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SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
$ IWORK, INFO )
*
* -- LAPACK driver 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
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
$ V( LDV, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGGSVD computes the generalized singular value decomposition (GSVD)
* of an M-by-N real matrix A and P-by-N real matrix B:
*
* U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
*
* where U, V and Q are orthogonal matrices.
* Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
* then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
* D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
* following structures, respectively:
*
* If M-K-L >= 0,
*
* K L
* D1 = K ( I 0 )
* L ( 0 C )
* M-K-L ( 0 0 )
*
* K L
* D2 = L ( 0 S )
* P-L ( 0 0 )
*
* N-K-L K L
* ( 0 R ) = K ( 0 R11 R12 )
* L ( 0 0 R22 )
*
* where
*
* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
* S = diag( BETA(K+1), ... , BETA(K+L) ),
* C**2 + S**2 = I.
*
* R is stored in A(1:K+L,N-K-L+1:N) on exit.
*
* If M-K-L < 0,
*
* K M-K K+L-M
* D1 = K ( I 0 0 )
* M-K ( 0 C 0 )
*
* K M-K K+L-M
* D2 = M-K ( 0 S 0 )
* K+L-M ( 0 0 I )
* P-L ( 0 0 0 )
*
* N-K-L K M-K K+L-M
* ( 0 R ) = K ( 0 R11 R12 R13 )
* M-K ( 0 0 R22 R23 )
* K+L-M ( 0 0 0 R33 )
*
* where
*
* C = diag( ALPHA(K+1), ... , ALPHA(M) ),
* S = diag( BETA(K+1), ... , BETA(M) ),
* C**2 + S**2 = I.
*
* (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
* ( 0 R22 R23 )
* in B(M-K+1:L,N+M-K-L+1:N) on exit.
*
* The routine computes C, S, R, and optionally the orthogonal
* transformation matrices U, V and Q.
*
* In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
* A and B implicitly gives the SVD of A*inv(B):
* A*inv(B) = U*(D1*inv(D2))*V**T.
* If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is
* also equal to the CS decomposition of A and B. Furthermore, the GSVD
* can be used to derive the solution of the eigenvalue problem:
* A**T*A x = lambda* B**T*B x.
* In some literature, the GSVD of A and B is presented in the form
* U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
* where U and V are orthogonal and X is nonsingular, D1 and D2 are
* ``diagonal''. The former GSVD form can be converted to the latter
* form by taking the nonsingular matrix X as
*
* X = Q*( I 0 )
* ( 0 inv(R) ).
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* = 'U': Orthogonal matrix U is computed;
* = 'N': U is not computed.
*
* JOBV (input) CHARACTER*1
* = 'V': Orthogonal matrix V is computed;
* = 'N': V is not computed.
*
* JOBQ (input) CHARACTER*1
* = 'Q': Orthogonal matrix Q is computed;
* = 'N': Q is not computed.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* K (output) INTEGER
* L (output) INTEGER
* On exit, K and L specify the dimension of the subblocks
* described in the Purpose section.
* K + L = effective numerical rank of (A**T,B**T)**T.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A contains the triangular matrix R, or part of R.
* See Purpose for details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
* On entry, the P-by-N matrix B.
* On exit, B contains the triangular matrix R if M-K-L < 0.
* See Purpose for details.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* ALPHA (output) DOUBLE PRECISION array, dimension (N)
* BETA (output) DOUBLE PRECISION array, dimension (N)
* On exit, ALPHA and BETA contain the generalized singular
* value pairs of A and B;
* ALPHA(1:K) = 1,
* BETA(1:K) = 0,
* and if M-K-L >= 0,
* ALPHA(K+1:K+L) = C,
* BETA(K+1:K+L) = S,
* or if M-K-L < 0,
* ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
* BETA(K+1:M) =S, BETA(M+1:K+L) =1
* and
* ALPHA(K+L+1:N) = 0
* BETA(K+L+1:N) = 0
*
* U (output) DOUBLE PRECISION array, dimension (LDU,M)
* If JOBU = 'U', U contains the M-by-M orthogonal 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) DOUBLE PRECISION array, dimension (LDV,P)
* If JOBV = 'V', V contains the P-by-P orthogonal 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) DOUBLE PRECISION array, dimension (LDQ,N)
* If JOBQ = 'Q', Q contains the N-by-N orthogonal 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.
*
* WORK (workspace) DOUBLE PRECISION array,
* dimension (max(3*N,M,P)+N)
*
* IWORK (workspace/output) INTEGER array, dimension (N)
* On exit, IWORK stores the sorting information. More
* precisely, the following loop will sort ALPHA
* for I = K+1, min(M,K+L)
* swap ALPHA(I) and ALPHA(IWORK(I))
* endfor
* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = 1, the Jacobi-type procedure failed to
* converge. For further details, see subroutine DTGSJA.
*
* Internal Parameters
* ===================
*
* TOLA DOUBLE PRECISION
* TOLB DOUBLE PRECISION
* TOLA and TOLB are the thresholds to determine the effective
* rank of (A',B')**T. 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.
*
* Further Details
* ===============
*
* 2-96 Based on modifications by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL WANTQ, WANTU, WANTV
INTEGER I, IBND, ISUB, J, NCYCLE
DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTU = LSAME( JOBU, 'U' )
WANTV = LSAME( JOBV, 'V' )
WANTQ = LSAME( JOBQ, 'Q' )
*
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( N.LT.0 ) THEN
INFO = -5
ELSE IF( P.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -12
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( 'DGGSVD', -INFO )
RETURN
END IF
*
* Compute the Frobenius norm of matrices A and B
*
ANORM = DLANGE( '1', M, N, A, LDA, WORK )
BNORM = DLANGE( '1', P, N, B, LDB, WORK )
*
* Get machine precision and set up threshold for determining
* the effective numerical rank of the matrices A and B.
*
ULP = DLAMCH( 'Precision' )
UNFL = DLAMCH( 'Safe Minimum' )
TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
*
* Preprocessing
*
CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
$ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
$ WORK( N+1 ), INFO )
*
* Compute the GSVD of two upper "triangular" matrices
*
CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
$ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
$ WORK, NCYCLE, INFO )
*
* Sort the singular values and store the pivot indices in IWORK
* Copy ALPHA to WORK, then sort ALPHA in WORK
*
CALL DCOPY( N, ALPHA, 1, WORK, 1 )
IBND = MIN( L, M-K )
DO 20 I = 1, IBND
*
* Scan for largest ALPHA(K+I)
*
ISUB = I
SMAX = WORK( K+I )
DO 10 J = I + 1, IBND
TEMP = WORK( K+J )
IF( TEMP.GT.SMAX ) THEN
ISUB = J
SMAX = TEMP
END IF
10 CONTINUE
IF( ISUB.NE.I ) THEN
WORK( K+ISUB ) = WORK( K+I )
WORK( K+I ) = SMAX
IWORK( K+I ) = K + ISUB
ELSE
IWORK( K+I ) = K + I
END IF
20 CONTINUE
*
RETURN
*
* End of DGGSVD
*
END
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