1: RECURSIVE SUBROUTINE DORCSD( 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, IWORK, INFO )
6: IMPLICIT NONE
7: *
8: * -- LAPACK routine (version 3.3.0) --
9: *
10: * -- Contributed by Brian Sutton of the Randolph-Macon College --
11: * -- November 2010
12: *
13: * -- LAPACK is a software package provided by Univ. of Tennessee, --
14: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
15: *
16: * .. Scalar Arguments ..
17: CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
18: INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
19: $ LDX21, LDX22, LWORK, M, P, Q
20: * ..
21: * .. Array Arguments ..
22: INTEGER IWORK( * )
23: DOUBLE PRECISION THETA( * )
24: DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
25: $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
26: $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
27: $ * )
28: * ..
29: *
30: * Purpose
31: * =======
32: *
33: * DORCSD computes the CS decomposition of an M-by-M partitioned
34: * orthogonal matrix X:
35: *
36: * [ I 0 0 | 0 0 0 ]
37: * [ 0 C 0 | 0 -S 0 ]
38: * [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
39: * X = [-----------] = [---------] [---------------------] [---------] .
40: * [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
41: * [ 0 S 0 | 0 C 0 ]
42: * [ 0 0 I | 0 0 0 ]
43: *
44: * X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
45: * (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
46: * R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
47: * which R = MIN(P,M-P,Q,M-Q).
48: *
49: * Arguments
50: * =========
51: *
52: * JOBU1 (input) CHARACTER
53: * = 'Y': U1 is computed;
54: * otherwise: U1 is not computed.
55: *
56: * JOBU2 (input) CHARACTER
57: * = 'Y': U2 is computed;
58: * otherwise: U2 is not computed.
59: *
60: * JOBV1T (input) CHARACTER
61: * = 'Y': V1T is computed;
62: * otherwise: V1T is not computed.
63: *
64: * JOBV2T (input) CHARACTER
65: * = 'Y': V2T is computed;
66: * otherwise: V2T is not computed.
67: *
68: * TRANS (input) CHARACTER
69: * = 'T': X, U1, U2, V1T, and V2T are stored in row-major
70: * order;
71: * otherwise: X, U1, U2, V1T, and V2T are stored in column-
72: * major order.
73: *
74: * SIGNS (input) CHARACTER
75: * = 'O': The lower-left block is made nonpositive (the
76: * "other" convention);
77: * otherwise: The upper-right block is made nonpositive (the
78: * "default" convention).
79: *
80: * M (input) INTEGER
81: * The number of rows and columns in X.
82: *
83: * P (input) INTEGER
84: * The number of rows in X11 and X12. 0 <= P <= M.
85: *
86: * Q (input) INTEGER
87: * The number of columns in X11 and X21. 0 <= Q <= M.
88: *
89: * X (input/workspace) DOUBLE PRECISION array, dimension (LDX,M)
90: * On entry, the orthogonal matrix whose CSD is desired.
91: *
92: * LDX (input) INTEGER
93: * The leading dimension of X. LDX >= MAX(1,M).
94: *
95: * THETA (output) DOUBLE PRECISION array, dimension (R), in which R =
96: * MIN(P,M-P,Q,M-Q).
97: * C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
98: * S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
99: *
100: * U1 (output) DOUBLE PRECISION array, dimension (P)
101: * If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
102: *
103: * LDU1 (input) INTEGER
104: * The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
105: * MAX(1,P).
106: *
107: * U2 (output) DOUBLE PRECISION array, dimension (M-P)
108: * If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
109: * matrix U2.
110: *
111: * LDU2 (input) INTEGER
112: * The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
113: * MAX(1,M-P).
114: *
115: * V1T (output) DOUBLE PRECISION array, dimension (Q)
116: * If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
117: * matrix V1**T.
118: *
119: * LDV1T (input) INTEGER
120: * The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
121: * MAX(1,Q).
122: *
123: * V2T (output) DOUBLE PRECISION array, dimension (M-Q)
124: * If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
125: * matrix V2**T.
126: *
127: * LDV2T (input) INTEGER
128: * The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
129: * MAX(1,M-Q).
130: *
131: * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
132: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
133: * If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
134: * ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
135: * define the matrix in intermediate bidiagonal-block form
136: * remaining after nonconvergence. INFO specifies the number
137: * of nonzero PHI's.
138: *
139: * LWORK (input) INTEGER
140: * The dimension of the array WORK.
141: *
142: * If LWORK = -1, then a workspace query is assumed; the routine
143: * only calculates the optimal size of the WORK array, returns
144: * this value as the first entry of the work array, and no error
145: * message related to LWORK is issued by XERBLA.
146: *
147: * IWORK (workspace) INTEGER array, dimension (M-Q)
148: *
149: * INFO (output) INTEGER
150: * = 0: successful exit.
151: * < 0: if INFO = -i, the i-th argument had an illegal value.
152: * > 0: DBBCSD did not converge. See the description of WORK
153: * above for details.
