Annotation of rpl/lapack/lapack/zhsein.f, revision 1.9
1.8 bertrand 1: *> \brief \b ZHSEIN
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHSEIN + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhsein.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhsein.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhsein.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHSEIN( 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: * DOUBLE PRECISION RWORK( * )
33: * COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
34: * $ W( * ), WORK( * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZHSEIN 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 ZHSEQR; 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 ZHSEIN 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, ZHSEIN 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*16 array, dimension (LDH,N)
106: *> The upper Hessenberg matrix H.
107: *> \endverbatim
108: *>
109: *> \param[in] LDH
110: *> \verbatim
111: *> LDH is INTEGER
112: *> The leading dimension of the array H. LDH >= max(1,N).
113: *> \endverbatim
114: *>
115: *> \param[in,out] W
116: *> \verbatim
117: *> W is COMPLEX*16 array, dimension (N)
118: *> On entry, the eigenvalues of H.
119: *> On exit, the real parts of W may have been altered since
120: *> close eigenvalues are perturbed slightly in searching for
121: *> independent eigenvectors.
122: *> \endverbatim
123: *>
124: *> \param[in,out] VL
125: *> \verbatim
126: *> VL is COMPLEX*16 array, dimension (LDVL,MM)
127: *> On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
128: *> contain starting vectors for the inverse iteration for the
129: *> left eigenvectors; the starting vector for each eigenvector
130: *> must be in the same column in which the eigenvector will be
131: *> stored.
132: *> On exit, if SIDE = 'L' or 'B', the left eigenvectors
133: *> specified by SELECT will be stored consecutively in the
134: *> columns of VL, in the same order as their eigenvalues.
135: *> If SIDE = 'R', VL is not referenced.
136: *> \endverbatim
137: *>
138: *> \param[in] LDVL
139: *> \verbatim
140: *> LDVL is INTEGER
141: *> The leading dimension of the array VL.
142: *> LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
143: *> \endverbatim
144: *>
145: *> \param[in,out] VR
146: *> \verbatim
147: *> VR is COMPLEX*16 array, dimension (LDVR,MM)
148: *> On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
149: *> contain starting vectors for the inverse iteration for the
150: *> right eigenvectors; the starting vector for each eigenvector
151: *> must be in the same column in which the eigenvector will be
152: *> stored.
153: *> On exit, if SIDE = 'R' or 'B', the right eigenvectors
154: *> specified by SELECT will be stored consecutively in the
155: *> columns of VR, in the same order as their eigenvalues.
156: *> If SIDE = 'L', VR is not referenced.
157: *> \endverbatim
158: *>
159: *> \param[in] LDVR
160: *> \verbatim
161: *> LDVR is INTEGER
162: *> The leading dimension of the array VR.
163: *> LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
164: *> \endverbatim
165: *>
166: *> \param[in] MM
167: *> \verbatim
168: *> MM is INTEGER
169: *> The number of columns in the arrays VL and/or VR. MM >= M.
170: *> \endverbatim
171: *>
172: *> \param[out] M
173: *> \verbatim
174: *> M is INTEGER
175: *> The number of columns in the arrays VL and/or VR required to
176: *> store the eigenvectors (= the number of .TRUE. elements in
177: *> SELECT).
178: *> \endverbatim
179: *>
180: *> \param[out] WORK
181: *> \verbatim
182: *> WORK is COMPLEX*16 array, dimension (N*N)
183: *> \endverbatim
184: *>
185: *> \param[out] RWORK
186: *> \verbatim
187: *> RWORK is DOUBLE PRECISION array, dimension (N)
188: *> \endverbatim
189: *>
190: *> \param[out] IFAILL
191: *> \verbatim
192: *> IFAILL is INTEGER array, dimension (MM)
193: *> If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
194: *> eigenvector in the i-th column of VL (corresponding to the
195: *> eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
196: *> eigenvector converged satisfactorily.
197: *> If SIDE = 'R', IFAILL is not referenced.
198: *> \endverbatim
199: *>
200: *> \param[out] IFAILR
201: *> \verbatim
202: *> IFAILR is INTEGER array, dimension (MM)
203: *> If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
204: *> eigenvector in the i-th column of VR (corresponding to the
205: *> eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
206: *> eigenvector converged satisfactorily.
