sggsvd.f

          SUBROUTINE SGGSVD( 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 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBQ, JOBU, JOBV
          INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          REAL               A( LDA, * ), ALPHA( * ), B( LDB, * ),
         $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
         $                   V( LDV, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGGSVD computes the generalized singular value decomposition (GSVD)
    *  of an M-by-N real matrix A and P-by-N real matrix B:
    *
    *      U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
    *
    *  where U, V and Q are orthogonal matrices, and Z' is the transpose
    *  of Z.  Let K+L = the effective numerical rank of the matrix (A',B')',
    *  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'.
    *  If ( A',B')' 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'*A x = lambda* B'*B x.
    *  In some literature, the GSVD of A and B is presented in the form
    *                   U'*A*X = ( 0 D1 ),   V'*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',B')'.
    *
    *  A       (input/output) REAL 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) REAL 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) REAL array, dimension (N)
    *  BETA    (output) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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 STGSJA.
    *
    *  Internal Parameters
    *  ===================
    *
    *  TOLA    REAL
    *  TOLB    REAL
    *          TOLA and TOLB are the thresholds to determine the effective
    *          rank of (A',B')'. 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.
    *
    *  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
          REAL               ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          REAL               SLAMCH, SLANGE
          EXTERNAL           LSAME, SLAMCH, SLANGE
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SCOPY, SGGSVP, STGSJA, 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( 'SGGSVD', -INFO )
             RETURN
          END IF
    *
    *     Compute the Frobenius norm of matrices A and B
    *
          ANORM = SLANGE( '1', M, N, A, LDA, WORK )
          BNORM = SLANGE( '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 = SLAMCH( 'Precision' )
          UNFL = SLAMCH( 'Safe Minimum' )
          TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
          TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
    *
    *     Preprocessing
    *
          CALL SGGSVP( 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 STGSJA( 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 SCOPY( 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 SGGSVD
    *
          END