154: *
155: * Reference
156: * =========
157: *
158: * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
159: * Algorithms, 50(1):33-65, 2009.
160: *
161: * ===================================================================
162: *
163: * .. Parameters ..
164: DOUBLE PRECISION REALONE
165: PARAMETER ( REALONE = 1.0D0 )
166: DOUBLE PRECISION NEGONE, ONE, PIOVER2, ZERO
167: PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0,
168: $ PIOVER2 = 1.57079632679489662D0,
169: $ ZERO = 0.0D0 )
170: * ..
171: * .. Local Scalars ..
172: CHARACTER TRANST, SIGNST
173: INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
174: $ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
175: $ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
176: $ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
177: $ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
178: $ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
179: $ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
180: $ LORGQRWORKOPT, LWORKMIN, LWORKOPT
181: LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
182: $ WANTV1T, WANTV2T
183: * ..
184: * .. External Subroutines ..
185: EXTERNAL DBBCSD, DLACPY, DLAPMR, DLAPMT, DLASCL, DLASET,
186: $ DORBDB, DORGLQ, DORGQR, XERBLA
187: * ..
188: * .. External Functions ..
189: LOGICAL LSAME
190: EXTERNAL LSAME
191: * ..
192: * .. Intrinsic Functions
193: INTRINSIC COS, INT, MAX, MIN, SIN
194: * ..
195: * .. Executable Statements ..
196: *
197: * Test input arguments
198: *
199: INFO = 0
200: WANTU1 = LSAME( JOBU1, 'Y' )
201: WANTU2 = LSAME( JOBU2, 'Y' )
202: WANTV1T = LSAME( JOBV1T, 'Y' )
203: WANTV2T = LSAME( JOBV2T, 'Y' )
204: COLMAJOR = .NOT. LSAME( TRANS, 'T' )
205: DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' )
206: LQUERY = LWORK .EQ. -1
207: IF( M .LT. 0 ) THEN
208: INFO = -7
209: ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
210: INFO = -8
211: ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
212: INFO = -9
213: ELSE IF( ( COLMAJOR .AND. LDX11 .LT. MAX(1,P) ) .OR.
214: $ ( .NOT.COLMAJOR .AND. LDX11 .LT. MAX(1,Q) ) ) THEN
215: INFO = -11
216: ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
217: INFO = -14
218: ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
219: INFO = -16
220: ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
221: INFO = -18
222: ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
223: INFO = -20
224: END IF
225: *
226: * Work with transpose if convenient
227: *
228: IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN
229: IF( COLMAJOR ) THEN
230: TRANST = 'T'
231: ELSE
232: TRANST = 'N'
233: END IF
234: IF( DEFAULTSIGNS ) THEN
235: SIGNST = 'O'
236: ELSE
237: SIGNST = 'D'
238: END IF
239: CALL DORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST, M,
240: $ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22,
241: $ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1,
242: $ U2, LDU2, WORK, LWORK, IWORK, INFO )
243: RETURN
244: END IF
245: *
246: * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
247: * convenient
248: *
249: IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN
250: IF( DEFAULTSIGNS ) THEN
251: SIGNST = 'O'
252: ELSE
253: SIGNST = 'D'
254: END IF
255: CALL DORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M,
256: $ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11,
257: $ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T,
258: $ LDV1T, WORK, LWORK, IWORK, INFO )
259: RETURN
260: END IF
261: *
262: * Compute workspace
263: *
264: IF( INFO .EQ. 0 ) THEN
265: *
266: IPHI = 2
267: ITAUP1 = IPHI + MAX( 1, Q - 1 )
268: ITAUP2 = ITAUP1 + MAX( 1, P )
269: ITAUQ1 = ITAUP2 + MAX( 1, M - P )
270: ITAUQ2 = ITAUQ1 + MAX( 1, Q )
271: IORGQR = ITAUQ2 + MAX( 1, M - Q )
272: CALL DORGQR( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
273: $ CHILDINFO )
274: LORGQRWORKOPT = INT( WORK(1) )
275: LORGQRWORKMIN = MAX( 1, M - Q )
276: IORGLQ = ITAUQ2 + MAX( 1, M - Q )
277: CALL DORGLQ( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
278: $ CHILDINFO )
279: LORGLQWORKOPT = INT( WORK(1) )
280: LORGLQWORKMIN = MAX( 1, M - Q )
281: IORBDB = ITAUQ2 + MAX( 1, M - Q )
282: CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
283: $ X21, LDX21, X22, LDX22, 0, 0, 0, 0, 0, 0, WORK,
284: $ -1, CHILDINFO )
285: LORBDBWORKOPT = INT( WORK(1) )
286: LORBDBWORKMIN = LORBDBWORKOPT
287: IB11D = ITAUQ2 + MAX( 1, M - Q )
288: IB11E = IB11D + MAX( 1, Q )
289: IB12D = IB11E + MAX( 1, Q - 1 )
290: IB12E = IB12D + MAX( 1, Q )
291: IB21D = IB12E + MAX( 1, Q - 1 )
292: IB21E = IB21D + MAX( 1, Q )
293: IB22D = IB21E + MAX( 1, Q - 1 )
294: IB22E = IB22D + MAX( 1, Q )
295: IBBCSD = IB22E + MAX( 1, Q - 1 )
296: CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 0,
297: $ 0, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, 0,
298: $ 0, 0, 0, 0, 0, 0, 0, WORK, -1, CHILDINFO )