207: *> If SIDE = 'L', IFAILR is not referenced.
208: *> \endverbatim
209: *>
210: *> \param[out] INFO
211: *> \verbatim
212: *> INFO is INTEGER
213: *> = 0: successful exit
214: *> < 0: if INFO = -i, the i-th argument had an illegal value
215: *> > 0: if INFO = i, i is the number of eigenvectors which
216: *> failed to converge; see IFAILL and IFAILR for further
217: *> details.
218: *> \endverbatim
219: *
220: * Authors:
221: * ========
222: *
223: *> \author Univ. of Tennessee
224: *> \author Univ. of California Berkeley
225: *> \author Univ. of Colorado Denver
226: *> \author NAG Ltd.
227: *
228: *> \date November 2011
229: *
230: *> \ingroup complex16OTHERcomputational
231: *
232: *> \par Further Details:
233: * =====================
234: *>
235: *> \verbatim
236: *>
237: *> Each eigenvector is normalized so that the element of largest
238: *> magnitude has magnitude 1; here the magnitude of a complex number
239: *> (x,y) is taken to be |x|+|y|.
240: *> \endverbatim
241: *>
242: * =====================================================================
1.1 bertrand 243: SUBROUTINE ZHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
244: $ LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
245: $ IFAILR, INFO )
246: *
1.8 bertrand 247: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 248: * -- LAPACK is a software package provided by Univ. of Tennessee, --
249: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 bertrand 250: * November 2011
1.1 bertrand 251: *
252: * .. Scalar Arguments ..
253: CHARACTER EIGSRC, INITV, SIDE
254: INTEGER INFO, LDH, LDVL, LDVR, M, MM, N
255: * ..
256: * .. Array Arguments ..
257: LOGICAL SELECT( * )
258: INTEGER IFAILL( * ), IFAILR( * )
259: DOUBLE PRECISION RWORK( * )
260: COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
261: $ W( * ), WORK( * )
262: * ..
263: *
264: * =====================================================================
265: *
266: * .. Parameters ..
267: COMPLEX*16 ZERO
268: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) )
269: DOUBLE PRECISION RZERO
270: PARAMETER ( RZERO = 0.0D+0 )
271: * ..
272: * .. Local Scalars ..
273: LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, RIGHTV
274: INTEGER I, IINFO, K, KL, KLN, KR, KS, LDWORK
275: DOUBLE PRECISION EPS3, HNORM, SMLNUM, ULP, UNFL
276: COMPLEX*16 CDUM, WK
277: * ..
278: * .. External Functions ..
279: LOGICAL LSAME
280: DOUBLE PRECISION DLAMCH, ZLANHS
281: EXTERNAL LSAME, DLAMCH, ZLANHS
282: * ..
283: * .. External Subroutines ..
284: EXTERNAL XERBLA, ZLAEIN
285: * ..
286: * .. Intrinsic Functions ..
287: INTRINSIC ABS, DBLE, DIMAG, MAX
288: * ..
289: * .. Statement Functions ..
290: DOUBLE PRECISION CABS1
291: * ..
292: * .. Statement Function definitions ..
293: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
294: * ..
295: * .. Executable Statements ..
296: *
297: * Decode and test the input parameters.
298: *
299: BOTHV = LSAME( SIDE, 'B' )
300: RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
301: LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
302: *
303: FROMQR = LSAME( EIGSRC, 'Q' )
304: *
305: NOINIT = LSAME( INITV, 'N' )
306: *
307: * Set M to the number of columns required to store the selected
308: * eigenvectors.
309: *
310: M = 0
311: DO 10 K = 1, N
312: IF( SELECT( K ) )
313: $ M = M + 1
314: 10 CONTINUE
315: *
316: INFO = 0
317: IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
318: INFO = -1
319: ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN
320: INFO = -2
321: ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN
322: INFO = -3
323: ELSE IF( N.LT.0 ) THEN
324: INFO = -5
325: ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
326: INFO = -7
327: ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
328: INFO = -10
329: ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
330: INFO = -12
331: ELSE IF( MM.LT.M ) THEN
332: INFO = -13
333: END IF
334: IF( INFO.NE.0 ) THEN
335: CALL XERBLA( 'ZHSEIN', -INFO )
336: RETURN
337: END IF
338: *
339: * Quick return if possible.
