LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ zhsein()

subroutine zhsein ( character  SIDE,
character  EIGSRC,
character  INITV,
logical, dimension( * )  SELECT,
integer  N,
complex*16, dimension( ldh, * )  H,
integer  LDH,
complex*16, dimension( * )  W,
complex*16, dimension( ldvl, * )  VL,
integer  LDVL,
complex*16, dimension( ldvr, * )  VR,
integer  LDVR,
integer  MM,
integer  M,
complex*16, dimension( * )  WORK,
double precision, dimension( * )  RWORK,
integer, dimension( * )  IFAILL,
integer, dimension( * )  IFAILR,
integer  INFO 
)

ZHSEIN

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Purpose:
 ZHSEIN uses inverse iteration to find specified right and/or left
 eigenvectors of a complex upper Hessenberg matrix H.

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

              H * x = w * x,     y**h * H = w * y**h

 where y**h denotes the conjugate transpose of the vector y.
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]EIGSRC
          EIGSRC is CHARACTER*1
          Specifies the source of eigenvalues supplied in W:
          = 'Q': the eigenvalues were found using ZHSEQR; thus, if
                 H has zero subdiagonal elements, and so is
                 block-triangular, then the j-th eigenvalue can be
                 assumed to be an eigenvalue of the block containing
                 the j-th row/column.  This property allows ZHSEIN to
                 perform inverse iteration on just one diagonal block.
          = 'N': no assumptions are made on the correspondence
                 between eigenvalues and diagonal blocks.  In this
                 case, ZHSEIN must always perform inverse iteration
                 using the whole matrix H.
[in]INITV
          INITV is CHARACTER*1
          = 'N': no initial vectors are supplied;
          = 'U': user-supplied initial vectors are stored in the arrays
                 VL and/or VR.
[in]SELECT
          SELECT is LOGICAL array, dimension (N)
          Specifies the eigenvectors to be computed. To select the
          eigenvector corresponding to the eigenvalue W(j),
          SELECT(j) must be set to .TRUE..
[in]N
          N is INTEGER
          The order of the matrix H.  N >= 0.
[in]H
          H is COMPLEX*16 array, dimension (LDH,N)
          The upper Hessenberg matrix H.
          If a NaN is detected in H, the routine will return with INFO=-6.
[in]LDH
          LDH is INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).
[in,out]W
          W is COMPLEX*16 array, dimension (N)
          On entry, the eigenvalues of H.
          On exit, the real parts of W may have been altered since
          close eigenvalues are perturbed slightly in searching for
          independent eigenvectors.
[in,out]VL
          VL is COMPLEX*16 array, dimension (LDVL,MM)
          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
          contain starting vectors for the inverse iteration for the
          left eigenvectors; the starting vector for each eigenvector
          must be in the same column in which the eigenvector will be
          stored.
          On exit, if SIDE = 'L' or 'B', the left eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VL, in the same order as their eigenvalues.
          If SIDE = 'R', VL is not referenced.
[in]LDVL
          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
[in,out]VR
          VR is COMPLEX*16 array, dimension (LDVR,MM)
          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
          contain starting vectors for the inverse iteration for the
          right eigenvectors; the starting vector for each eigenvector
          must be in the same column in which the eigenvector will be
          stored.
          On exit, if SIDE = 'R' or 'B', the right eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VR, in the same order as their eigenvalues.
          If SIDE = 'L', VR is not referenced.
[in]LDVR
          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
[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 required to
          store the eigenvectors (= the number of .TRUE. elements in
          SELECT).
[out]WORK
          WORK is COMPLEX*16 array, dimension (N*N)
[out]RWORK
          RWORK is DOUBLE PRECISION array, dimension (N)
[out]IFAILL
          IFAILL is INTEGER array, dimension (MM)
          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
          eigenvector in the i-th column of VL (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
          eigenvector converged satisfactorily.
          If SIDE = 'R', IFAILL is not referenced.
[out]IFAILR
          IFAILR is INTEGER array, dimension (MM)
          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
          eigenvector in the i-th column of VR (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
          eigenvector converged satisfactorily.
          If SIDE = 'L', IFAILR is not referenced.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, i is the number of eigenvectors which
                failed to converge; see IFAILL and IFAILR for further
                details.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  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 242 of file zhsein.f.

