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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, 2: $ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, 3: $ Q, LDQ, WORK, NCYCLE, INFO ) 4: * 5: * -- LAPACK routine (version 3.2.1) -- 6: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 8: * -- April 2009 -- 9: * 10: * .. Scalar Arguments .. 11: CHARACTER JOBQ, JOBU, JOBV 12: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, 13: $ NCYCLE, P 14: DOUBLE PRECISION TOLA, TOLB 15: * .. 16: * .. Array Arguments .. 17: DOUBLE PRECISION ALPHA( * ), BETA( * ) 18: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 19: $ U( LDU, * ), V( LDV, * ), WORK( * ) 20: * .. 21: * 22: * Purpose 23: * ======= 24: * 25: * ZTGSJA computes the generalized singular value decomposition (GSVD) 26: * of two complex upper triangular (or trapezoidal) matrices A and B. 27: * 28: * On entry, it is assumed that matrices A and B have the following 29: * forms, which may be obtained by the preprocessing subroutine ZGGSVP 30: * from a general M-by-N matrix A and P-by-N matrix B: 31: * 32: * N-K-L K L 33: * A = K ( 0 A12 A13 ) if M-K-L >= 0; 34: * L ( 0 0 A23 ) 35: * M-K-L ( 0 0 0 ) 36: * 37: * N-K-L K L 38: * A = K ( 0 A12 A13 ) if M-K-L < 0; 39: * M-K ( 0 0 A23 ) 40: * 41: * N-K-L K L 42: * B = L ( 0 0 B13 ) 43: * P-L ( 0 0 0 ) 44: * 45: * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular 46: * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, 47: * otherwise A23 is (M-K)-by-L upper trapezoidal. 48: * 49: * On exit, 50: * 51: * U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), 52: * 53: * where U, V and Q are unitary matrices, Z' denotes the conjugate 54: * transpose of Z, R is a nonsingular upper triangular matrix, and D1 55: * and D2 are ``diagonal'' matrices, which are of the following 56: * structures: 57: * 58: * If M-K-L >= 0, 59: * 60: * K L 61: * D1 = K ( I 0 ) 62: * L ( 0 C ) 63: * M-K-L ( 0 0 ) 64: * 65: * K L 66: * D2 = L ( 0 S ) 67: * P-L ( 0 0 ) 68: * 69: * N-K-L K L 70: * ( 0 R ) = K ( 0 R11 R12 ) K 71: * L ( 0 0 R22 ) L 72: * 73: * where 74: * 75: * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), 76: * S = diag( BETA(K+1), ... , BETA(K+L) ), 77: * C**2 + S**2 = I. 78: * 79: * R is stored in A(1:K+L,N-K-L+1:N) on exit. 80: * 81: * If M-K-L < 0, 82: * 83: * K M-K K+L-M 84: * D1 = K ( I 0 0 ) 85: * M-K ( 0 C 0 ) 86: * 87: * K M-K K+L-M 88: * D2 = M-K ( 0 S 0 ) 89: * K+L-M ( 0 0 I ) 90: * P-L ( 0 0 0 ) 91: * 92: * N-K-L K M-K K+L-M 93: * ( 0 R ) = K ( 0 R11 R12 R13 ) 94: * M-K ( 0 0 R22 R23 ) 95: * K+L-M ( 0 0 0 R33 ) 96: * 97: * where 98: * C = diag( ALPHA(K+1), ... , ALPHA(M) ), 99: * S = diag( BETA(K+1), ... , BETA(M) ), 100: * C**2 + S**2 = I. 101: * 102: * R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored 103: * ( 0 R22 R23 ) 104: * in B(M-K+1:L,N+M-K-L+1:N) on exit. 105: * 106: * The computation of the unitary transformation matrices U, V or Q 107: * is optional. These matrices may either be formed explicitly, or they 108: * may be postmultiplied into input matrices U1, V1, or Q1. 109: * 110: * Arguments 111: * ========= 112: * 113: * JOBU (input) CHARACTER*1 114: * = 'U': U must contain a unitary matrix U1 on entry, and 115: * the product U1*U is returned; 116: * = 'I': U is initialized to the unit matrix, and the 117: * unitary matrix U is returned; 118: * = 'N': U is not computed. 119: * 120: * JOBV (input) CHARACTER*1 121: * = 'V': V must contain a unitary matrix V1 on entry, and 122: * the product V1*V is returned; 123: * = 'I': V is initialized to the unit matrix, and the 124: * unitary matrix V is returned; 125: * = 'N': V is not computed. 126: * 127: * JOBQ (input) CHARACTER*1 128: * = 'Q': Q must contain a unitary matrix Q1 on entry, and 129: * the product Q1*Q is returned; 130: * = 'I': Q is initialized to the unit matrix, and the 131: * unitary matrix Q is returned; 132: * = 'N': Q is not computed. 