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
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11: *> [TGZ]</a>
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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: *> 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*16 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*16 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*16 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*16 array, dimension (N*N)
184: *> \endverbatim
185: *>
186: *> \param[out] RWORK
187: *> \verbatim
188: *> RWORK is DOUBLE PRECISION 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 complex16OTHERcomputational
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 ZHSEIN( 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: 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: *
465: END
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