LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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chsein.f
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1*> \brief \b CHSEIN
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CHSEIN + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chsein.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chsein.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chsein.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
22* LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
23* IFAILR, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER EIGSRC, INITV, SIDE
27* INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
28* ..
29* .. Array Arguments ..
30* LOGICAL SELECT( * )
31* INTEGER IFAILL( * ), IFAILR( * )
32* REAL RWORK( * )
33* COMPLEX H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
34* $ W( * ), WORK( * )
35* ..
36*
37*
38*> \par Purpose:
39* =============
40*>
41*> \verbatim
42*>
43*> CHSEIN uses inverse iteration to find specified right and/or left
44*> eigenvectors of a complex upper Hessenberg matrix H.
45*>
46*> The right eigenvector x and the left eigenvector y of the matrix H
47*> corresponding to an eigenvalue w are defined by:
48*>
49*> H * x = w * x, y**h * H = w * y**h
50*>
51*> where y**h denotes the conjugate transpose of the vector y.
52*> \endverbatim
53*
54* Arguments:
55* ==========
56*
57*> \param[in] SIDE
58*> \verbatim
59*> SIDE is CHARACTER*1
60*> = 'R': compute right eigenvectors only;
61*> = 'L': compute left eigenvectors only;
62*> = 'B': compute both right and left eigenvectors.
63*> \endverbatim
64*>
65*> \param[in] EIGSRC
66*> \verbatim
67*> EIGSRC is CHARACTER*1
68*> Specifies the source of eigenvalues supplied in W:
69*> = 'Q': the eigenvalues were found using CHSEQR; thus, if
70*> H has zero subdiagonal elements, and so is
71*> block-triangular, then the j-th eigenvalue can be
72*> assumed to be an eigenvalue of the block containing
73*> the j-th row/column. This property allows CHSEIN to
74*> perform inverse iteration on just one diagonal block.
75*> = 'N': no assumptions are made on the correspondence
76*> between eigenvalues and diagonal blocks. In this
77*> case, CHSEIN must always perform inverse iteration
78*> using the whole matrix H.
79*> \endverbatim
80*>
81*> \param[in] INITV
82*> \verbatim
83*> INITV is CHARACTER*1
84*> = 'N': no initial vectors are supplied;
85*> = 'U': user-supplied initial vectors are stored in the arrays
86*> VL and/or VR.
87*> \endverbatim
88*>
89*> \param[in] SELECT
90*> \verbatim
91*> SELECT is LOGICAL array, dimension (N)
92*> Specifies the eigenvectors to be computed. To select the
93*> eigenvector corresponding to the eigenvalue W(j),
94*> SELECT(j) must be set to .TRUE..
95*> \endverbatim
96*>
97*> \param[in] N
98*> \verbatim
99*> N is INTEGER
100*> The order of the matrix H. N >= 0.
101*> \endverbatim
102*>
103*> \param[in] H
104*> \verbatim
105*> H is COMPLEX array, dimension (LDH,N)
106*> The upper Hessenberg matrix H.
107*> If a NaN is detected in H, the routine will return with INFO=-6.
108*> \endverbatim
109*>
110*> \param[in] LDH
111*> \verbatim
112*> LDH is INTEGER
113*> The leading dimension of the array H. LDH >= max(1,N).
114*> \endverbatim
115*>
116*> \param[in,out] W
117*> \verbatim
118*> W is COMPLEX array, dimension (N)
119*> On entry, the eigenvalues of H.
120*> On exit, the real parts of W may have been altered since
121*> close eigenvalues are perturbed slightly in searching for
122*> independent eigenvectors.
123*> \endverbatim
124*>
125*> \param[in,out] VL
126*> \verbatim
127*> VL is COMPLEX array, dimension (LDVL,MM)
128*> On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
129*> contain starting vectors for the inverse iteration for the
130*> left eigenvectors; the starting vector for each eigenvector
131*> must be in the same column in which the eigenvector will be
132*> stored.