133: * 134: * M (input) INTEGER 135: * The number of rows of the matrix A. M >= 0. 136: * 137: * P (input) INTEGER 138: * The number of rows of the matrix B. P >= 0. 139: * 140: * N (input) INTEGER 141: * The number of columns of the matrices A and B. N >= 0. 142: * 143: * K (input) INTEGER 144: * L (input) INTEGER 145: * K and L specify the subblocks in the input matrices A and B: 146: * A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N) 147: * of A and B, whose GSVD is going to be computed by ZTGSJA. 148: * See Further Details. 149: * 150: * A (input/output) COMPLEX*16 array, dimension (LDA,N) 151: * On entry, the M-by-N matrix A. 152: * On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular 153: * matrix R or part of R. See Purpose for details. 154: * 155: * LDA (input) INTEGER 156: * The leading dimension of the array A. LDA >= max(1,M). 157: * 158: * B (input/output) COMPLEX*16 array, dimension (LDB,N) 159: * On entry, the P-by-N matrix B. 160: * On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains 161: * a part of R. See Purpose for details. 162: * 163: * LDB (input) INTEGER 164: * The leading dimension of the array B. LDB >= max(1,P). 165: * 166: * TOLA (input) DOUBLE PRECISION 167: * TOLB (input) DOUBLE PRECISION 168: * TOLA and TOLB are the convergence criteria for the Jacobi- 169: * Kogbetliantz iteration procedure. Generally, they are the 170: * same as used in the preprocessing step, say 171: * TOLA = MAX(M,N)*norm(A)*MAZHEPS, 172: * TOLB = MAX(P,N)*norm(B)*MAZHEPS. 173: * 174: * ALPHA (output) DOUBLE PRECISION array, dimension (N) 175: * BETA (output) DOUBLE PRECISION array, dimension (N) 176: * On exit, ALPHA and BETA contain the generalized singular 177: * value pairs of A and B; 178: * ALPHA(1:K) = 1, 179: * BETA(1:K) = 0, 180: * and if M-K-L >= 0, 181: * ALPHA(K+1:K+L) = diag(C), 182: * BETA(K+1:K+L) = diag(S), 183: * or if M-K-L < 0, 184: * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 185: * BETA(K+1:M) = S, BETA(M+1:K+L) = 1. 186: * Furthermore, if K+L < N, 187: * ALPHA(K+L+1:N) = 0 188: * BETA(K+L+1:N) = 0. 189: * 190: * U (input/output) COMPLEX*16 array, dimension (LDU,M) 191: * On entry, if JOBU = 'U', U must contain a matrix U1 (usually 192: * the unitary matrix returned by ZGGSVP). 193: * On exit, 194: * if JOBU = 'I', U contains the unitary matrix U; 195: * if JOBU = 'U', U contains the product U1*U. 196: * If JOBU = 'N', U is not referenced. 197: * 198: * LDU (input) INTEGER 199: * The leading dimension of the array U. LDU >= max(1,M) if 200: * JOBU = 'U'; LDU >= 1 otherwise. 201: * 202: * V (input/output) COMPLEX*16 array, dimension (LDV,P) 203: * On entry, if JOBV = 'V', V must contain a matrix V1 (usually 204: * the unitary matrix returned by ZGGSVP). 205: * On exit, 206: * if JOBV = 'I', V contains the unitary matrix V; 207: * if JOBV = 'V', V contains the product V1*V. 208: * If JOBV = 'N', V is not referenced. 209: * 210: * LDV (input) INTEGER 211: * The leading dimension of the array V. LDV >= max(1,P) if 212: * JOBV = 'V'; LDV >= 1 otherwise. 213: * 214: * Q (input/output) COMPLEX*16 array, dimension (LDQ,N) 215: * On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually 216: * the unitary matrix returned by ZGGSVP). 217: * On exit, 218: * if JOBQ = 'I', Q contains the unitary matrix Q; 219: * if JOBQ = 'Q', Q contains the product Q1*Q. 220: * If JOBQ = 'N', Q is not referenced. 221: * 222: * LDQ (input) INTEGER 223: * The leading dimension of the array Q. LDQ >= max(1,N) if 224: * JOBQ = 'Q'; LDQ >= 1 otherwise. 225: * 226: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 227: * 228: * NCYCLE (output) INTEGER 229: * The number of cycles required for convergence. 230: * 231: * INFO (output) INTEGER 232: * = 0: successful exit 233: * < 0: if INFO = -i, the i-th argument had an illegal value. 234: * = 1: the procedure does not converge after MAXIT cycles. 