 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.```
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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