1: RECURSIVE SUBROUTINE ZUNCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS,
2: $ SIGNS, M, P, Q, X11, LDX11, X12,
3: $ LDX12, X21, LDX21, X22, LDX22, THETA,
4: $ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
5: $ LDV2T, WORK, LWORK, RWORK, LRWORK,
6: $ IWORK, INFO )
7: IMPLICIT NONE
8: *
9: * -- LAPACK routine (version 3.3.1) --
10: *
11: * -- Contributed by Brian Sutton of the Randolph-Macon College --
12: * -- November 2010
13: *
14: * -- LAPACK is a software package provided by Univ. of Tennessee, --
15: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
16: *
17: * @precisions normal z -> c
18: *
19: * .. Scalar Arguments ..
20: CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
21: INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
22: $ LDX21, LDX22, LRWORK, LWORK, M, P, Q
23: * ..
24: * .. Array Arguments ..
25: INTEGER IWORK( * )
26: DOUBLE PRECISION THETA( * )
27: DOUBLE PRECISION RWORK( * )
28: COMPLEX*16 U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
29: $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
30: $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
31: $ * )
32: * ..
33: *
34: * Purpose
35: * =======
36: *
37: * ZUNCSD computes the CS decomposition of an M-by-M partitioned
38: * unitary matrix X:
39: *
40: * [ I 0 0 | 0 0 0 ]
41: * [ 0 C 0 | 0 -S 0 ]
42: * [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**H
43: * X = [-----------] = [---------] [---------------------] [---------] .
44: * [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
45: * [ 0 S 0 | 0 C 0 ]
46: * [ 0 0 I | 0 0 0 ]
47: *
48: * X11 is P-by-Q. The unitary matrices U1, U2, V1, and V2 are P-by-P,
49: * (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
50: * R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
51: * which R = MIN(P,M-P,Q,M-Q).
52: *
53: * Arguments
54: * =========
55: *
56: * JOBU1 (input) CHARACTER
57: * = 'Y': U1 is computed;
58: * otherwise: U1 is not computed.
59: *
60: * JOBU2 (input) CHARACTER
61: * = 'Y': U2 is computed;
62: * otherwise: U2 is not computed.
63: *
64: * JOBV1T (input) CHARACTER
65: * = 'Y': V1T is computed;
66: * otherwise: V1T is not computed.
67: *
68: * JOBV2T (input) CHARACTER
69: * = 'Y': V2T is computed;
70: * otherwise: V2T is not computed.
71: *
72: * TRANS (input) CHARACTER
73: * = 'T': X, U1, U2, V1T, and V2T are stored in row-major
74: * order;
75: * otherwise: X, U1, U2, V1T, and V2T are stored in column-
76: * major order.
77: *
78: * SIGNS (input) CHARACTER
79: * = 'O': The lower-left block is made nonpositive (the
80: * "other" convention);
81: * otherwise: The upper-right block is made nonpositive (the
82: * "default" convention).
83: *
84: * M (input) INTEGER
85: * The number of rows and columns in X.
86: *
87: * P (input) INTEGER
88: * The number of rows in X11 and X12. 0 <= P <= M.
89: *
90: * Q (input) INTEGER
91: * The number of columns in X11 and X21. 0 <= Q <= M.
92: *
93: * X (input/workspace) COMPLEX*16 array, dimension (LDX,M)
94: * On entry, the unitary matrix whose CSD is desired.
95: *
96: * LDX (input) INTEGER
97: * The leading dimension of X. LDX >= MAX(1,M).
98: *
99: * THETA (output) DOUBLE PRECISION array, dimension (R), in which R =
100: * MIN(P,M-P,Q,M-Q).
101: * C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
102: * S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
103: *
104: * U1 (output) COMPLEX*16 array, dimension (P)
105: * If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1.
106: *
107: * LDU1 (input) INTEGER
108: * The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
109: * MAX(1,P).
110: *
111: * U2 (output) COMPLEX*16 array, dimension (M-P)
112: * If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
113: * matrix U2.
114: *
115: * LDU2 (input) INTEGER
116: * The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
117: * MAX(1,M-P).
118: *
119: * V1T (output) COMPLEX*16 array, dimension (Q)
120: * If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
121: * matrix V1**H.
122: *
123: * LDV1T (input) INTEGER
124: * The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
125: * MAX(1,Q).
126: *
127: * V2T (output) COMPLEX*16 array, dimension (M-Q)
128: * If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary
129: * matrix V2**H.