245 *
246 * -- LAPACK computational routine --
247 * -- LAPACK is a software package provided by Univ. of Tennessee, --
248 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
249 *
250 * .. Scalar Arguments ..
251  CHARACTER EIGSRC, INITV, SIDE
252  INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
253 * ..
254 * .. Array Arguments ..
255  LOGICAL SELECT( * )
256  INTEGER IFAILL( * ), IFAILR( * )
257  DOUBLE PRECISION RWORK( * )
258  COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
259  $ W( * ), WORK( * )
260 * ..
261 *
262 * =====================================================================
263 *
264 * .. Parameters ..
265  COMPLEX*16 ZERO
266  parameter( zero = ( 0.0d+0, 0.0d+0 ) )
267  DOUBLE PRECISION RZERO
268  parameter( rzero = 0.0d+0 )
269 * ..
270 * .. Local Scalars ..
271  LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, RIGHTV
272  INTEGER I, IINFO, K, KL, KLN, KR, KS, LDWORK
273  DOUBLE PRECISION EPS3, HNORM, SMLNUM, ULP, UNFL
274  COMPLEX*16 CDUM, WK
275 * ..
276 * .. External Functions ..
277  LOGICAL LSAME, DISNAN
278  DOUBLE PRECISION DLAMCH, ZLANHS
279  EXTERNAL lsame, dlamch, zlanhs, disnan
280 * ..
281 * .. External Subroutines ..
282  EXTERNAL xerbla, zlaein
283 * ..
284 * .. Intrinsic Functions ..
285  INTRINSIC abs, dble, dimag, max
286 * ..
287 * .. Statement Functions ..
288  DOUBLE PRECISION CABS1
289 * ..
290 * .. Statement Function definitions ..
291  cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
292 * ..
293 * .. Executable Statements ..
294 *
295 * Decode and test the input parameters.
296 *
297  bothv = lsame( side, 'B' )
298  rightv = lsame( side, 'R' ) .OR. bothv
299  leftv = lsame( side, 'L' ) .OR. bothv
300 *
301  fromqr = lsame( eigsrc, 'Q' )
302 *
303  noinit = lsame( initv, 'N' )
304 *
305 * Set M to the number of columns required to store the selected
306 * eigenvectors.
307 *
308  m = 0
309  DO 10 k = 1, n
310  IF( SELECT( k ) )
311  $ m = m + 1
312  10 CONTINUE
313 *
314  info = 0
315  IF( .NOT.rightv .AND. .NOT.leftv ) THEN
316  info = -1
317  ELSE IF( .NOT.fromqr .AND. .NOT.lsame( eigsrc, 'N' ) ) THEN
318  info = -2
319  ELSE IF( .NOT.noinit .AND. .NOT.lsame( initv, 'U' ) ) THEN
320  info = -3
321  ELSE IF( n.LT.0 ) THEN
322  info = -5
323  ELSE IF( ldh.LT.max( 1, n ) ) THEN
324  info = -7
325  ELSE IF( ldvl.LT.1 .OR. ( leftv .AND. ldvl.LT.n ) ) THEN
326  info = -10
327  ELSE IF( ldvr.LT.1 .OR. ( rightv .AND. ldvr.LT.n ) ) THEN
328  info = -12
329  ELSE IF( mm.LT.m ) THEN
330  info = -13
331  END IF
332  IF( info.NE.0 ) THEN
333  CALL xerbla( 'ZHSEIN', -info )
334  RETURN
335  END IF
336 *
337 * Quick return if possible.
338 *
339  IF( n.EQ.0 )
340  $ RETURN
341 *
342 * Set machine-dependent constants.
343 *
344  unfl = dlamch( 'Safe minimum' )
345  ulp = dlamch( 'Precision' )
346  smlnum = unfl*( n / ulp )
347 *
348  ldwork = n
349 *
350  kl = 1
351  kln = 0
352  IF( fromqr ) THEN
353  kr = 0
354  ELSE
355  kr = n
356  END IF
357  ks = 1
358 *
359  DO 100 k = 1, n
360  IF( SELECT( k ) ) THEN
361 *
362 * Compute eigenvector(s) corresponding to W(K).
363 *
364  IF( fromqr ) THEN
365 *
366 * If affiliation of eigenvalues is known, check whether
367 * the matrix splits.
368 *
369 * Determine KL and KR such that 1 <= KL <= K <= KR <= N
370 * and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
371 * KR = N).
372 *
373 * Then inverse iteration can be performed with the
374 * submatrix H(KL:N,KL:N) for a left eigenvector, and with
375 * the submatrix H(1:KR,1:KR) for a right eigenvector.
376 *
377  DO 20 i = k, kl + 1, -1
378  IF( h( i, i-1 ).EQ.zero )
379  $ GO TO 30
380  20 CONTINUE
381  30 CONTINUE
382  kl = i
383  IF( k.GT.kr ) THEN
384  DO 40 i = k, n - 1
385  IF( h( i+1, i ).EQ.zero )
386  $ GO TO 50
387  40 CONTINUE
388  50 CONTINUE
389  kr = i
390  END IF
391  END IF
392 *
393  IF( kl.NE.kln ) THEN
394  kln = kl
395 *
396 * Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
397 * has not ben computed before.
398 *
399  hnorm = zlanhs( 'I', kr-kl+1, h( kl, kl ), ldh, rwork )
400  IF( disnan( hnorm ) ) THEN
401  info = -6
402  RETURN
403  ELSE IF( hnorm.GT.rzero ) THEN
404  eps3 = hnorm*ulp
405  ELSE
406  eps3 = smlnum
407  END IF
408  END IF
409 *
410 * Perturb eigenvalue if it is close to any previous
411 * selected eigenvalues affiliated to the submatrix
412 * H(KL:KR,KL:KR). Close roots are modified by EPS3.
413 *
414  wk = w( k )
415  60 CONTINUE
416  DO 70 i = k - 1, kl, -1
417  IF( SELECT( i ) .AND. cabs1( w( i )-wk ).LT.eps3 ) THEN
418  wk = wk + eps3
419  GO TO 60
420  END IF
421  70 CONTINUE
422  w( k ) = wk
423 *
424  IF( leftv ) THEN
425 *
426 * Compute left eigenvector.
427 *
428  CALL zlaein( .false., noinit, n-kl+1, h( kl, kl ), ldh,
429  $ wk, vl( kl, ks ), work, ldwork, rwork, eps3,
430  $ smlnum, iinfo )
431  IF( iinfo.GT.0 ) THEN
432  info = info + 1
433  ifaill( ks ) = k
434  ELSE
435  ifaill( ks ) = 0
436  END IF
437  DO 80 i = 1, kl - 1
438  vl( i, ks ) = zero
439  80 CONTINUE
440  END IF
441  IF( rightv ) THEN
442 *
443 * Compute right eigenvector.
444 *
445  CALL zlaein( .true., noinit, kr, h, ldh, wk, vr( 1, ks ),
446  $ work, ldwork, rwork, eps3, smlnum, iinfo )
447  IF( iinfo.GT.0 ) THEN
448  info = info + 1
449  ifailr( ks ) = k
450  ELSE
451  ifailr( ks ) = 0
452  END IF
453  DO 90 i = kr + 1, n
454  vr( i, ks ) = zero
455  90 CONTINUE
456  END IF
457  ks = ks + 1
458  END IF
459  100 CONTINUE
460 *
461  RETURN
462 *
463 * End of ZHSEIN
464 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:59
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zlaein(RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iterat...
Definition: zlaein.f:149
double precision function zlanhs(NORM, N, A, LDA, WORK)
ZLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlanhs.f:109
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