Annotation of rpl/lapack/lapack/ztrevc.f, revision 1.16
1.9 bertrand 1: *> \brief \b ZTREVC
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 9: *> Download ZTREVC + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztrevc.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztrevc.f">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
22: * LDVR, MM, M, WORK, RWORK, INFO )
1.15 bertrand 23: *
1.9 bertrand 24: * .. Scalar Arguments ..
25: * CHARACTER HOWMNY, SIDE
26: * INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
27: * ..
28: * .. Array Arguments ..
29: * LOGICAL SELECT( * )
30: * DOUBLE PRECISION RWORK( * )
31: * COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
32: * $ WORK( * )
33: * ..
1.15 bertrand 34: *
1.9 bertrand 35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZTREVC computes some or all of the right and/or left eigenvectors of
42: *> a complex upper triangular matrix T.
43: *> Matrices of this type are produced by the Schur factorization of
44: *> a complex general matrix: A = Q*T*Q**H, as computed by ZHSEQR.
1.15 bertrand 45: *>
1.9 bertrand 46: *> The right eigenvector x and the left eigenvector y of T corresponding
47: *> to an eigenvalue w are defined by:
1.15 bertrand 48: *>
1.9 bertrand 49: *> T*x = w*x, (y**H)*T = w*(y**H)
1.15 bertrand 50: *>
1.9 bertrand 51: *> where y**H denotes the conjugate transpose of the vector y.
52: *> The eigenvalues are not input to this routine, but are read directly
53: *> from the diagonal of T.
1.15 bertrand 54: *>
1.9 bertrand 55: *> This routine returns the matrices X and/or Y of right and left
56: *> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
57: *> input matrix. If Q is the unitary factor that reduces a matrix A to
58: *> Schur form T, then Q*X and Q*Y are the matrices of right and left
59: *> eigenvectors of A.
60: *> \endverbatim
61: *
62: * Arguments:
63: * ==========
64: *
65: *> \param[in] SIDE
66: *> \verbatim
67: *> SIDE is CHARACTER*1
68: *> = 'R': compute right eigenvectors only;
69: *> = 'L': compute left eigenvectors only;
70: *> = 'B': compute both right and left eigenvectors.
71: *> \endverbatim
72: *>
73: *> \param[in] HOWMNY
74: *> \verbatim
75: *> HOWMNY is CHARACTER*1
76: *> = 'A': compute all right and/or left eigenvectors;
77: *> = 'B': compute all right and/or left eigenvectors,
78: *> backtransformed using the matrices supplied in
79: *> VR and/or VL;
80: *> = 'S': compute selected right and/or left eigenvectors,
81: *> as indicated by the logical array SELECT.
82: *> \endverbatim
83: *>
84: *> \param[in] SELECT
85: *> \verbatim
86: *> SELECT is LOGICAL array, dimension (N)
87: *> If HOWMNY = 'S', SELECT specifies the eigenvectors to be
88: *> computed.
89: *> The eigenvector corresponding to the j-th eigenvalue is
90: *> computed if SELECT(j) = .TRUE..
91: *> Not referenced if HOWMNY = 'A' or 'B'.
92: *> \endverbatim
93: *>
94: *> \param[in] N
95: *> \verbatim
96: *> N is INTEGER
97: *> The order of the matrix T. N >= 0.
98: *> \endverbatim
99: *>
100: *> \param[in,out] T
101: *> \verbatim
102: *> T is COMPLEX*16 array, dimension (LDT,N)
103: *> The upper triangular matrix T. T is modified, but restored
104: *> on exit.
105: *> \endverbatim
106: *>
107: *> \param[in] LDT
108: *> \verbatim
109: *> LDT is INTEGER
110: *> The leading dimension of the array T. LDT >= max(1,N).
111: *> \endverbatim
112: *>
113: *> \param[in,out] VL
114: *> \verbatim
115: *> VL is COMPLEX*16 array, dimension (LDVL,MM)
116: *> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
117: *> contain an N-by-N matrix Q (usually the unitary matrix Q of
118: *> Schur vectors returned by ZHSEQR).
119: *> On exit, if SIDE = 'L' or 'B', VL contains:
120: *> if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
121: *> if HOWMNY = 'B', the matrix Q*Y;
122: *> if HOWMNY = 'S', the left eigenvectors of T specified by
123: *> SELECT, stored consecutively in the columns
124: *> of VL, in the same order as their
125: *> eigenvalues.
