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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, 2: $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, 3: $ IWORK, RWORK, 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 RWORK( * ) 18: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 19: $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) 20: * .. 21: * 22: * Purpose 23: * ======= 24: * 25: * ZGGSVP computes unitary matrices U, V and Q such that 26: * 27: * N-K-L K L 28: * U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; 29: * L ( 0 0 A23 ) 30: * M-K-L ( 0 0 0 ) 31: * 32: * N-K-L K L 33: * = K ( 0 A12 A13 ) if M-K-L < 0; 34: * M-K ( 0 0 A23 ) 35: * 36: * N-K-L K L 37: * V'*B*Q = L ( 0 0 B13 ) 38: * P-L ( 0 0 0 ) 39: * 40: * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular 41: * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, 42: * otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective 43: * numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the 44: * conjugate transpose of Z. 45: * 46: * This decomposition is the preprocessing step for computing the 47: * Generalized Singular Value Decomposition (GSVD), see subroutine 48: * ZGGSVD. 49: * 50: * Arguments 51: * ========= 52: * 53: * JOBU (input) CHARACTER*1 54: * = 'U': Unitary matrix U is computed; 55: * = 'N': U is not computed. 56: * 57: * JOBV (input) CHARACTER*1 58: * = 'V': Unitary matrix V is computed; 59: * = 'N': V is not computed. 60: * 61: * JOBQ (input) CHARACTER*1 62: * = 'Q': Unitary matrix Q is computed; 63: * = 'N': Q is not computed. 64: * 65: * M (input) INTEGER 66: * The number of rows of the matrix A. M >= 0. 67: * 68: * P (input) INTEGER 69: * The number of rows of the matrix B. P >= 0. 70: * 71: * N (input) INTEGER 72: * The number of columns of the matrices A and B. N >= 0. 73: * 74: * A (input/output) COMPLEX*16 array, dimension (LDA,N) 75: * On entry, the M-by-N matrix A. 76: * On exit, A contains the triangular (or trapezoidal) matrix 77: * described in the Purpose section. 78: * 79: * LDA (input) INTEGER 80: * The leading dimension of the array A. LDA >= max(1,M). 81: * 82: * B (input/output) COMPLEX*16 array, dimension (LDB,N) 83: * On entry, the P-by-N matrix B. 84: * On exit, B contains the triangular matrix described in 85: * the Purpose section. 86: * 87: * LDB (input) INTEGER 88: * The leading dimension of the array B. LDB >= max(1,P). 89: * 90: * TOLA (input) DOUBLE PRECISION 91: * TOLB (input) DOUBLE PRECISION 92: * TOLA and TOLB are the thresholds to determine the effective 93: * numerical rank of matrix B and a subblock of A. Generally, 94: * they are set to 95: * TOLA = MAX(M,N)*norm(A)*MAZHEPS, 96: * TOLB = MAX(P,N)*norm(B)*MAZHEPS. 97: * The size of TOLA and TOLB may affect the size of backward 98: * errors of the decomposition. 99: * 100: * K (output) INTEGER 101: * L (output) INTEGER 102: * On exit, K and L specify the dimension of the subblocks 103: * described in Purpose section. 104: * K + L = effective numerical rank of (A',B')'. 105: * 106: * U (output) COMPLEX*16 array, dimension (LDU,M) 107: * If JOBU = 'U', U contains the unitary matrix U. 108: * If JOBU = 'N', U is not referenced. 109: * 110: * LDU (input) INTEGER 111: * The leading dimension of the array U. LDU >= max(1,M) if 112: * JOBU = 'U'; LDU >= 1 otherwise. 113: * 114: * V (output) COMPLEX*16 array, dimension (LDV,P) 115: * If JOBV = 'V', V contains the unitary matrix V. 116: * If JOBV = 'N', V is not referenced. 117: * 118: * LDV (input) INTEGER 119: * The leading dimension of the array V. LDV >= max(1,P) if 120: * JOBV = 'V'; LDV >= 1 otherwise. 121: * 122: * Q (output) COMPLEX*16 array, dimension (LDQ,N) 123: * If JOBQ = 'Q', Q contains the unitary matrix Q. 124: * If JOBQ = 'N', Q is not referenced. 