Annotation of rpl/lapack/lapack/dtgsja.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DTGSJA( 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 A( LDA, * ), ALPHA( * ), B( LDB, * ),
! 18: $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
! 19: $ V( LDV, * ), WORK( * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DTGSJA computes the generalized singular value decomposition (GSVD)
! 26: * of two real 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 DGGSVP
! 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 orthogonal matrices, Z' denotes the transpose
! 54: * of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
! 55: * ``diagonal'' matrices, which are of the following structures:
! 56: *
! 57: * If M-K-L >= 0,
! 58: *
! 59: * K L
! 60: * D1 = K ( I 0 )
! 61: * L ( 0 C )
! 62: * M-K-L ( 0 0 )
! 63: *
! 64: * K L
! 65: * D2 = L ( 0 S )
! 66: * P-L ( 0 0 )
! 67: *
! 68: * N-K-L K L
! 69: * ( 0 R ) = K ( 0 R11 R12 ) K
! 70: * L ( 0 0 R22 ) L
! 71: *
! 72: * where
! 73: *
! 74: * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
! 75: * S = diag( BETA(K+1), ... , BETA(K+L) ),
! 76: * C**2 + S**2 = I.
! 77: *
! 78: * R is stored in A(1:K+L,N-K-L+1:N) on exit.
! 79: *
! 80: * If M-K-L < 0,
! 81: *
! 82: * K M-K K+L-M
! 83: * D1 = K ( I 0 0 )
! 84: * M-K ( 0 C 0 )
! 85: *
! 86: * K M-K K+L-M
! 87: * D2 = M-K ( 0 S 0 )
! 88: * K+L-M ( 0 0 I )
! 89: * P-L ( 0 0 0 )
! 90: *
! 91: * N-K-L K M-K K+L-M
! 92: * ( 0 R ) = K ( 0 R11 R12 R13 )
! 93: * M-K ( 0 0 R22 R23 )
! 94: * K+L-M ( 0 0 0 R33 )
! 95: *
! 96: * where
! 97: * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
! 98: * S = diag( BETA(K+1), ... , BETA(M) ),
! 99: * C**2 + S**2 = I.
! 100: *
! 101: * R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
! 102: * ( 0 R22 R23 )
! 103: * in B(M-K+1:L,N+M-K-L+1:N) on exit.
! 104: *
! 105: * The computation of the orthogonal transformation matrices U, V or Q
! 106: * is optional. These matrices may either be formed explicitly, or they
! 107: * may be postmultiplied into input matrices U1, V1, or Q1.
! 108: *
! 109: * Arguments
! 110: * =========
! 111: *
! 112: * JOBU (input) CHARACTER*1
! 113: * = 'U': U must contain an orthogonal matrix U1 on entry, and
! 114: * the product U1*U is returned;
! 115: * = 'I': U is initialized to the unit matrix, and the
! 116: * orthogonal matrix U is returned;
! 117: * = 'N': U is not computed.
! 118: *
! 119: * JOBV (input) CHARACTER*1
! 120: * = 'V': V must contain an orthogonal matrix V1 on entry, and
! 121: * the product V1*V is returned;
! 122: * = 'I': V is initialized to the unit matrix, and the
! 123: * orthogonal matrix V is returned;
! 124: * = 'N': V is not computed.
! 125: *
! 126: * JOBQ (input) CHARACTER*1
! 127: * = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
! 128: * the product Q1*Q is returned;
! 129: * = 'I': Q is initialized to the unit matrix, and the
! 130: * orthogonal matrix Q is returned;
! 131: * = 'N': Q is not computed.
! 132: *
! 133: * M (input) INTEGER
! 134: * The number of rows of the matrix A. M >= 0.
! 135: *
! 136: * P (input) INTEGER
! 137: * The number of rows of the matrix B. P >= 0.
! 138: *
! 139: * N (input) INTEGER
! 140: * The number of columns of the matrices A and B. N >= 0.
! 141: *
! 142: * K (input) INTEGER
! 143: * L (input) INTEGER
! 144: * K and L specify the subblocks in the input matrices A and B:
! 145: * A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
! 146: * of A and B, whose GSVD is going to be computed by DTGSJA.
! 147: * See Further Details.
! 148: *
! 149: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
! 150: * On entry, the M-by-N matrix A.
