LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
logical function disnan(din)
DISNAN tests input for NaN.
Definition disnan.f:59
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 dlamch(cmach)
DLAMCH
Definition dlamch.f:69
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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