126: *> Not referenced if SIDE = 'R'.
127: *> \endverbatim
128: *>
129: *> \param[in] LDVL
130: *> \verbatim
131: *> LDVL is INTEGER
132: *> The leading dimension of the array VL. LDVL >= 1, and if
133: *> SIDE = 'L' or 'B', LDVL >= N.
134: *> \endverbatim
135: *>
136: *> \param[in,out] VR
137: *> \verbatim
138: *> VR is COMPLEX*16 array, dimension (LDVR,MM)
139: *> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
140: *> contain an N-by-N matrix Q (usually the unitary matrix Q of
141: *> Schur vectors returned by ZHSEQR).
142: *> On exit, if SIDE = 'R' or 'B', VR contains:
143: *> if HOWMNY = 'A', the matrix X of right eigenvectors of T;
144: *> if HOWMNY = 'B', the matrix Q*X;
145: *> if HOWMNY = 'S', the right eigenvectors of T specified by
146: *> SELECT, stored consecutively in the columns
147: *> of VR, in the same order as their
148: *> eigenvalues.
149: *> Not referenced if SIDE = 'L'.
150: *> \endverbatim
151: *>
152: *> \param[in] LDVR
153: *> \verbatim
154: *> LDVR is INTEGER
155: *> The leading dimension of the array VR. LDVR >= 1, and if
156: *> SIDE = 'R' or 'B'; LDVR >= N.
157: *> \endverbatim
158: *>
159: *> \param[in] MM
160: *> \verbatim
161: *> MM is INTEGER
162: *> The number of columns in the arrays VL and/or VR. MM >= M.
163: *> \endverbatim
164: *>
165: *> \param[out] M
166: *> \verbatim
167: *> M is INTEGER
168: *> The number of columns in the arrays VL and/or VR actually
169: *> used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
170: *> is set to N. Each selected eigenvector occupies one
171: *> column.
172: *> \endverbatim
173: *>
174: *> \param[out] WORK
175: *> \verbatim
176: *> WORK is COMPLEX*16 array, dimension (2*N)
177: *> \endverbatim
178: *>
179: *> \param[out] RWORK
180: *> \verbatim
181: *> RWORK is DOUBLE PRECISION array, dimension (N)
182: *> \endverbatim
183: *>
184: *> \param[out] INFO
185: *> \verbatim
186: *> INFO is INTEGER
187: *> = 0: successful exit
188: *> < 0: if INFO = -i, the i-th argument had an illegal value
189: *> \endverbatim
190: *
191: * Authors:
192: * ========
193: *
1.15 bertrand 194: *> \author Univ. of Tennessee
195: *> \author Univ. of California Berkeley
196: *> \author Univ. of Colorado Denver
197: *> \author NAG Ltd.
1.9 bertrand 198: *
1.15 bertrand 199: *> \date December 2016
1.9 bertrand 200: *
201: *> \ingroup complex16OTHERcomputational
202: *
203: *> \par Further Details:
204: * =====================
205: *>
206: *> \verbatim
207: *>
208: *> The algorithm used in this program is basically backward (forward)
209: *> substitution, with scaling to make the the code robust against
210: *> possible overflow.
211: *>
212: *> Each eigenvector is normalized so that the element of largest
213: *> magnitude has magnitude 1; here the magnitude of a complex number
214: *> (x,y) is taken to be |x| + |y|.
215: *> \endverbatim
216: *>
217: * =====================================================================
1.1 bertrand 218: SUBROUTINE ZTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
219: $ LDVR, MM, M, WORK, RWORK, INFO )
220: *
1.15 bertrand 221: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 222: * -- LAPACK is a software package provided by Univ. of Tennessee, --
223: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.15 bertrand 224: * December 2016
1.1 bertrand 225: *
226: * .. Scalar Arguments ..
227: CHARACTER HOWMNY, SIDE
228: INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
229: * ..
230: * .. Array Arguments ..
231: LOGICAL SELECT( * )
232: DOUBLE PRECISION RWORK( * )
233: COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
234: $ WORK( * )
235: * ..
236: *
237: * =====================================================================
238: *
239: * .. Parameters ..
240: DOUBLE PRECISION ZERO, ONE
241: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
242: COMPLEX*16 CMZERO, CMONE
243: PARAMETER ( CMZERO = ( 0.0D+0, 0.0D+0 ),
244: $ CMONE = ( 1.0D+0, 0.0D+0 ) )
245: * ..