133*> On exit, if SIDE = 'L' or 'B', the left eigenvectors
134*> specified by SELECT will be stored consecutively in the
135*> columns of VL, in the same order as their eigenvalues.
136*> If SIDE = 'R', VL is not referenced.
137*> \endverbatim
138*>
139*> \param[in] LDVL
140*> \verbatim
141*> LDVL is INTEGER
142*> The leading dimension of the array VL.
143*> LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
144*> \endverbatim
145*>
146*> \param[in,out] VR
147*> \verbatim
148*> VR is COMPLEX array, dimension (LDVR,MM)
149*> On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
150*> contain starting vectors for the inverse iteration for the
151*> right eigenvectors; the starting vector for each eigenvector
152*> must be in the same column in which the eigenvector will be
153*> stored.
154*> On exit, if SIDE = 'R' or 'B', the right eigenvectors
155*> specified by SELECT will be stored consecutively in the
156*> columns of VR, in the same order as their eigenvalues.
157*> If SIDE = 'L', VR is not referenced.
158*> \endverbatim
159*>
160*> \param[in] LDVR
161*> \verbatim
162*> LDVR is INTEGER
163*> The leading dimension of the array VR.
164*> LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
165*> \endverbatim
166*>
167*> \param[in] MM
168*> \verbatim
169*> MM is INTEGER
170*> The number of columns in the arrays VL and/or VR. MM >= M.
171*> \endverbatim
172*>
173*> \param[out] M
174*> \verbatim
175*> M is INTEGER
176*> The number of columns in the arrays VL and/or VR required to
177*> store the eigenvectors (= the number of .TRUE. elements in
178*> SELECT).
179*> \endverbatim
180*>
181*> \param[out] WORK
182*> \verbatim
183*> WORK is COMPLEX array, dimension (N*N)
184*> \endverbatim
185*>
186*> \param[out] RWORK
187*> \verbatim
188*> RWORK is REAL array, dimension (N)
189*> \endverbatim
190*>
191*> \param[out] IFAILL
192*> \verbatim
193*> IFAILL is INTEGER array, dimension (MM)
194*> If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
195*> eigenvector in the i-th column of VL (corresponding to the
196*> eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
197*> eigenvector converged satisfactorily.
198*> If SIDE = 'R', IFAILL is not referenced.
199*> \endverbatim
200*>
201*> \param[out] IFAILR
202*> \verbatim
203*> IFAILR is INTEGER array, dimension (MM)
204*> If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
205*> eigenvector in the i-th column of VR (corresponding to the
206*> eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
207*> eigenvector converged satisfactorily.
208*> If SIDE = 'L', IFAILR is not referenced.
209*> \endverbatim
210*>
211*> \param[out] INFO
212*> \verbatim
213*> INFO is INTEGER
214*> = 0: successful exit
215*> < 0: if INFO = -i, the i-th argument had an illegal value
216*> > 0: if INFO = i, i is the number of eigenvectors which
217*> failed to converge; see IFAILL and IFAILR for further
218*> details.
219*> \endverbatim
220*
221* Authors:
222* ========
223*
224*> \author Univ. of Tennessee
225*> \author Univ. of California Berkeley
226*> \author Univ. of Colorado Denver
227*> \author NAG Ltd.
228*
229*> \ingroup hsein
230*
231*> \par Further Details:
232* =====================
233*>
234*> \verbatim
235*>
236*> Each eigenvector is normalized so that the element of largest
237*> magnitude has magnitude 1; here the magnitude of a complex number
238*> (x,y) is taken to be |x|+|y|.
239*> \endverbatim
240*>
241* =====================================================================
242 SUBROUTINE chsein( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
243 $ LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
244 $ IFAILR, INFO )
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*
465 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine chsein(side, eigsrc, initv, select, n, h, ldh, w, vl, ldvl, vr, ldvr, mm, m, work, rwork, ifaill, ifailr, info)
CHSEIN
Definition chsein.f:245
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