299: LBBCSDWORKOPT = INT( WORK(1) )
300: LBBCSDWORKMIN = LBBCSDWORKOPT
301: LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
302: $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1
303: LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
304: $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1
305: WORK(1) = LWORKOPT
306: *
307: IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
308: INFO = -22
309: ELSE
310: LORGQRWORK = LWORK - IORGQR + 1
311: LORGLQWORK = LWORK - IORGLQ + 1
312: LORBDBWORK = LWORK - IORBDB + 1
313: LBBCSDWORK = LWORK - IBBCSD + 1
314: END IF
315: END IF
316: *
317: * Abort if any illegal arguments
318: *
319: IF( INFO .NE. 0 ) THEN
320: CALL XERBLA( 'DORCSD', -INFO )
321: RETURN
322: ELSE IF( LQUERY ) THEN
323: RETURN
324: END IF
325: *
326: * Transform to bidiagonal block form
327: *
328: CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
329: $ LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1),
330: $ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2),
331: $ WORK(IORBDB), LORBDBWORK, CHILDINFO )
332: *
333: * Accumulate Householder reflectors
334: *
335: IF( COLMAJOR ) THEN
336: IF( WANTU1 .AND. P .GT. 0 ) THEN
337: CALL DLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
338: CALL DORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR),
339: $ LORGQRWORK, INFO)
340: END IF
341: IF( WANTU2 .AND. M-P .GT. 0 ) THEN
342: CALL DLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
343: CALL DORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
344: $ WORK(IORGQR), LORGQRWORK, INFO )
345: END IF
346: IF( WANTV1T .AND. Q .GT. 0 ) THEN
347: CALL DLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2),
348: $ LDV1T )
349: V1T(1, 1) = ONE
350: DO J = 2, Q
351: V1T(1,J) = ZERO
352: V1T(J,1) = ZERO
353: END DO
354: CALL DORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
355: $ WORK(IORGLQ), LORGLQWORK, INFO )
356: END IF
357: IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
358: CALL DLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
359: CALL DLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
360: $ V2T(P+1,P+1), LDV2T )
361: CALL DORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
362: $ WORK(IORGLQ), LORGLQWORK, INFO )
363: END IF
364: ELSE
365: IF( WANTU1 .AND. P .GT. 0 ) THEN
366: CALL DLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
367: CALL DORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ),
368: $ LORGLQWORK, INFO)
369: END IF
370: IF( WANTU2 .AND. M-P .GT. 0 ) THEN
371: CALL DLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
372: CALL DORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
373: $ WORK(IORGLQ), LORGLQWORK, INFO )
374: END IF
375: IF( WANTV1T .AND. Q .GT. 0 ) THEN
376: CALL DLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2),
377: $ LDV1T )
378: V1T(1, 1) = ONE
379: DO J = 2, Q
380: V1T(1,J) = ZERO
381: V1T(J,1) = ZERO
382: END DO
383: CALL DORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
384: $ WORK(IORGQR), LORGQRWORK, INFO )
385: END IF
386: IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
387: CALL DLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
388: CALL DLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
389: $ V2T(P+1,P+1), LDV2T )
390: CALL DORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
391: $ WORK(IORGQR), LORGQRWORK, INFO )
392: END IF
393: END IF
394: *
395: * Compute the CSD of the matrix in bidiagonal-block form
396: *
397: CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA,
398: $ WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
399: $ LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D),
400: $ WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D),
401: $ WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO )
402: *
403: * Permute rows and columns to place identity submatrices in top-
404: * left corner of (1,1)-block and/or bottom-right corner of (1,2)-
405: * block and/or bottom-right corner of (2,1)-block and/or top-left
406: * corner of (2,2)-block
407: *
408: IF( Q .GT. 0 .AND. WANTU2 ) THEN
409: DO I = 1, Q
410: IWORK(I) = M - P - Q + I
411: END DO
412: DO I = Q + 1, M - P
413: IWORK(I) = I - Q
414: END DO
415: IF( COLMAJOR ) THEN
416: CALL DLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
417: ELSE
418: CALL DLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK )
419: END IF
420: END IF
421: IF( M .GT. 0 .AND. WANTV2T ) THEN
422: DO I = 1, P
423: IWORK(I) = M - P - Q + I
424: END DO
425: DO I = P + 1, M - Q
426: IWORK(I) = I - P
427: END DO
428: IF( .NOT. COLMAJOR ) THEN
429: CALL DLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
430: ELSE
431: CALL DLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
432: END IF
433: END IF
434: *
435: RETURN
436: *
437: * End DORCSD
438: *
439: END
440:
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