246: * .. Local Scalars ..
247: LOGICAL ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
248: INTEGER I, II, IS, J, K, KI
249: DOUBLE PRECISION OVFL, REMAX, SCALE, SMIN, SMLNUM, ULP, UNFL
250: COMPLEX*16 CDUM
251: * ..
252: * .. External Functions ..
253: LOGICAL LSAME
254: INTEGER IZAMAX
255: DOUBLE PRECISION DLAMCH, DZASUM
256: EXTERNAL LSAME, IZAMAX, DLAMCH, DZASUM
257: * ..
258: * .. External Subroutines ..
259: EXTERNAL XERBLA, ZCOPY, ZDSCAL, ZGEMV, ZLATRS
260: * ..
261: * .. Intrinsic Functions ..
262: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
263: * ..
264: * .. Statement Functions ..
265: DOUBLE PRECISION CABS1
266: * ..
267: * .. Statement Function definitions ..
268: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
269: * ..
270: * .. Executable Statements ..
271: *
272: * Decode and test the input parameters
273: *
274: BOTHV = LSAME( SIDE, 'B' )
275: RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
276: LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
277: *
278: ALLV = LSAME( HOWMNY, 'A' )
279: OVER = LSAME( HOWMNY, 'B' )
280: SOMEV = LSAME( HOWMNY, 'S' )
281: *
282: * Set M to the number of columns required to store the selected
283: * eigenvectors.
284: *
285: IF( SOMEV ) THEN
286: M = 0
287: DO 10 J = 1, N
288: IF( SELECT( J ) )
289: $ M = M + 1
290: 10 CONTINUE
291: ELSE
292: M = N
293: END IF
294: *
295: INFO = 0
296: IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
297: INFO = -1
298: ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
299: INFO = -2
300: ELSE IF( N.LT.0 ) THEN
301: INFO = -4
302: ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
303: INFO = -6
304: ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
305: INFO = -8
306: ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
307: INFO = -10
308: ELSE IF( MM.LT.M ) THEN
309: INFO = -11
310: END IF
311: IF( INFO.NE.0 ) THEN
312: CALL XERBLA( 'ZTREVC', -INFO )
313: RETURN
314: END IF
315: *
316: * Quick return if possible.
317: *
318: IF( N.EQ.0 )
319: $ RETURN
320: *
321: * Set the constants to control overflow.
322: *
323: UNFL = DLAMCH( 'Safe minimum' )
324: OVFL = ONE / UNFL
325: CALL DLABAD( UNFL, OVFL )
326: ULP = DLAMCH( 'Precision' )
327: SMLNUM = UNFL*( N / ULP )
328: *
329: * Store the diagonal elements of T in working array WORK.
330: *
331: DO 20 I = 1, N
332: WORK( I+N ) = T( I, I )
333: 20 CONTINUE
334: *
335: * Compute 1-norm of each column of strictly upper triangular
336: * part of T to control overflow in triangular solver.
337: *
338: RWORK( 1 ) = ZERO
339: DO 30 J = 2, N
340: RWORK( J ) = DZASUM( J-1, T( 1, J ), 1 )
341: 30 CONTINUE
342: *
343: IF( RIGHTV ) THEN
344: *
345: * Compute right eigenvectors.
346: *
347: IS = M
348: DO 80 KI = N, 1, -1
349: *
350: IF( SOMEV ) THEN
351: IF( .NOT.SELECT( KI ) )
352: $ GO TO 80
353: END IF
354: SMIN = MAX( ULP*( CABS1( T( KI, KI ) ) ), SMLNUM )
355: *
356: WORK( 1 ) = CMONE
357: *
358: * Form right-hand side.
359: *
360: DO 40 K = 1, KI - 1
361: WORK( K ) = -T( K, KI )
362: 40 CONTINUE
363: *
364: * Solve the triangular system:
365: * (T(1:KI-1,1:KI-1) - T(KI,KI))*X = SCALE*WORK.
366: *
367: DO 50 K = 1, KI - 1
368: T( K, K ) = T( K, K ) - T( KI, KI )
369: IF( CABS1( T( K, K ) ).LT.SMIN )
370: $ T( K, K ) = SMIN
371: 50 CONTINUE
372: *
373: IF( KI.GT.1 ) THEN
374: CALL ZLATRS( 'Upper', 'No transpose', 'Non-unit', 'Y',
375: $ KI-1, T, LDT, WORK( 1 ), SCALE, RWORK,
376: $ INFO )
377: WORK( KI ) = SCALE
378: END IF
379: *
380: * Copy the vector x or Q*x to VR and normalize.