340: *
341: IF( N.EQ.0 )
342: $ RETURN
343: *
344: * Set machine-dependent constants.
345: *
346: UNFL = DLAMCH( 'Safe minimum' )
347: ULP = DLAMCH( 'Precision' )
348: SMLNUM = UNFL*( N / ULP )
349: *
350: LDWORK = N
351: *
352: KL = 1
353: KLN = 0
354: IF( FROMQR ) THEN
355: KR = 0
356: ELSE
357: KR = N
358: END IF
359: KS = 1
360: *
361: DO 100 K = 1, N
362: IF( SELECT( K ) ) THEN
363: *
364: * Compute eigenvector(s) corresponding to W(K).
365: *
366: IF( FROMQR ) THEN
367: *
368: * If affiliation of eigenvalues is known, check whether
369: * the matrix splits.
370: *
371: * Determine KL and KR such that 1 <= KL <= K <= KR <= N
372: * and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
373: * KR = N).
374: *
375: * Then inverse iteration can be performed with the
376: * submatrix H(KL:N,KL:N) for a left eigenvector, and with
377: * the submatrix H(1:KR,1:KR) for a right eigenvector.
378: *
379: DO 20 I = K, KL + 1, -1
380: IF( H( I, I-1 ).EQ.ZERO )
381: $ GO TO 30
382: 20 CONTINUE
383: 30 CONTINUE
384: KL = I
385: IF( K.GT.KR ) THEN
386: DO 40 I = K, N - 1
387: IF( H( I+1, I ).EQ.ZERO )
388: $ GO TO 50
389: 40 CONTINUE
390: 50 CONTINUE
391: KR = I
392: END IF
393: END IF
394: *
395: IF( KL.NE.KLN ) THEN
396: KLN = KL
397: *
398: * Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
399: * has not ben computed before.
400: *
401: HNORM = ZLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, RWORK )
402: IF( HNORM.GT.RZERO ) THEN
403: EPS3 = HNORM*ULP
404: ELSE
405: EPS3 = SMLNUM
406: END IF
407: END IF
408: *
409: * Perturb eigenvalue if it is close to any previous
410: * selected eigenvalues affiliated to the submatrix
411: * H(KL:KR,KL:KR). Close roots are modified by EPS3.
412: *
413: WK = W( K )
414: 60 CONTINUE
415: DO 70 I = K - 1, KL, -1
416: IF( SELECT( I ) .AND. CABS1( W( I )-WK ).LT.EPS3 ) THEN
417: WK = WK + EPS3
418: GO TO 60
419: END IF
420: 70 CONTINUE
421: W( K ) = WK
422: *
423: IF( LEFTV ) THEN
424: *
425: * Compute left eigenvector.
426: *
427: CALL ZLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ), LDH,
428: $ WK, VL( KL, KS ), WORK, LDWORK, RWORK, EPS3,
429: $ SMLNUM, IINFO )
430: IF( IINFO.GT.0 ) THEN
431: INFO = INFO + 1
432: IFAILL( KS ) = K
433: ELSE
434: IFAILL( KS ) = 0
435: END IF
436: DO 80 I = 1, KL - 1
437: VL( I, KS ) = ZERO
438: 80 CONTINUE
439: END IF
440: IF( RIGHTV ) THEN
441: *
442: * Compute right eigenvector.
443: *
444: CALL ZLAEIN( .TRUE., NOINIT, KR, H, LDH, WK, VR( 1, KS ),
445: $ WORK, LDWORK, RWORK, EPS3, SMLNUM, IINFO )
446: IF( IINFO.GT.0 ) THEN
447: INFO = INFO + 1
448: IFAILR( KS ) = K
449: ELSE
450: IFAILR( KS ) = 0
451: END IF
452: DO 90 I = KR + 1, N
453: VR( I, KS ) = ZERO
454: 90 CONTINUE
455: END IF
456: KS = KS + 1
457: END IF
458: 100 CONTINUE
459: *
460: RETURN
461: *
462: * End of ZHSEIN
463: *
464: END
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