130: *
131: * LDV2T (input) INTEGER
132: * The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
133: * MAX(1,M-Q).
134: *
135: * WORK (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
136: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
137: *
138: * LWORK (input) INTEGER
139: * The dimension of the array WORK.
140: *
141: * If LWORK = -1, then a workspace query is assumed; the routine
142: * only calculates the optimal size of the WORK array, returns
143: * this value as the first entry of the work array, and no error
144: * message related to LWORK is issued by XERBLA.
145: *
146: * RWORK (workspace) DOUBLE PRECISION array, dimension MAX(1,LRWORK)
147: * On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
148: * If INFO > 0 on exit, RWORK(2:R) contains the values PHI(1),
149: * ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
150: * define the matrix in intermediate bidiagonal-block form
151: * remaining after nonconvergence. INFO specifies the number
152: * of nonzero PHI's.
153: *
154: * LRWORK (input) INTEGER
155: * The dimension of the array RWORK.
156: *
157: * If LRWORK = -1, then a workspace query is assumed; the routine
158: * only calculates the optimal size of the RWORK array, returns
159: * this value as the first entry of the work array, and no error
160: * message related to LRWORK is issued by XERBLA.
161: *
162: * IWORK (workspace) INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
163: *
164: * INFO (output) INTEGER
165: * = 0: successful exit.
166: * < 0: if INFO = -i, the i-th argument had an illegal value.
167: * > 0: ZBBCSD did not converge. See the description of RWORK
168: * above for details.
169: *
170: * Reference
171: * =========
172: *
173: * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
174: * Algorithms, 50(1):33-65, 2009.
175: *
176: * ===================================================================
177: *
178: * .. Parameters ..
179: DOUBLE PRECISION REALONE
180: PARAMETER ( REALONE = 1.0D0 )
181: COMPLEX*16 NEGONE, ONE, PIOVER2, ZERO
182: PARAMETER ( NEGONE = (-1.0D0,0.0D0), ONE = (1.0D0,0.0D0),
183: $ PIOVER2 = 1.57079632679489662D0,
184: $ ZERO = (0.0D0,0.0D0) )
185: * ..
186: * .. Local Scalars ..
187: CHARACTER TRANST, SIGNST
188: INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
189: $ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
190: $ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
191: $ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
192: $ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
193: $ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
194: $ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
195: $ LORGQRWORKOPT, LWORKMIN, LWORKOPT
196: LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
197: $ WANTV1T, WANTV2T
198: INTEGER LRWORKMIN, LRWORKOPT
199: LOGICAL LRQUERY
200: * ..
201: * .. External Subroutines ..
202: EXTERNAL XERBLA, ZBBCSD, ZLACPY, ZLAPMR, ZLAPMT, ZLASCL,
203: $ ZLASET, ZUNBDB, ZUNGLQ, ZUNGQR
204: * ..
205: * .. External Functions ..
206: LOGICAL LSAME
207: EXTERNAL LSAME
208: * ..
209: * .. Intrinsic Functions
210: INTRINSIC COS, INT, MAX, MIN, SIN
211: * ..
212: * .. Executable Statements ..
213: *
214: * Test input arguments
215: *
216: INFO = 0
217: WANTU1 = LSAME( JOBU1, 'Y' )
218: WANTU2 = LSAME( JOBU2, 'Y' )
219: WANTV1T = LSAME( JOBV1T, 'Y' )
220: WANTV2T = LSAME( JOBV2T, 'Y' )
221: COLMAJOR = .NOT. LSAME( TRANS, 'T' )
222: DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' )
223: LQUERY = LWORK .EQ. -1
224: LRQUERY = LRWORK .EQ. -1
225: IF( M .LT. 0 ) THEN
226: INFO = -7
227: ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
228: INFO = -8
229: ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
230: INFO = -9
231: ELSE IF( ( COLMAJOR .AND. LDX11 .LT. MAX(1,P) ) .OR.