! 151: * On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
! 152: * matrix R or part of R. See Purpose for details.
! 153: *
! 154: * LDA (input) INTEGER
! 155: * The leading dimension of the array A. LDA >= max(1,M).
! 156: *
! 157: * B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
! 158: * On entry, the P-by-N matrix B.
! 159: * On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
! 160: * a part of R. See Purpose for details.
! 161: *
! 162: * LDB (input) INTEGER
! 163: * The leading dimension of the array B. LDB >= max(1,P).
! 164: *
! 165: * TOLA (input) DOUBLE PRECISION
! 166: * TOLB (input) DOUBLE PRECISION
! 167: * TOLA and TOLB are the convergence criteria for the Jacobi-
! 168: * Kogbetliantz iteration procedure. Generally, they are the
! 169: * same as used in the preprocessing step, say
! 170: * TOLA = max(M,N)*norm(A)*MAZHEPS,
! 171: * TOLB = max(P,N)*norm(B)*MAZHEPS.
! 172: *
! 173: * ALPHA (output) DOUBLE PRECISION array, dimension (N)
! 174: * BETA (output) DOUBLE PRECISION array, dimension (N)
! 175: * On exit, ALPHA and BETA contain the generalized singular
! 176: * value pairs of A and B;
! 177: * ALPHA(1:K) = 1,
! 178: * BETA(1:K) = 0,
! 179: * and if M-K-L >= 0,
! 180: * ALPHA(K+1:K+L) = diag(C),
! 181: * BETA(K+1:K+L) = diag(S),
! 182: * or if M-K-L < 0,
! 183: * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
! 184: * BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
! 185: * Furthermore, if K+L < N,
! 186: * ALPHA(K+L+1:N) = 0 and
! 187: * BETA(K+L+1:N) = 0.
! 188: *
! 189: * U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
! 190: * On entry, if JOBU = 'U', U must contain a matrix U1 (usually
! 191: * the orthogonal matrix returned by DGGSVP).
! 192: * On exit,
! 193: * if JOBU = 'I', U contains the orthogonal matrix U;
! 194: * if JOBU = 'U', U contains the product U1*U.
! 195: * If JOBU = 'N', U is not referenced.
! 196: *
! 197: * LDU (input) INTEGER
! 198: * The leading dimension of the array U. LDU >= max(1,M) if
! 199: * JOBU = 'U'; LDU >= 1 otherwise.
! 200: *
! 201: * V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
! 202: * On entry, if JOBV = 'V', V must contain a matrix V1 (usually
! 203: * the orthogonal matrix returned by DGGSVP).
! 204: * On exit,
! 205: * if JOBV = 'I', V contains the orthogonal matrix V;
! 206: * if JOBV = 'V', V contains the product V1*V.
! 207: * If JOBV = 'N', V is not referenced.
! 208: *
! 209: * LDV (input) INTEGER
! 210: * The leading dimension of the array V. LDV >= max(1,P) if
! 211: * JOBV = 'V'; LDV >= 1 otherwise.
! 212: *
! 213: * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
! 214: * On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
! 215: * the orthogonal matrix returned by DGGSVP).
! 216: * On exit,
! 217: * if JOBQ = 'I', Q contains the orthogonal matrix Q;
! 218: * if JOBQ = 'Q', Q contains the product Q1*Q.
! 219: * If JOBQ = 'N', Q is not referenced.
! 220: *
! 221: * LDQ (input) INTEGER
! 222: * The leading dimension of the array Q. LDQ >= max(1,N) if
! 223: * JOBQ = 'Q'; LDQ >= 1 otherwise.
! 224: *
! 225: * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
! 226: *
! 227: * NCYCLE (output) INTEGER
! 228: * The number of cycles required for convergence.
! 229: *
! 230: * INFO (output) INTEGER
! 231: * = 0: successful exit
! 232: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 233: * = 1: the procedure does not converge after MAXIT cycles.
! 234: *
! 235: * Internal Parameters
! 236: * ===================
! 237: *
! 238: * MAXIT INTEGER
! 239: * MAXIT specifies the total loops that the iterative procedure
! 240: * may take. If after MAXIT cycles, the routine fails to
! 241: * converge, we return INFO = 1.
