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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, 2: $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, 3: $ IWORK, TAU, WORK, INFO ) 4: * 5: * -- LAPACK routine (version 3.2) -- 6: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 8: * November 2006 9: * 10: * .. Scalar Arguments .. 11: CHARACTER JOBQ, JOBU, JOBV 12: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P 13: DOUBLE PRECISION TOLA, TOLB 14: * .. 15: * .. Array Arguments .. 16: INTEGER IWORK( * ) 17: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 18: $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * DGGSVP computes orthogonal matrices U, V and Q such that 25: * 26: * N-K-L K L 27: * U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; 28: * L ( 0 0 A23 ) 29: * M-K-L ( 0 0 0 ) 30: * 31: * N-K-L K L 32: * = K ( 0 A12 A13 ) if M-K-L < 0; 33: * M-K ( 0 0 A23 ) 34: * 35: * N-K-L K L 36: * V'*B*Q = L ( 0 0 B13 ) 37: * P-L ( 0 0 0 ) 38: * 39: * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular 40: * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, 41: * otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective 42: * numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the 43: * transpose of Z. 44: * 45: * This decomposition is the preprocessing step for computing the 46: * Generalized Singular Value Decomposition (GSVD), see subroutine 47: * DGGSVD. 48: * 49: * Arguments 50: * ========= 51: * 52: * JOBU (input) CHARACTER*1 53: * = 'U': Orthogonal matrix U is computed; 54: * = 'N': U is not computed. 55: * 56: * JOBV (input) CHARACTER*1 57: * = 'V': Orthogonal matrix V is computed; 58: * = 'N': V is not computed. 59: * 60: * JOBQ (input) CHARACTER*1 61: * = 'Q': Orthogonal matrix Q is computed; 62: * = 'N': Q is not computed. 63: * 64: * M (input) INTEGER 65: * The number of rows of the matrix A. M >= 0. 66: * 67: * P (input) INTEGER 68: * The number of rows of the matrix B. P >= 0. 69: * 70: * N (input) INTEGER 71: * The number of columns of the matrices A and B. N >= 0. 72: * 73: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 74: * On entry, the M-by-N matrix A. 75: * On exit, A contains the triangular (or trapezoidal) matrix 76: * described in the Purpose section. 77: * 78: * LDA (input) INTEGER 79: * The leading dimension of the array A. LDA >= max(1,M). 80: * 81: * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) 82: * On entry, the P-by-N matrix B. 83: * On exit, B contains the triangular matrix described in 84: * the Purpose section. 85: * 86: * LDB (input) INTEGER 87: * The leading dimension of the array B. LDB >= max(1,P). 88: * 89: * TOLA (input) DOUBLE PRECISION 90: * TOLB (input) DOUBLE PRECISION 91: * TOLA and TOLB are the thresholds to determine the effective 92: * numerical rank of matrix B and a subblock of A. Generally, 93: * they are set to 94: * TOLA = MAX(M,N)*norm(A)*MAZHEPS, 95: * TOLB = MAX(P,N)*norm(B)*MAZHEPS. 96: * The size of TOLA and TOLB may affect the size of backward 97: * errors of the decomposition. 98: * 99: * K (output) INTEGER 100: * L (output) INTEGER 101: * On exit, K and L specify the dimension of the subblocks 102: * described in Purpose. 103: * K + L = effective numerical rank of (A',B')'. 104: * 105: * U (output) DOUBLE PRECISION array, dimension (LDU,M) 106: * If JOBU = 'U', U contains the orthogonal matrix U. 107: * If JOBU = 'N', U is not referenced. 108: * 109: * LDU (input) INTEGER 110: * The leading dimension of the array U. LDU >= max(1,M) if 111: * JOBU = 'U'; LDU >= 1 otherwise. 112: * 113: * V (output) DOUBLE PRECISION array, dimension (LDV,P) 114: * If JOBV = 'V', V contains the orthogonal matrix V. 115: * If JOBV = 'N', V is not referenced. 