232: $ ( .NOT.COLMAJOR .AND. LDX11 .LT. MAX(1,Q) ) ) THEN
233: INFO = -11
234: ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
235: INFO = -14
236: ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
237: INFO = -16
238: ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
239: INFO = -18
240: ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
241: INFO = -20
242: END IF
243: *
244: * Work with transpose if convenient
245: *
246: IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN
247: IF( COLMAJOR ) THEN
248: TRANST = 'T'
249: ELSE
250: TRANST = 'N'
251: END IF
252: IF( DEFAULTSIGNS ) THEN
253: SIGNST = 'O'
254: ELSE
255: SIGNST = 'D'
256: END IF
257: CALL ZUNCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST, M,
258: $ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22,
259: $ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1,
260: $ U2, LDU2, WORK, LWORK, RWORK, LRWORK, IWORK,
261: $ INFO )
262: RETURN
263: END IF
264: *
265: * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
266: * convenient
267: *
268: IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN
269: IF( DEFAULTSIGNS ) THEN
270: SIGNST = 'O'
271: ELSE
272: SIGNST = 'D'
273: END IF
274: CALL ZUNCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M,
275: $ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11,
276: $ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T,
277: $ LDV1T, WORK, LWORK, RWORK, LRWORK, IWORK, INFO )
278: RETURN
279: END IF
280: *
281: * Compute workspace
282: *
283: IF( INFO .EQ. 0 ) THEN
284: *
285: * Real workspace
286: *
287: IPHI = 2
288: IB11D = IPHI + MAX( 1, Q - 1 )
289: IB11E = IB11D + MAX( 1, Q )
290: IB12D = IB11E + MAX( 1, Q - 1 )
291: IB12E = IB12D + MAX( 1, Q )
292: IB21D = IB12E + MAX( 1, Q - 1 )
293: IB21E = IB21D + MAX( 1, Q )
294: IB22D = IB21E + MAX( 1, Q - 1 )
295: IB22E = IB22D + MAX( 1, Q )
296: IBBCSD = IB22E + MAX( 1, Q - 1 )
297: CALL ZBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 0,
298: $ 0, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, 0,
299: $ 0, 0, 0, 0, 0, 0, 0, RWORK, -1, CHILDINFO )
300: LBBCSDWORKOPT = INT( RWORK(1) )
301: LBBCSDWORKMIN = LBBCSDWORKOPT
302: LRWORKOPT = IBBCSD + LBBCSDWORKOPT - 1
303: LRWORKMIN = IBBCSD + LBBCSDWORKMIN - 1
304: RWORK(1) = LRWORKOPT
305: *
306: * Complex workspace
307: *
308: ITAUP1 = 2
309: ITAUP2 = ITAUP1 + MAX( 1, P )
310: ITAUQ1 = ITAUP2 + MAX( 1, M - P )
311: ITAUQ2 = ITAUQ1 + MAX( 1, Q )
312: IORGQR = ITAUQ2 + MAX( 1, M - Q )
313: CALL ZUNGQR( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
314: $ CHILDINFO )
315: LORGQRWORKOPT = INT( WORK(1) )
316: LORGQRWORKMIN = MAX( 1, M - Q )
317: IORGLQ = ITAUQ2 + MAX( 1, M - Q )
318: CALL ZUNGLQ( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
319: $ CHILDINFO )
320: LORGLQWORKOPT = INT( WORK(1) )
321: LORGLQWORKMIN = MAX( 1, M - Q )
322: IORBDB = ITAUQ2 + MAX( 1, M - Q )
323: CALL ZUNBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
324: $ X21, LDX21, X22, LDX22, 0, 0, 0, 0, 0, 0, WORK,
325: $ -1, CHILDINFO )
326: LORBDBWORKOPT = INT( WORK(1) )
327: LORBDBWORKMIN = LORBDBWORKOPT
328: LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
329: $ IORBDB + LORBDBWORKOPT ) - 1
330: LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
331: $ IORBDB + LORBDBWORKMIN ) - 1
332: WORK(1) = MAX(LWORKOPT,LWORKMIN)
333: *
334: IF( LWORK .LT. LWORKMIN
335: $ .AND. .NOT. ( LQUERY .OR. LRQUERY ) ) THEN
336: INFO = -22
337: ELSE IF( LRWORK .LT. LRWORKMIN
338: $ .AND. .NOT. ( LQUERY .OR. LRQUERY ) ) THEN
339: INFO = -24
340: ELSE
341: LORGQRWORK = LWORK - IORGQR + 1
342: LORGLQWORK = LWORK - IORGLQ + 1
343: LORBDBWORK = LWORK - IORBDB + 1
344: LBBCSDWORK = LRWORK - IBBCSD + 1
345: END IF
346: END IF
347: *
348: * Abort if any illegal arguments
349: *
350: IF( INFO .NE. 0 ) THEN
351: CALL XERBLA( 'ZUNCSD', -INFO )
352: RETURN
353: ELSE IF( LQUERY .OR. LRQUERY ) THEN
354: RETURN
355: END IF
356: *
357: * Transform to bidiagonal block form
358: *