! 242: *
! 243: * Further Details
! 244: * ===============
! 245: *
! 246: * DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
! 247: * min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
! 248: * matrix B13 to the form:
! 249: *
! 250: * U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
! 251: *
! 252: * where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
! 253: * of Z. C1 and S1 are diagonal matrices satisfying
! 254: *
! 255: * C1**2 + S1**2 = I,
! 256: *
! 257: * and R1 is an L-by-L nonsingular upper triangular matrix.
! 258: *
! 259: * =====================================================================
! 260: *
! 261: * .. Parameters ..
! 262: INTEGER MAXIT
! 263: PARAMETER ( MAXIT = 40 )
! 264: DOUBLE PRECISION ZERO, ONE
! 265: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 266: * ..
! 267: * .. Local Scalars ..
! 268: *
! 269: LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
! 270: INTEGER I, J, KCYCLE
! 271: DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
! 272: $ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
! 273: * ..
! 274: * .. External Functions ..
! 275: LOGICAL LSAME
! 276: EXTERNAL LSAME
! 277: * ..
! 278: * .. External Subroutines ..
! 279: EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT,
! 280: $ DSCAL, XERBLA
! 281: * ..
! 282: * .. Intrinsic Functions ..
! 283: INTRINSIC ABS, MAX, MIN
! 284: * ..
! 285: * .. Executable Statements ..
! 286: *
! 287: * Decode and test the input parameters
! 288: *
! 289: INITU = LSAME( JOBU, 'I' )
! 290: WANTU = INITU .OR. LSAME( JOBU, 'U' )
! 291: *
! 292: INITV = LSAME( JOBV, 'I' )
! 293: WANTV = INITV .OR. LSAME( JOBV, 'V' )
! 294: *
! 295: INITQ = LSAME( JOBQ, 'I' )
! 296: WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
! 297: *
! 298: INFO = 0
! 299: IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
! 300: INFO = -1
! 301: ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
! 302: INFO = -2
! 303: ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
! 304: INFO = -3
! 305: ELSE IF( M.LT.0 ) THEN
! 306: INFO = -4
! 307: ELSE IF( P.LT.0 ) THEN
! 308: INFO = -5
! 309: ELSE IF( N.LT.0 ) THEN
! 310: INFO = -6
! 311: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 312: INFO = -10
! 313: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
! 314: INFO = -12
! 315: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
! 316: INFO = -18
! 317: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
! 318: INFO = -20
! 319: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
! 320: INFO = -22
! 321: END IF
! 322: IF( INFO.NE.0 ) THEN
! 323: CALL XERBLA( 'DTGSJA', -INFO )
! 324: RETURN
! 325: END IF
! 326: *
! 327: * Initialize U, V and Q, if necessary
! 328: *
! 329: IF( INITU )
! 330: $ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
! 331: IF( INITV )
! 332: $ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
! 333: IF( INITQ )
! 334: $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
! 335: *
! 336: * Loop until convergence
! 337: *
! 338: UPPER = .FALSE.
! 339: DO 40 KCYCLE = 1, MAXIT
! 340: *
! 341: UPPER = .NOT.UPPER
! 342: *
! 343: DO 20 I = 1, L - 1
! 344: DO 10 J = I + 1, L
! 345: *
! 346: A1 = ZERO
! 347: A2 = ZERO
! 348: A3 = ZERO
! 349: IF( K+I.LE.M )
! 350: $ A1 = A( K+I, N-L+I )
! 351: IF( K+J.LE.M )
! 352: $ A3 = A( K+J, N-L+J )
! 353: *
! 354: B1 = B( I, N-L+I )
! 355: B3 = B( J, N-L+J )
! 356: *
! 357: IF( UPPER ) THEN
! 358: IF( K+I.LE.M )
! 359: $ A2 = A( K+I, N-L+J )
! 360: B2 = B( I, N-L+J )
! 361: ELSE
! 362: IF( K+J.LE.M )
! 363: $ A2 = A( K+J, N-L+I )
! 364: B2 = B( J, N-L+I )
! 365: END IF
! 366: *
! 367: CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
! 368: $ CSV, SNV, CSQ, SNQ )
! 369: *
! 370: * Update (K+I)-th and (K+J)-th rows of matrix A: U'*A
! 371: *
! 372: IF( K+J.LE.M )
! 373: $ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
! 374: $ LDA, CSU, SNU )
! 375: *
! 376: * Update I-th and J-th rows of matrix B: V'*B
! 377: *
! 378: CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
! 379: $ CSV, SNV )
! 380: *
! 381: * Update (N-L+I)-th and (N-L+J)-th columns of matrices
! 382: * A and B: A*Q and B*Q
! 383: *
! 384: CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
! 385: $ A( 1, N-L+I ), 1, CSQ, SNQ )
! 386: *
! 387: CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
! 388: $ SNQ )
! 389: *
! 390: IF( UPPER ) THEN
! 391: IF( K+I.LE.M )
! 392: $ A( K+I, N-L+J ) = ZERO
! 393: B( I, N-L+J ) = ZERO
! 394: ELSE
! 395: IF( K+J.LE.M )
! 396: $ A( K+J, N-L+I ) = ZERO
! 397: B( J, N-L+I ) = ZERO
! 398: END IF
! 399: *
! 400: * Update orthogonal matrices U, V, Q, if desired.
