*> \brief \b SGGSVP3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGGSVP3 + dependencies
*>
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*
* Definition:
* ===========
*
* SUBROUTINE SGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
* TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
* IWORK, 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 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGGSVP3 computes orthogonal matrices U, V and Q such that
*>
*> N-K-L K L
*> U**T*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**T*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**T,B**T)**T.
*>
*> This decomposition is the preprocessing step for computing the
*> Generalized Singular Value Decomposition (GSVD), see subroutine
*> SGGSVD3.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'U': Orthogonal matrix U is computed;
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'V': Orthogonal matrix V is computed;
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*> JOBQ is CHARACTER*1
*> = 'Q': Orthogonal 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 REAL 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 REAL 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**T,B**T)**T.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is REAL array, dimension (LDU,M)
*> If JOBU = 'U', U contains the orthogonal 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 REAL array, dimension (LDV,P)
*> If JOBV = 'V', V contains the orthogonal 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 REAL array, dimension (LDQ,N)
*> If JOBQ = 'Q', Q contains the orthogonal 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] TAU
*> \verbatim
*> TAU is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION 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 realOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The subroutine uses LAPACK subroutine SGEQP3 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.
*>
*> SGGSVP3 replaces the deprecated subroutine SGGSVP.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
$ IWORK, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.6.0) --
* -- 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 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL FORWRD, WANTQ, WANTU, WANTV, LQUERY
INTEGER I, J, LWKOPT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SGEQP3, SGEQR2, SGERQ2, SLACPY, SLAPMT,
$ SLASET, SORG2R, SORM2R, SORMR2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. 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 SGEQP3( P, N, B, LDB, IWORK, TAU, WORK, -1, 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 SGEQP3( M, N, A, LDA, IWORK, TAU, WORK, -1, INFO )
LWKOPT = MAX( LWKOPT, INT( WORK ( 1 ) ) )
LWKOPT = MAX( 1, LWKOPT )
WORK( 1 ) = REAL( LWKOPT )
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGGSVP3', -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 SGEQP3( P, N, B, LDB, IWORK, TAU, WORK, LWORK, INFO )
*
* Update A := A*P
*
CALL SLAPMT( 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 SLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
IF( P.GT.1 )
$ CALL SLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
$ LDV )
CALL SORG2R( 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 ) = ZERO
30 CONTINUE
40 CONTINUE
IF( P.GT.L )
$ CALL SLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
*
IF( WANTQ ) THEN
*
* Set Q = I and Update Q := Q*P
*
CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
CALL SLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
END IF
*
IF( P.GE.L .AND. N.NE.L ) THEN
*
* RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
*
CALL SGERQ2( L, N, B, LDB, TAU, WORK, INFO )
*
* Update A := A*Z**T
*
CALL SORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
$ LDA, WORK, INFO )
*
IF( WANTQ ) THEN
*
* Update Q := Q*Z**T
*
CALL SORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
$ LDQ, WORK, INFO )
END IF
*
* Clean up B
*
CALL SLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
DO 60 J = N - L + 1, N
DO 50 I = J - N + L + 1, L
B( I, J ) = ZERO
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**T
* ( 0 0 )
*
DO 70 I = 1, N - L
IWORK( I ) = 0
70 CONTINUE
CALL SGEQP3( M, N-L, A, LDA, IWORK, TAU, WORK, LWORK, 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**T*A12, where A12 = A( 1:M, N-L+1:N )
*
CALL SORM2R( 'Left', '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 SLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
IF( M.GT.1 )
$ CALL SLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
$ LDU )
CALL SORG2R( 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 SLAPMT( 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 ) = ZERO
90 CONTINUE
100 CONTINUE
IF( M.GT.K )
$ CALL SLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
*
IF( N-L.GT.K ) THEN
*
* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
*
CALL SGERQ2( 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**T
*
CALL SORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
$ Q, LDQ, WORK, INFO )
END IF
*
* Clean up A
*
CALL SLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
DO 120 J = N - L - K + 1, N - L
DO 110 I = J - N + L + K + 1, K
A( I, J ) = ZERO
110 CONTINUE
120 CONTINUE
*
END IF
*
IF( M.GT.K ) THEN
*
* QR factorization of A( K+1:M,N-L+1:N )
*
CALL SGEQR2( 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 SORM2R( '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 ) = ZERO
130 CONTINUE
140 CONTINUE
*
END IF
*
WORK( 1 ) = REAL( LWKOPT )
RETURN
*
* End of SGGSVP3
*
END