 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ chsein()

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

CHSEIN

Purpose:
``` CHSEIN 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 CHSEQR; 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 CHSEIN 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, CHSEIN 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 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 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 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 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 array, dimension (N*N)` [out] RWORK ` RWORK is REAL 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 chsein.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 REAL RWORK( * )
258 COMPLEX H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
259 \$ W( * ), WORK( * )
260* ..
261*
262* =====================================================================
263*
264* .. Parameters ..
265 COMPLEX ZERO
266 parameter( zero = ( 0.0e+0, 0.0e+0 ) )
267 REAL RZERO
268 parameter( rzero = 0.0e+0 )
269* ..
270* .. Local Scalars ..
271 LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, RIGHTV
272 INTEGER I, IINFO, K, KL, KLN, KR, KS, LDWORK
273 REAL EPS3, HNORM, SMLNUM, ULP, UNFL
274 COMPLEX CDUM, WK
275* ..
276* .. External Functions ..
277 LOGICAL LSAME, SISNAN
278 REAL CLANHS, SLAMCH
279 EXTERNAL lsame, clanhs, slamch, sisnan
280* ..
281* .. External Subroutines ..
282 EXTERNAL claein, xerbla
283* ..
284* .. Intrinsic Functions ..
285 INTRINSIC abs, aimag, max, real
286* ..
287* .. Statement Functions ..
288 REAL CABS1
289* ..
290* .. Statement Function definitions ..
291 cabs1( cdum ) = abs( real( cdum ) ) + abs( aimag( 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( 'CHSEIN', -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 = slamch( 'Safe minimum' )
345 ulp = slamch( '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 = clanhs( 'I', kr-kl+1, h( kl, kl ), ldh, rwork )
400 IF( sisnan( 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 claein( .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 claein( .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 CHSEIN
464*
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:59
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function clanhs(NORM, N, A, LDA, WORK)
CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clanhs.f:109
subroutine claein(RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iterat...
Definition: claein.f:149
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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