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SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, $ IWORK, RWORK, TAU, WORK, INFO ) * * -- LAPACK routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P DOUBLE PRECISION TOLA, TOLB * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * * Purpose * ======= * * ZGGSVP computes unitary matrices U, V and Q such that * * N-K-L K L * U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; * L ( 0 0 A23 ) * M-K-L ( 0 0 0 ) * * N-K-L K L * = K ( 0 A12 A13 ) if M-K-L < 0; * M-K ( 0 0 A23 ) * * N-K-L K L * V**H*B*Q = L ( 0 0 B13 ) * P-L ( 0 0 0 ) * * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, * otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective * numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H. * * This decomposition is the preprocessing step for computing the * Generalized Singular Value Decomposition (GSVD), see subroutine * ZGGSVD. * * Arguments * ========= * * JOBU (input) CHARACTER*1 * = 'U': Unitary matrix U is computed; * = 'N': U is not computed. * * JOBV (input) CHARACTER*1 * = 'V': Unitary matrix V is computed; * = 'N': V is not computed. * * JOBQ (input) CHARACTER*1 * = 'Q': Unitary matrix Q is computed; * = 'N': Q is not computed. * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * P (input) INTEGER * The number of rows of the matrix B. P >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * A (input/output) COMPLEX*16 array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, A contains the triangular (or trapezoidal) matrix * described in the Purpose section. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) COMPLEX*16 array, dimension (LDB,N) * On entry, the P-by-N matrix B. * On exit, B contains the triangular matrix described in * the Purpose section. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,P). * * TOLA (input) DOUBLE PRECISION * TOLB (input) DOUBLE PRECISION * TOLA and TOLB are the thresholds to determine the effective * numerical rank of matrix B and a subblock of A. Generally, * they are set to * TOLA = MAX(M,N)*norm(A)*MAZHEPS, * TOLB = MAX(P,N)*norm(B)*MAZHEPS. * The size of TOLA and TOLB may affect the size of backward * errors of the decomposition. * * K (output) INTEGER * L (output) INTEGER * On exit, K and L specify the dimension of the subblocks * described in Purpose section. * K + L = effective numerical rank of (A**H,B**H)**H. * * U (output) COMPLEX*16 array, dimension (LDU,M) * If JOBU = 'U', U contains the unitary matrix U. * If JOBU = 'N', U is not referenced. * * LDU (input) INTEGER * The leading dimension of the array U. LDU >= max(1,M) if * JOBU = 'U'; LDU >= 1 otherwise. * * V (output) COMPLEX*16 array, dimension (LDV,P) * If JOBV = 'V', V contains the unitary matrix V. * If JOBV = 'N', V is not referenced. * * LDV (input) INTEGER * The leading dimension of the array V. LDV >= max(1,P) if * JOBV = 'V'; LDV >= 1 otherwise. * * Q (output) COMPLEX*16 array, dimension (LDQ,N) * If JOBQ = 'Q', Q contains the unitary matrix Q. * If JOBQ = 'N', Q is not referenced. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= max(1,N) if * JOBQ = 'Q'; LDQ >= 1 otherwise. * * IWORK (workspace) INTEGER array, dimension (N) * * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) * * TAU (workspace) COMPLEX*16 array, dimension (N) * * WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization * with column pivoting to detect the effective numerical rank of the * a matrix. It may be replaced by a better rank determination strategy. * * ===================================================================== * * .. Parameters .. COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), $ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL FORWRD, WANTQ, WANTU, WANTV INTEGER I, J COMPLEX*16 T * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, ZGEQPF, ZGEQR2, ZGERQ2, ZLACPY, ZLAPMT, $ ZLASET, ZUNG2R, ZUNM2R, ZUNMR2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DIMAG, MAX, MIN * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) ) * .. * .. Executable Statements .. * * Test the input parameters * WANTU = LSAME( JOBU, 'U' ) WANTV = LSAME( JOBV, 'V' ) WANTQ = LSAME( JOBQ, 'Q' ) FORWRD = .TRUE. * INFO = 0 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( P.LT.0 ) THEN INFO = -5 ELSE IF( N.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -10 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN INFO = -16 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN INFO = -18 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -20 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGGSVP', -INFO ) RETURN END IF * * QR with column pivoting of B: B*P = V*( S11 S12 ) * ( 0 0 ) * DO 10 I = 1, N IWORK( I ) = 0 10 CONTINUE CALL ZGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO ) * * Update A := A*P * CALL ZLAPMT( FORWRD, M, N, A, LDA, IWORK ) * * Determine the effective rank of matrix B. * L = 0 DO 20 I = 1, MIN( P, N ) IF( CABS1( B( I, I ) ).GT.TOLB ) $ L = L + 1 20 CONTINUE * IF( WANTV ) THEN * * Copy the details of V, and form V. * CALL ZLASET( 'Full', P, P, CZERO, CZERO, V, LDV ) IF( P.GT.1 ) $ CALL ZLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ), $ LDV ) CALL ZUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO ) END IF * * Clean up B * DO 40 J = 1, L - 1 DO 30 I = J + 1, L B( I, J ) = CZERO 30 CONTINUE 40 CONTINUE IF( P.GT.L ) $ CALL ZLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB ) * IF( WANTQ ) THEN * * Set Q = I and Update Q := Q*P * CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) CALL ZLAPMT( FORWRD, N, N, Q, LDQ, IWORK ) END IF * IF( P.GE.L .AND. N.NE.L ) THEN * * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z * CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO ) * * Update A := A*Z**H * CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB, $ TAU, A, LDA, WORK, INFO ) IF( WANTQ ) THEN * * Update Q := Q*Z**H * CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B, $ LDB, TAU, Q, LDQ, WORK, INFO ) END IF * * Clean up B * CALL ZLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB ) DO 60 J = N - L + 1, N DO 50 I = J - N + L + 1, L B( I, J ) = CZERO 50 CONTINUE 60 CONTINUE * END IF * * Let N-L L * A = ( A11 A12 ) M, * * then the following does the complete QR decomposition of A11: * * A11 = U*( 0 T12 )*P1**H * ( 0 0 ) * DO 70 I = 1, N - L IWORK( I ) = 0 70 CONTINUE CALL ZGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO ) * * Determine the effective rank of A11 * K = 0 DO 80 I = 1, MIN( M, N-L ) IF( CABS1( A( I, I ) ).GT.TOLA ) $ K = K + 1 80 CONTINUE * * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N ) * CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ), $ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) * IF( WANTU ) THEN * * Copy the details of U, and form U * CALL ZLASET( 'Full', M, M, CZERO, CZERO, U, LDU ) IF( M.GT.1 ) $ CALL ZLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ), $ LDU ) CALL ZUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO ) END IF * IF( WANTQ ) THEN * * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 * CALL ZLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK ) END IF * * Clean up A: set the strictly lower triangular part of * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. * DO 100 J = 1, K - 1 DO 90 I = J + 1, K A( I, J ) = CZERO 90 CONTINUE 100 CONTINUE IF( M.GT.K ) $ CALL ZLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA ) * IF( N-L.GT.K ) THEN * * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 * CALL ZGERQ2( K, N-L, A, LDA, TAU, WORK, INFO ) * IF( WANTQ ) THEN * * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H * CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A, $ LDA, TAU, Q, LDQ, WORK, INFO ) END IF * * Clean up A * CALL ZLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA ) DO 120 J = N - L - K + 1, N - L DO 110 I = J - N + L + K + 1, K A( I, J ) = CZERO 110 CONTINUE 120 CONTINUE * END IF * IF( M.GT.K ) THEN * * QR factorization of A( K+1:M,N-L+1:N ) * CALL ZGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO ) * IF( WANTU ) THEN * * Update U(:,K+1:M) := U(:,K+1:M)*U1 * CALL ZUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ), $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU, $ WORK, INFO ) END IF * * Clean up * DO 140 J = N - L + 1, N DO 130 I = J - N + K + L + 1, M A( I, J ) = CZERO 130 CONTINUE 140 CONTINUE * END IF * RETURN * * End of ZGGSVP * END |