359: CALL ZUNBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
360: $ LDX21, X22, LDX22, THETA, RWORK(IPHI), WORK(ITAUP1),
361: $ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2),
362: $ WORK(IORBDB), LORBDBWORK, CHILDINFO )
363: *
364: * Accumulate Householder reflectors
365: *
366: IF( COLMAJOR ) THEN
367: IF( WANTU1 .AND. P .GT. 0 ) THEN
368: CALL ZLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
369: CALL ZUNGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR),
370: $ LORGQRWORK, INFO)
371: END IF
372: IF( WANTU2 .AND. M-P .GT. 0 ) THEN
373: CALL ZLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
374: CALL ZUNGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
375: $ WORK(IORGQR), LORGQRWORK, INFO )
376: END IF
377: IF( WANTV1T .AND. Q .GT. 0 ) THEN
378: CALL ZLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2),
379: $ LDV1T )
380: V1T(1, 1) = ONE
381: DO J = 2, Q
382: V1T(1,J) = ZERO
383: V1T(J,1) = ZERO
384: END DO
385: CALL ZUNGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
386: $ WORK(IORGLQ), LORGLQWORK, INFO )
387: END IF
388: IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
389: CALL ZLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
390: CALL ZLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
391: $ V2T(P+1,P+1), LDV2T )
392: CALL ZUNGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
393: $ WORK(IORGLQ), LORGLQWORK, INFO )
394: END IF
395: ELSE
396: IF( WANTU1 .AND. P .GT. 0 ) THEN
397: CALL ZLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
398: CALL ZUNGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ),
399: $ LORGLQWORK, INFO)
400: END IF
401: IF( WANTU2 .AND. M-P .GT. 0 ) THEN
402: CALL ZLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
403: CALL ZUNGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
404: $ WORK(IORGLQ), LORGLQWORK, INFO )
405: END IF
406: IF( WANTV1T .AND. Q .GT. 0 ) THEN
407: CALL ZLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2),
408: $ LDV1T )
409: V1T(1, 1) = ONE
410: DO J = 2, Q
411: V1T(1,J) = ZERO
412: V1T(J,1) = ZERO
413: END DO
414: CALL ZUNGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
415: $ WORK(IORGQR), LORGQRWORK, INFO )
416: END IF
417: IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
418: CALL ZLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
419: CALL ZLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
420: $ V2T(P+1,P+1), LDV2T )
421: CALL ZUNGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
422: $ WORK(IORGQR), LORGQRWORK, INFO )
423: END IF
424: END IF
425: *
426: * Compute the CSD of the matrix in bidiagonal-block form
427: *
428: CALL ZBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA,
429: $ RWORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
430: $ LDV2T, RWORK(IB11D), RWORK(IB11E), RWORK(IB12D),
431: $ RWORK(IB12E), RWORK(IB21D), RWORK(IB21E),
432: $ RWORK(IB22D), RWORK(IB22E), RWORK(IBBCSD),
433: $ LBBCSDWORK, INFO )
434: *
435: * Permute rows and columns to place identity submatrices in top-
436: * left corner of (1,1)-block and/or bottom-right corner of (1,2)-
437: * block and/or bottom-right corner of (2,1)-block and/or top-left
438: * corner of (2,2)-block
439: *
440: IF( Q .GT. 0 .AND. WANTU2 ) THEN
441: DO I = 1, Q
442: IWORK(I) = M - P - Q + I
443: END DO
444: DO I = Q + 1, M - P
445: IWORK(I) = I - Q
446: END DO
447: IF( COLMAJOR ) THEN
448: CALL ZLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
449: ELSE
450: CALL ZLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK )
451: END IF
452: END IF
453: IF( M .GT. 0 .AND. WANTV2T ) THEN
454: DO I = 1, P
455: IWORK(I) = M - P - Q + I
456: END DO
457: DO I = P + 1, M - Q
458: IWORK(I) = I - P
459: END DO
460: IF( .NOT. COLMAJOR ) THEN
461: CALL ZLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
462: ELSE
463: CALL ZLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
464: END IF
465: END IF
466: *
467: RETURN
468: *
469: * End ZUNCSD
470: *
471: END
472:
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