Annotation of rpl/lapack/lapack/zggsvp.f, revision 1.1
1.1 ! bertrand 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
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