235: * 236: * Internal Parameters 237: * =================== 238: * 239: * MAXIT INTEGER 240: * MAXIT specifies the total loops that the iterative procedure 241: * may take. If after MAXIT cycles, the routine fails to 242: * converge, we return INFO = 1. 243: * 244: * Further Details 245: * =============== 246: * 247: * ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce 248: * min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L 249: * matrix B13 to the form: 250: * 251: * U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, 252: * 253: * where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate 254: * transpose of Z. C1 and S1 are diagonal matrices satisfying 255: * 256: * C1**2 + S1**2 = I, 257: * 258: * and R1 is an L-by-L nonsingular upper triangular matrix. 259: * 260: * ===================================================================== 261: * 262: * .. Parameters .. 263: INTEGER MAXIT 264: PARAMETER ( MAXIT = 40 ) 265: DOUBLE PRECISION ZERO, ONE 266: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 267: COMPLEX*16 CZERO, CONE 268: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), 269: $ CONE = ( 1.0D+0, 0.0D+0 ) ) 270: * .. 271: * .. Local Scalars .. 272: * 273: LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV 274: INTEGER I, J, KCYCLE 275: DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA, 276: $ RWK, SSMIN 277: COMPLEX*16 A2, B2, SNQ, SNU, SNV 278: * .. 279: * .. External Functions .. 280: LOGICAL LSAME 281: EXTERNAL LSAME 282: * .. 283: * .. External Subroutines .. 284: EXTERNAL DLARTG, XERBLA, ZCOPY, ZDSCAL, ZLAGS2, ZLAPLL, 285: $ ZLASET, ZROT 286: * .. 287: * .. Intrinsic Functions .. 288: INTRINSIC ABS, DBLE, DCONJG, MAX, MIN 289: * .. 290: * .. Executable Statements .. 291: * 292: * Decode and test the input parameters 293: * 294: INITU = LSAME( JOBU, 'I' ) 295: WANTU = INITU .OR. LSAME( JOBU, 'U' ) 296: * 297: INITV = LSAME( JOBV, 'I' ) 298: WANTV = INITV .OR. LSAME( JOBV, 'V' ) 299: * 300: INITQ = LSAME( JOBQ, 'I' ) 301: WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' ) 302: * 303: INFO = 0 304: IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN 305: INFO = -1 306: ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN 307: INFO = -2 308: ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN 309: INFO = -3 310: ELSE IF( M.LT.0 ) THEN 311: INFO = -4 312: ELSE IF( P.LT.0 ) THEN 313: INFO = -5 314: ELSE IF( N.LT.0 ) THEN 315: INFO = -6 316: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 317: INFO = -10 318: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN 319: INFO = -12 320: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN 321: INFO = -18 322: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN 323: INFO = -20 324: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 325: INFO = -22 326: END IF 327: IF( INFO.NE.0 ) THEN 328: CALL XERBLA( 'ZTGSJA', -INFO ) 329: RETURN 330: END IF 331: * 332: * Initialize U, V and Q, if necessary 333: * 334: IF( INITU ) 335: $ CALL ZLASET( 'Full', M, M, CZERO, CONE, U, LDU ) 336: IF( INITV ) 337: $ CALL ZLASET( 'Full', P, P, CZERO, CONE, V, LDV ) 338: IF( INITQ ) 339: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) 340: * 341: * Loop until convergence 342: * 343: UPPER = .FALSE. 344: DO 40 KCYCLE = 1, MAXIT 345: * 346: UPPER = .NOT.UPPER 347: * 348: DO 20 I = 1, L - 1 349: DO 10 J = I + 1, L 350: * 351: A1 = ZERO 352: A2 = CZERO 353: A3 = ZERO 354: IF( K+I.LE.M ) 355: $ A1 = DBLE( A( K+I, N-L+I ) ) 356: IF( K+J.LE.M ) 357: $ A3 = DBLE( A( K+J, N-L+J ) ) 358: * 359: B1 = DBLE( B( I, N-L+I ) ) 360: B3 = DBLE( B( J, N-L+J ) ) 361: * 362: IF( UPPER ) THEN 363: IF( K+I.LE.M ) 364: $ A2 = A( K+I, N-L+J ) 365: B2 = B( I, N-L+J ) 366: ELSE 367: IF( K+J.LE.M ) 368: $ A2 = A( K+J, N-L+I ) 369: B2 = B( J, N-L+I ) 370: END IF 371: * 372: CALL ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, 373: $ CSV, SNV, CSQ, SNQ ) 374: * 375: * Update (K+I)-th and (K+J)-th rows of matrix