125: * 126: * LDQ (input) INTEGER 127: * The leading dimension of the array Q. LDQ >= max(1,N) if 128: * JOBQ = 'Q'; LDQ >= 1 otherwise. 129: * 130: * IWORK (workspace) INTEGER array, dimension (N) 131: * 132: * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) 133: * 134: * TAU (workspace) COMPLEX*16 array, dimension (N) 135: * 136: * WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)) 137: * 138: * INFO (output) INTEGER 139: * = 0: successful exit 140: * < 0: if INFO = -i, the i-th argument had an illegal value. 141: * 142: * Further Details 143: * =============== 144: * 145: * The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization 146: * with column pivoting to detect the effective numerical rank of the 147: * a matrix. It may be replaced by a better rank determination strategy. 148: * 149: * ===================================================================== 150: * 151: * .. Parameters .. 152: COMPLEX*16 CZERO, CONE 153: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), 154: $ CONE = ( 1.0D+0, 0.0D+0 ) ) 155: * .. 156: * .. Local Scalars .. 157: LOGICAL FORWRD, WANTQ, WANTU, WANTV 158: INTEGER I, J 159: COMPLEX*16 T 160: * .. 161: * .. External Functions .. 162: LOGICAL LSAME 163: EXTERNAL LSAME 164: * .. 165: * .. External Subroutines .. 166: EXTERNAL XERBLA, ZGEQPF, ZGEQR2, ZGERQ2, ZLACPY, ZLAPMT, 167: $ ZLASET, ZUNG2R, ZUNM2R, ZUNMR2 168: * .. 169: * .. Intrinsic Functions .. 170: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 171: * .. 172: * .. Statement Functions .. 173: DOUBLE PRECISION CABS1 174: * .. 175: * .. Statement Function definitions .. 176: CABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) ) 177: * .. 178: * .. Executable Statements .. 179: * 180: * Test the input parameters 181: * 182: WANTU = LSAME( JOBU, 'U' ) 183: WANTV = LSAME( JOBV, 'V' ) 184: WANTQ = LSAME( JOBQ, 'Q' ) 185: FORWRD = .TRUE. 186: * 187: INFO = 0 188: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN 189: INFO = -1 190: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN 191: INFO = -2 192: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN 193: INFO = -3 194: ELSE IF( M.LT.0 ) THEN 195: INFO = -4 196: ELSE IF( P.LT.0 ) THEN 197: INFO = -5 198: ELSE IF( N.LT.0 ) THEN 199: INFO = -6 200: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 201: INFO = -8 202: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN 203: INFO = -10 204: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN 205: INFO = -16 206: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN 207: INFO = -18 208: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 209: INFO = -20 210: END IF 211: IF( INFO.NE.0 ) THEN 212: CALL XERBLA( 'ZGGSVP', -INFO ) 213: RETURN 214: END IF 215: * 216: * QR with column pivoting of B: B*P = V*( S11 S12 ) 217: * ( 0 0 ) 218: * 219: DO 10 I = 1, N 220: IWORK( I ) = 0 221: 10 CONTINUE 222: CALL ZGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO ) 223: * 224: * Update A := A*P 225: * 226: CALL ZLAPMT( FORWRD, M, N, A, LDA, IWORK ) 227: * 228: * Determine the effective rank of matrix B. 229: * 230: L = 0 231: DO 20 I = 1, MIN( P, N ) 232: IF( CABS1( B( I, I ) ).GT.TOLB ) 233: $ L = L + 1 234: 20 CONTINUE 235: * 236: IF( WANTV ) THEN 237: * 238: * Copy the details of V, and form V. 239: * 240: CALL ZLASET( 'Full', P, P, CZERO, CZERO, V, LDV ) 241: IF( P.GT.1 ) 242: $ CALL ZLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ), 243: $ LDV ) 244: CALL ZUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO ) 245: END IF 246: * 247: * Clean up B 248: * 249: DO 40 J = 1, L - 1 250: DO 30 I = J + 1, L 251: B( I, J ) = CZERO 252: 30 CONTINUE 253: 40 CONTINUE 254: IF( P.GT.L ) 255: $ CALL ZLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB ) 256: * 257: IF( WANTQ ) THEN 258: * 259: * Set Q = I and Update Q := Q*P 260: * 261: CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) 262: CALL ZLAPMT( FORWRD, N, N, Q, LDQ, IWORK ) 263: END IF 264: * 265: IF( P.GE.L .AND. N.NE.L ) THEN 266: * 267: * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z 268: * 269: CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO ) 270: * 271: * Update A := A*Z' 272: * 273: CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB, 274: $ TAU, A, LDA, WORK, INFO ) 275: IF( WANTQ ) THEN 276: * 277: * Update Q := Q*Z' 278: * 279: CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B, 280: $ LDB, TAU, Q, LDQ, WORK, INFO ) 281: END IF 282: * 283: * Clean up B 284: * 285: CALL ZLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB ) 286: DO 60 J = N - L + 1, N 287: DO 50 I = J - N + L + 1, L 288: B( I, J ) = CZERO 289: 50 CONTINUE 290: 60 CONTINUE 291: * 292: END IF 293: * 294: * Let N-L L 295: * A = ( A11 A12 ) M, 296: * 297: * then the following does the complete QR decomposition of A11: 298: * 299: * A11 = U*( 0 T12 )*P1' 300: * ( 0 0 ) 301: * 302: DO 70 I = 1, N - L 303: IWORK( I ) = 0 304: 70 CONTINUE 305: CALL ZGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO ) 306: * 307: * Determine the effective rank of A11 308: * 309: K = 0 310: DO 80 I = 1, MIN( M, N-L ) 311: IF( CABS1( A( I, I ) ).GT.TOLA ) 312: $ K = K + 1 313: 80 CONTINUE 314: * 315: * Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N ) 316: * 317: CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ), 318: $ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) 319: * 320: IF( WANTU ) THEN 321: * 322: * Copy the details of U, and form U 323: * 324: CALL ZLASET( 'Full', M, M, CZERO, CZERO, U, LDU ) 325: IF( M.GT.1 ) 326: $ CALL ZLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ), 327: $ LDU ) 328: CALL ZUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO ) 329: END IF 330: * 331: IF( WANTQ ) THEN 332: * 333: * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 334: * 335: CALL ZLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK ) 336: END IF 337: * 338: * Clean up A: set the strictly lower triangular part of 339: * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. 340: * 341: DO 100 J = 1, K - 1 342: DO 90 I = J + 1, K 343: A( I, J ) = CZERO 344: 90 CONTINUE 345: 100 CONTINUE 346: IF( M.GT.K ) 347: $ CALL ZLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA ) 348: * 349: IF( N-L.GT.K ) THEN 350: * 351: * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 352: * 353: CALL ZGERQ2( K, N-L, A, LDA, TAU, WORK, INFO ) 354: * 355: IF( WANTQ ) THEN 356: * 357: * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' 358: * 359: CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A, 360: $ LDA, TAU, Q, LDQ, WORK, INFO ) 361: END IF 362: * 363: * Clean up A 364: * 365: CALL ZLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA ) 366: DO 120 J = N - L - K + 1, N - L 367: DO 110 I = J - N + L + K + 1, K 368: A( I, J ) = CZERO 369: 110 CONTINUE 370: 120 CONTINUE 371: * 372: END IF 373: * 374: IF( M.GT.K ) THEN 375: * 376: * QR factorization of A( K+1:M,N-L+1:N ) 377: * 378: CALL ZGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO ) 379: * 380: IF( WANTU ) THEN 381: * 382: * Update U(:,K+1:M) := U(:,K+1:M)*U1 383: * 384: CALL ZUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ), 385: $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU, 386: $ WORK, INFO ) 387: END IF 388: * 389: * Clean up 390: * 391: DO 140 J = N - L + 1, N 392: DO 130 I = J - N + K + L + 1, M 393: A( I, J ) = CZERO 394: 130 CONTINUE 395: 140 CONTINUE 396: * 397: END IF 398: * 399: RETURN 400: * 401: * End of ZGGSVP 402: * 403: END