! 401: *
! 402: IF( WANTU .AND. K+J.LE.M )
! 403: $ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
! 404: $ SNU )
! 405: *
! 406: IF( WANTV )
! 407: $ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
! 408: *
! 409: IF( WANTQ )
! 410: $ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
! 411: $ SNQ )
! 412: *
! 413: 10 CONTINUE
! 414: 20 CONTINUE
! 415: *
! 416: IF( .NOT.UPPER ) THEN
! 417: *
! 418: * The matrices A13 and B13 were lower triangular at the start
! 419: * of the cycle, and are now upper triangular.
! 420: *
! 421: * Convergence test: test the parallelism of the corresponding
! 422: * rows of A and B.
! 423: *
! 424: ERROR = ZERO
! 425: DO 30 I = 1, MIN( L, M-K )
! 426: CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
! 427: CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
! 428: CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
! 429: ERROR = MAX( ERROR, SSMIN )
! 430: 30 CONTINUE
! 431: *
! 432: IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
! 433: $ GO TO 50
! 434: END IF
! 435: *
! 436: * End of cycle loop
! 437: *
! 438: 40 CONTINUE
! 439: *
! 440: * The algorithm has not converged after MAXIT cycles.
! 441: *
! 442: INFO = 1
! 443: GO TO 100
! 444: *
! 445: 50 CONTINUE
! 446: *
! 447: * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
! 448: * Compute the generalized singular value pairs (ALPHA, BETA), and
! 449: * set the triangular matrix R to array A.
! 450: *
! 451: DO 60 I = 1, K
! 452: ALPHA( I ) = ONE
! 453: BETA( I ) = ZERO
! 454: 60 CONTINUE
! 455: *
! 456: DO 70 I = 1, MIN( L, M-K )
! 457: *
! 458: A1 = A( K+I, N-L+I )
! 459: B1 = B( I, N-L+I )
! 460: *
! 461: IF( A1.NE.ZERO ) THEN
! 462: GAMMA = B1 / A1
! 463: *
! 464: * change sign if necessary
! 465: *
! 466: IF( GAMMA.LT.ZERO ) THEN
! 467: CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
! 468: IF( WANTV )
! 469: $ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
! 470: END IF
! 471: *
! 472: CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
! 473: $ RWK )
! 474: *
! 475: IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
! 476: CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
! 477: $ LDA )
! 478: ELSE
! 479: CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
! 480: $ LDB )
! 481: CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
! 482: $ LDA )
! 483: END IF
! 484: *
! 485: ELSE
! 486: *
! 487: ALPHA( K+I ) = ZERO
! 488: BETA( K+I ) = ONE
! 489: CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
! 490: $ LDA )
! 491: *
! 492: END IF
! 493: *
! 494: 70 CONTINUE
! 495: *
! 496: * Post-assignment
! 497: *
! 498: DO 80 I = M + 1, K + L
! 499: ALPHA( I ) = ZERO
! 500: BETA( I ) = ONE
! 501: 80 CONTINUE
! 502: *
! 503: IF( K+L.LT.N ) THEN
! 504: DO 90 I = K + L + 1, N
! 505: ALPHA( I ) = ZERO
! 506: BETA( I ) = ZERO
! 507: 90 CONTINUE
! 508: END IF
! 509: *
! 510: 100 CONTINUE
! 511: NCYCLE = KCYCLE
! 512: RETURN
! 513: *
! 514: * End of DTGSJA
! 515: *
! 516: END
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