A: U'*A 376: * 377: IF( K+J.LE.M ) 378: $ CALL ZROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ), 379: $ LDA, CSU, DCONJG( SNU ) ) 380: * 381: * Update I-th and J-th rows of matrix B: V'*B 382: * 383: CALL ZROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB, 384: $ CSV, DCONJG( SNV ) ) 385: * 386: * Update (N-L+I)-th and (N-L+J)-th columns of matrices 387: * A and B: A*Q and B*Q 388: * 389: CALL ZROT( MIN( K+L, M ), A( 1, N-L+J ), 1, 390: $ A( 1, N-L+I ), 1, CSQ, SNQ ) 391: * 392: CALL ZROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ, 393: $ SNQ ) 394: * 395: IF( UPPER ) THEN 396: IF( K+I.LE.M ) 397: $ A( K+I, N-L+J ) = CZERO 398: B( I, N-L+J ) = CZERO 399: ELSE 400: IF( K+J.LE.M ) 401: $ A( K+J, N-L+I ) = CZERO 402: B( J, N-L+I ) = CZERO 403: END IF 404: * 405: * Ensure that the diagonal elements of A and B are real. 406: * 407: IF( K+I.LE.M ) 408: $ A( K+I, N-L+I ) = DBLE( A( K+I, N-L+I ) ) 409: IF( K+J.LE.M ) 410: $ A( K+J, N-L+J ) = DBLE( A( K+J, N-L+J ) ) 411: B( I, N-L+I ) = DBLE( B( I, N-L+I ) ) 412: B( J, N-L+J ) = DBLE( B( J, N-L+J ) ) 413: * 414: * Update unitary matrices U, V, Q, if desired. 415: * 416: IF( WANTU .AND. K+J.LE.M ) 417: $ CALL ZROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU, 418: $ SNU ) 419: * 420: IF( WANTV ) 421: $ CALL ZROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV ) 422: * 423: IF( WANTQ ) 424: $ CALL ZROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ, 425: $ SNQ ) 426: * 427: 10 CONTINUE 428: 20 CONTINUE 429: * 430: IF( .NOT.UPPER ) THEN 431: * 432: * The matrices A13 and B13 were lower triangular at the start 433: * of the cycle, and are now upper triangular. 434: * 435: * Convergence test: test the parallelism of the corresponding 436: * rows of A and B. 437: * 438: ERROR = ZERO 439: DO 30 I = 1, MIN( L, M-K ) 440: CALL ZCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 ) 441: CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 ) 442: CALL ZLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN ) 443: ERROR = MAX( ERROR, SSMIN ) 444: 30 CONTINUE 445: * 446: IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) ) 447: $ GO TO 50 448: END IF 449: * 450: * End of cycle loop 451: * 452: 40 CONTINUE 453: * 454: * The algorithm has not converged after MAXIT cycles. 455: * 456: INFO = 1 457: GO TO 100 458: * 459: 50 CONTINUE 460: * 461: * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged. 462: * Compute the generalized singular value pairs (ALPHA, BETA), and 463: * set the triangular matrix R to array A. 464: * 465: DO 60 I = 1, K 466: ALPHA( I ) = ONE 467: BETA( I ) = ZERO 468: 60 CONTINUE 469: * 470: DO 70 I = 1, MIN( L, M-K ) 471: * 472: A1 = DBLE( A( K+I, N-L+I ) ) 473: B1 = DBLE( B( I, N-L+I ) ) 474: * 475: IF( A1.NE.ZERO ) THEN 476: GAMMA = B1 / A1 477: * 478: IF( GAMMA.LT.ZERO ) THEN 479: CALL ZDSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB ) 480: IF( WANTV ) 481: $ CALL ZDSCAL( P, -ONE, V( 1, I ), 1 ) 482: END IF 483: * 484: CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ), 485: $ RWK ) 486: * 487: IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN 488: CALL ZDSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ), 489: $ LDA ) 490: ELSE 491: CALL ZDSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ), 492: $ LDB ) 493: CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ), 494: $ LDA ) 495: END IF 496: * 497: ELSE 498: ALPHA( K+I ) = ZERO 499: BETA( K+I ) = ONE 500: CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ), 501: $ LDA ) 502: END IF 503: 70 CONTINUE 504: * 505: * Post-assignment 506: * 507: DO 80 I = M + 1, K + L 508: ALPHA( I ) = ZERO 509: BETA( I ) = ONE 510: 80 CONTINUE 511: * 512: IF( K+L.LT.N ) THEN 513: DO 90 I = K + L + 1, N 514: ALPHA( I ) = ZERO 515: BETA( I ) = ZERO 516: 90 CONTINUE 517: END IF 518: * 519: 100 CONTINUE 520: NCYCLE = KCYCLE 521: * 522: RETURN 523: * 524: * End of ZTGSJA 525: * 526: END