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Mise à jour de Lapack vers la version 3.3.0.
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: