*> \brief \b CGGSVP3 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CGGSVP3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, * TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, * IWORK, RWORK, TAU, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBQ, JOBU, JOBV * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK * REAL TOLA, TOLB * .. * .. Array Arguments .. * INTEGER IWORK( * ) * REAL RWORK( * ) * COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CGGSVP3 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 *> CGGSVD3. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBU *> \verbatim *> JOBU is CHARACTER*1 *> = 'U': Unitary matrix U is computed; *> = 'N': U is not computed. *> \endverbatim *> *> \param[in] JOBV *> \verbatim *> JOBV is CHARACTER*1 *> = 'V': Unitary matrix V is computed; *> = 'N': V is not computed. *> \endverbatim *> *> \param[in] JOBQ *> \verbatim *> JOBQ is CHARACTER*1 *> = 'Q': Unitary matrix Q is computed; *> = 'N': Q is not computed. *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] P *> \verbatim *> P is INTEGER *> The number of rows of the matrix B. P >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX array, dimension (LDB,N) *> On entry, the P-by-N matrix B. *> On exit, B contains the triangular matrix described in *> the Purpose section. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,P). *> \endverbatim *> *> \param[in] TOLA *> \verbatim *> TOLA is REAL *> \endverbatim *> *> \param[in] TOLB *> \verbatim *> TOLB is REAL *> *> 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)*MACHEPS, *> TOLB = MAX(P,N)*norm(B)*MACHEPS. *> The size of TOLA and TOLB may affect the size of backward *> errors of the decomposition. *> \endverbatim *> *> \param[out] K *> \verbatim *> K is INTEGER *> \endverbatim *> *> \param[out] L *> \verbatim *> L is 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. *> \endverbatim *> *> \param[out] U *> \verbatim *> U is COMPLEX array, dimension (LDU,M) *> If JOBU = 'U', U contains the unitary matrix U. *> If JOBU = 'N', U is not referenced. *> \endverbatim *> *> \param[in] LDU *> \verbatim *> LDU is INTEGER *> The leading dimension of the array U. LDU >= max(1,M) if *> JOBU = 'U'; LDU >= 1 otherwise. *> \endverbatim *> *> \param[out] V *> \verbatim *> V is COMPLEX array, dimension (LDV,P) *> If JOBV = 'V', V contains the unitary matrix V. *> If JOBV = 'N', V is not referenced. *> \endverbatim *> *> \param[in] LDV *> \verbatim *> LDV is INTEGER *> The leading dimension of the array V. LDV >= max(1,P) if *> JOBV = 'V'; LDV >= 1 otherwise. *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is COMPLEX array, dimension (LDQ,N) *> If JOBQ = 'Q', Q contains the unitary matrix Q. *> If JOBQ = 'N', Q is not referenced. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= max(1,N) if *> JOBQ = 'Q'; LDQ >= 1 otherwise. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is REAL array, dimension (2*N) *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is COMPLEX array, dimension (N) *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date August 2015 * *> \ingroup complexOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> The subroutine uses LAPACK subroutine CGEQP3 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. *> *> CGGSVP3 replaces the deprecated subroutine CGGSVP. *> *> \endverbatim *> * ===================================================================== SUBROUTINE CGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, $ IWORK, RWORK, TAU, WORK, LWORK, INFO ) * * -- LAPACK computational routine (version 3.6.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * August 2015 * IMPLICIT NONE * * .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, $ LWORK REAL TOLA, TOLB * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL RWORK( * ) COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX CZERO, CONE PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), $ CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL FORWRD, WANTQ, WANTU, WANTV, LQUERY INTEGER I, J, LWKOPT * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL CGEQP3, CGEQR2, CGERQ2, CLACPY, CLAPMT, $ CLASET, CUNG2R, CUNM2R, CUNMR2, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, AIMAG, MAX, MIN, REAL * .. * .. Executable Statements .. * * Test the input parameters * WANTU = LSAME( JOBU, 'U' ) WANTV = LSAME( JOBV, 'V' ) WANTQ = LSAME( JOBQ, 'Q' ) FORWRD = .TRUE. LQUERY = ( LWORK.EQ.-1 ) LWKOPT = 1 * * Test the input arguments * 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 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -24 END IF * * Compute workspace * IF( INFO.EQ.0 ) THEN CALL CGEQP3( P, N, B, LDB, IWORK, TAU, WORK, -1, RWORK, INFO ) LWKOPT = INT( WORK ( 1 ) ) IF( WANTV ) THEN LWKOPT = MAX( LWKOPT, P ) END IF LWKOPT = MAX( LWKOPT, MIN( N, P ) ) LWKOPT = MAX( LWKOPT, M ) IF( WANTQ ) THEN LWKOPT = MAX( LWKOPT, N ) END IF CALL CGEQP3( M, N, A, LDA, IWORK, TAU, WORK, -1, RWORK, INFO ) LWKOPT = MAX( LWKOPT, INT( WORK ( 1 ) ) ) LWKOPT = MAX( 1, LWKOPT ) WORK( 1 ) = CMPLX( LWKOPT ) END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CGGSVP3', -INFO ) RETURN END IF IF( LQUERY ) THEN RETURN ENDIF * * QR with column pivoting of B: B*P = V*( S11 S12 ) * ( 0 0 ) * DO 10 I = 1, N IWORK( I ) = 0 10 CONTINUE CALL CGEQP3( P, N, B, LDB, IWORK, TAU, WORK, LWORK, RWORK, INFO ) * * Update A := A*P * CALL CLAPMT( FORWRD, M, N, A, LDA, IWORK ) * * Determine the effective rank of matrix B. * L = 0 DO 20 I = 1, MIN( P, N ) IF( ABS( B( I, I ) ).GT.TOLB ) $ L = L + 1 20 CONTINUE * IF( WANTV ) THEN * * Copy the details of V, and form V. * CALL CLASET( 'Full', P, P, CZERO, CZERO, V, LDV ) IF( P.GT.1 ) $ CALL CLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ), $ LDV ) CALL CUNG2R( 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 CLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB ) * IF( WANTQ ) THEN * * Set Q = I and Update Q := Q*P * CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) CALL CLAPMT( 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 CGERQ2( L, N, B, LDB, TAU, WORK, INFO ) * * Update A := A*Z**H * CALL CUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB, $ TAU, A, LDA, WORK, INFO ) IF( WANTQ ) THEN * * Update Q := Q*Z**H * CALL CUNMR2( 'Right', 'Conjugate transpose', N, N, L, B, $ LDB, TAU, Q, LDQ, WORK, INFO ) END IF * * Clean up B * CALL CLASET( '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 CGEQP3( M, N-L, A, LDA, IWORK, TAU, WORK, LWORK, RWORK, $ INFO ) * * Determine the effective rank of A11 * K = 0 DO 80 I = 1, MIN( M, N-L ) IF( ABS( 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 CUNM2R( '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 CLASET( 'Full', M, M, CZERO, CZERO, U, LDU ) IF( M.GT.1 ) $ CALL CLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ), $ LDU ) CALL CUNG2R( 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 CLAPMT( 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 CLASET( '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 CGERQ2( 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 CUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A, $ LDA, TAU, Q, LDQ, WORK, INFO ) END IF * * Clean up A * CALL CLASET( '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 CGEQR2( 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 CUNM2R( '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 * WORK( 1 ) = CMPLX( LWKOPT ) RETURN * * End of CGGSVP3 * END