520  iv = 1
521  is = 1
522  DO 130 ki = 1, n
523 *
524  IF( somev ) THEN
525  IF( .NOT.SELECT( ki ) )
526  \$ GO TO 130
527  END IF
528  smin = max( ulp*( cabs1( t( ki, ki ) ) ), smlnum )
529 *
530 * --------------------------------------------------------
531 * Complex left eigenvector
532 *
533  work( ki + iv*n ) = cone
534 *
535 * Form right-hand side.
536 *
537  DO 90 k = ki + 1, n
538  work( k + iv*n ) = -conjg( t( ki, k ) )
539  90 CONTINUE
540 *
541 * Solve conjugate-transposed triangular system:
542 * [ T(KI+1:N,KI+1:N) - T(KI,KI) ]**H * X = SCALE*WORK.
543 *
544  DO 100 k = ki + 1, n
545  t( k, k ) = t( k, k ) - t( ki, ki )
546  IF( cabs1( t( k, k ) ).LT.smin )
547  \$ t( k, k ) = smin
548  100 CONTINUE
549 *
550  IF( ki.LT.n ) THEN
551  CALL zlatrs( 'Upper', 'Conjugate transpose', 'Non-unit',
552  \$ 'Y', n-ki, t( ki+1, ki+1 ), ldt,
553  \$ work( ki+1 + iv*n ), scale, rwork, info )
554  work( ki + iv*n ) = scale
555  END IF
556 *
557 * Copy the vector x or Q*x to VL and normalize.
558 *
559  IF( .NOT.over ) THEN
560 * ------------------------------
561 * no back-transform: copy x to VL and normalize.
562  CALL zcopy( n-ki+1, work( ki + iv*n ), 1, vl(ki,is), 1 )
563 *
564  ii = izamax( n-ki+1, vl( ki, is ), 1 ) + ki - 1
565  remax = one / cabs1( vl( ii, is ) )
566  CALL zdscal( n-ki+1, remax, vl( ki, is ), 1 )
567 *
568  DO 110 k = 1, ki - 1
569  vl( k, is ) = czero
570  110 CONTINUE
571 *
572  ELSE IF( nb.EQ.1 ) THEN
573 * ------------------------------
574 * version 1: back-transform each vector with GEMV, Q*x.
575  IF( ki.LT.n )
576  \$ CALL zgemv( 'N', n, n-ki, cone, vl( 1, ki+1 ), ldvl,
577  \$ work( ki+1 + iv*n ), 1, dcmplx( scale ),
578  \$ vl( 1, ki ), 1 )
579 *
580  ii = izamax( n, vl( 1, ki ), 1 )
581  remax = one / cabs1( vl( ii, ki ) )
582  CALL zdscal( n, remax, vl( 1, ki ), 1 )
583 *
584  ELSE
585 * ------------------------------
586 * version 2: back-transform block of vectors with GEMM
587 * zero out above vector
588 * could go from KI-NV+1 to KI-1
589  DO k = 1, ki - 1
590  work( k + iv*n ) = czero
591  END DO
592 *
593 * Columns 1:IV of work are valid vectors.
594 * When the number of vectors stored reaches NB,
595 * or if this was last vector, do the GEMM
596  IF( (iv.EQ.nb) .OR. (ki.EQ.n) ) THEN
597  CALL zgemm( 'N', 'N', n, iv, n-ki+iv, cone,
598  \$ vl( 1, ki-iv+1 ), ldvl,
599  \$ work( ki-iv+1 + (1)*n ), n,
600  \$ czero,
601  \$ work( 1 + (nb+1)*n ), n )
602 * normalize vectors
603  DO k = 1, iv
604  ii = izamax( n, work( 1 + (nb+k)*n ), 1 )
605  remax = one / cabs1( work( ii + (nb+k)*n ) )
606  CALL zdscal( n, remax, work( 1 + (nb+k)*n ), 1 )
607  END DO
608  CALL zlacpy( 'F', n, iv,
609  \$ work( 1 + (nb+1)*n ), n,
610  \$ vl( 1, ki-iv+1 ), ldvl )
611  iv = 1
612  ELSE
613  iv = iv + 1
614  END IF
615  END IF
616 *
617 * Restore the original diagonal elements of T.
618 *
619  DO 120 k = ki + 1, n
620  t( k, k ) = work( k )
621  120 CONTINUE
622 *
623  is = is + 1
624  130 CONTINUE
625  END IF
626 *
627  RETURN
628 *
629 * End of ZTREVC3
630 *
double precision function dzasum(N, ZX, INCX)
DZASUM
Definition: dzasum.f:74
subroutine zlatrs(UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow...
Definition: zlatrs.f:241
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: zlaset.f:108
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:189
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
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
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:73
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 zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
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