381: *
382: IF( .NOT.OVER ) THEN
383: CALL ZCOPY( KI, WORK( 1 ), 1, VR( 1, IS ), 1 )
384: *
385: II = IZAMAX( KI, VR( 1, IS ), 1 )
386: REMAX = ONE / CABS1( VR( II, IS ) )
387: CALL ZDSCAL( KI, REMAX, VR( 1, IS ), 1 )
388: *
389: DO 60 K = KI + 1, N
390: VR( K, IS ) = CMZERO
391: 60 CONTINUE
392: ELSE
393: IF( KI.GT.1 )
394: $ CALL ZGEMV( 'N', N, KI-1, CMONE, VR, LDVR, WORK( 1 ),
395: $ 1, DCMPLX( SCALE ), VR( 1, KI ), 1 )
396: *
397: II = IZAMAX( N, VR( 1, KI ), 1 )
398: REMAX = ONE / CABS1( VR( II, KI ) )
399: CALL ZDSCAL( N, REMAX, VR( 1, KI ), 1 )
400: END IF
401: *
402: * Set back the original diagonal elements of T.
403: *
404: DO 70 K = 1, KI - 1
405: T( K, K ) = WORK( K+N )
406: 70 CONTINUE
407: *
408: IS = IS - 1
409: 80 CONTINUE
410: END IF
411: *
412: IF( LEFTV ) THEN
413: *
414: * Compute left eigenvectors.
415: *
416: IS = 1
417: DO 130 KI = 1, N
418: *
419: IF( SOMEV ) THEN
420: IF( .NOT.SELECT( KI ) )
421: $ GO TO 130
422: END IF
423: SMIN = MAX( ULP*( CABS1( T( KI, KI ) ) ), SMLNUM )
424: *
425: WORK( N ) = CMONE
426: *
427: * Form right-hand side.
428: *
429: DO 90 K = KI + 1, N
430: WORK( K ) = -DCONJG( T( KI, K ) )
431: 90 CONTINUE
432: *
433: * Solve the triangular system:
1.8 bertrand 434: * (T(KI+1:N,KI+1:N) - T(KI,KI))**H * X = SCALE*WORK.
1.1 bertrand 435: *
436: DO 100 K = KI + 1, N
437: T( K, K ) = T( K, K ) - T( KI, KI )
438: IF( CABS1( T( K, K ) ).LT.SMIN )
439: $ T( K, K ) = SMIN
440: 100 CONTINUE
441: *
442: IF( KI.LT.N ) THEN
443: CALL ZLATRS( 'Upper', 'Conjugate transpose', 'Non-unit',
444: $ 'Y', N-KI, T( KI+1, KI+1 ), LDT,
445: $ WORK( KI+1 ), SCALE, RWORK, INFO )
446: WORK( KI ) = SCALE
447: END IF
448: *
449: * Copy the vector x or Q*x to VL and normalize.
450: *
451: IF( .NOT.OVER ) THEN
452: CALL ZCOPY( N-KI+1, WORK( KI ), 1, VL( KI, IS ), 1 )
453: *
454: II = IZAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
455: REMAX = ONE / CABS1( VL( II, IS ) )
456: CALL ZDSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
457: *
458: DO 110 K = 1, KI - 1
459: VL( K, IS ) = CMZERO
460: 110 CONTINUE
461: ELSE
462: IF( KI.LT.N )
463: $ CALL ZGEMV( 'N', N, N-KI, CMONE, VL( 1, KI+1 ), LDVL,
464: $ WORK( KI+1 ), 1, DCMPLX( SCALE ),
465: $ VL( 1, KI ), 1 )
466: *
467: II = IZAMAX( N, VL( 1, KI ), 1 )
468: REMAX = ONE / CABS1( VL( II, KI ) )
469: CALL ZDSCAL( N, REMAX, VL( 1, KI ), 1 )
470: END IF
471: *
472: * Set back the original diagonal elements of T.
473: *
474: DO 120 K = KI + 1, N
475: T( K, K ) = WORK( K+N )
476: 120 CONTINUE
477: *
478: IS = IS + 1
479: 130 CONTINUE
480: END IF
481: *
482: RETURN
483: *
484: * End of ZTREVC
485: *
486: END
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