116: * 117: * LDV (input) INTEGER 118: * The leading dimension of the array V. LDV >= max(1,P) if 119: * JOBV = 'V'; LDV >= 1 otherwise. 120: * 121: * Q (output) DOUBLE PRECISION array, dimension (LDQ,N) 122: * If JOBQ = 'Q', Q contains the orthogonal matrix Q. 123: * If JOBQ = 'N', Q is not referenced. 124: * 125: * LDQ (input) INTEGER 126: * The leading dimension of the array Q. LDQ >= max(1,N) if 127: * JOBQ = 'Q'; LDQ >= 1 otherwise. 128: * 129: * IWORK (workspace) INTEGER array, dimension (N) 130: * 131: * TAU (workspace) DOUBLE PRECISION array, dimension (N) 132: * 133: * WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P)) 134: * 135: * INFO (output) INTEGER 136: * = 0: successful exit 137: * < 0: if INFO = -i, the i-th argument had an illegal value. 138: * 139: * 140: * Further Details 141: * =============== 142: * 143: * The subroutine uses LAPACK subroutine DGEQPF for the QR factorization 144: * with column pivoting to detect the effective numerical rank of the 145: * a matrix. It may be replaced by a better rank determination strategy. 146: * 147: * ===================================================================== 148: * 149: * .. Parameters .. 150: DOUBLE PRECISION ZERO, ONE 151: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 152: * .. 153: * .. Local Scalars .. 154: LOGICAL FORWRD, WANTQ, WANTU, WANTV 155: INTEGER I, J 156: * .. 157: * .. External Functions .. 158: LOGICAL LSAME 159: EXTERNAL LSAME 160: * .. 161: * .. External Subroutines .. 162: EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET, 163: $ DORG2R, DORM2R, DORMR2, XERBLA 164: * .. 165: * .. Intrinsic Functions .. 166: INTRINSIC ABS, MAX, MIN 167: * .. 168: * .. Executable Statements .. 169: * 170: * Test the input parameters 171: * 172: WANTU = LSAME( JOBU, 'U' ) 173: WANTV = LSAME( JOBV, 'V' ) 174: WANTQ = LSAME( JOBQ, 'Q' ) 175: FORWRD = .TRUE. 176: * 177: INFO = 0 178: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN 179: INFO = -1 180: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN 181: INFO = -2 182: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN 183: INFO = -3 184: ELSE IF( M.LT.0 ) THEN 185: INFO = -4 186: ELSE IF( P.LT.0 ) THEN 187: INFO = -5 188: ELSE IF( N.LT.0 ) THEN 189: INFO = -6 190: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 191: INFO = -8 192: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN 193: INFO = -10 194: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN 195: INFO = -16 196: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN 197: INFO = -18 198: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 199: INFO = -20 200: END IF 201: IF( INFO.NE.0 ) THEN 202: CALL XERBLA( 'DGGSVP', -INFO ) 203: RETURN 204: END IF 205: * 206: * QR with column pivoting of B: B*P = V*( S11 S12 ) 207: * ( 0 0 ) 208: * 209: DO 10 I = 1, N 210: IWORK( I ) = 0 211: 10 CONTINUE 212: CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO ) 213: * 214: * Update A := A*P 215: * 216: CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK ) 217: * 218: * Determine the effective rank of matrix B. 219: * 220: L = 0 221: DO 20 I = 1, MIN( P, N ) 222: IF( ABS( B( I, I ) ).GT.TOLB ) 223: $ L = L + 1 224: 20 CONTINUE 225: * 226: IF( WANTV ) THEN 227: * 228: * Copy the details of V, and form V. 229: * 230: CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV ) 231: IF( P.GT.1 ) 232: $ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ), 233: $ LDV ) 234: CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO ) 235: END IF 236: * 237: * Clean up B 238: * 239: DO 40 J = 1, L - 1 240: DO 30 I = J + 1, L 241: B( I, J ) = ZERO 242: 30 CONTINUE 243: 40 CONTINUE 244: IF( P.GT.L ) 245: $ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB ) 246: * 247: IF( WANTQ ) THEN 248: * 249: * Set Q = I and Update Q := Q*P 250: * 251: CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) 252: CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK ) 253: END IF 254: * 255: IF( P.GE.L .AND. N.NE.L ) THEN 256: * 257: * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z 258: * 259: CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO ) 260: * 261: * Update A := A*Z' 262: * 263: CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A, 264: $ LDA, WORK, INFO ) 265: * 266: IF( WANTQ ) THEN 267: * 268: * Update Q := Q*Z' 269: * 270: CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q, 271: $ LDQ, WORK, INFO ) 272: END IF 273: * 274: * Clean up B 275: * 276: CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB ) 277: DO 60 J = N - L + 1, N 278: DO 50 I = J - N + L + 1, L 279: B( I, J ) = ZERO 280: 50 CONTINUE 281: 60 CONTINUE 282: * 283: END IF 284: * 285: * Let N-L L 286: * A = ( A11 A12 ) M, 287: * 288: * then the following does the complete QR decomposition of A11: 289: * 290: * A11 = U*( 0 T12 )*P1' 291: * ( 0 0 ) 292: * 293: DO 70 I = 1, N - L 294: IWORK( I ) = 0 295: 70 CONTINUE 296: CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO ) 297: * 298: * Determine the effective rank of A11 299: * 300: K = 0 301: DO 80 I = 1, MIN( M, N-L ) 302: IF( ABS( A( I, I ) ).GT.TOLA ) 303: $ K = K + 1 304: 80 CONTINUE 305: * 306: * Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N ) 307: * 308: CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA, 309: $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) 310: * 311: IF( WANTU ) THEN 312: * 313: * Copy the details of U, and form U 314: * 315: CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU ) 316: IF( M.GT.1 ) 317: $ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ), 318: $ LDU ) 319: CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO ) 320: END IF 321: * 322: IF( WANTQ ) THEN 323: * 324: * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 325: * 326: CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK ) 327: END IF 328: * 329: * Clean up A: set the strictly lower triangular part of 330: * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. 331: * 332: DO 100 J = 1, K - 1 333: DO 90 I = J + 1, K 334: A( I, J ) = ZERO 335: 90 CONTINUE 336: 100 CONTINUE 337: IF( M.GT.K ) 338: $ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA ) 339: * 340: IF( N-L.GT.K ) THEN 341: * 342: * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 343: * 344: CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO ) 345: * 346: IF( WANTQ ) THEN 347: * 348: * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' 349: * 350: CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU, 351: $ Q, LDQ, WORK, INFO ) 352: END IF 353: * 354: * Clean up A 355: * 356: CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA ) 357: DO 120 J = N - L - K + 1, N - L 358: DO 110 I = J - N + L + K + 1, K 359: A( I, J ) = ZERO 360: 110 CONTINUE 361: 120 CONTINUE 362: * 363: END IF 364: * 365: IF( M.GT.K ) THEN 366: * 367: * QR factorization of A( K+1:M,N-L+1:N ) 368: * 369: CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO ) 370: * 371: IF( WANTU ) THEN 372: * 373: * Update U(:,K+1:M) := U(:,K+1:M)*U1 374: * 375: CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ), 376: $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU, 377: $ WORK, INFO ) 378: END IF 379: * 380: * Clean up 381: * 382: DO 140 J = N - L + 1, N 383: DO 130 I = J - N + K + L + 1, M 384: A( I, J ) = ZERO 385: 130 CONTINUE 386: 140 CONTINUE 387: * 388: END IF 389: * 390: RETURN 391: * 392: * End of DGGSVP 393: * 394: END