dggsvd.f man page

dggsvd.f —

Synopsis

Functions/Subroutines

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)
DGGSVD computes the singular value decomposition (SVD) for OTHER matrices

Function/Subroutine Documentation

subroutine dggsvd (characterJOBU, characterJOBV, characterJOBQ, integerM, integerN, integerP, integerK, integerL, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( * )ALPHA, double precision, dimension( * )BETA, double precision, dimension( ldu, * )U, integerLDU, double precision, dimension( ldv, * )V, integerLDV, double precision, dimension( ldq, * )Q, integerLDQ, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)

DGGSVD computes the singular value decomposition (SVD) for OTHER matrices

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) ).

Parameters:

JOBU

JOBU is CHARACTER*1
= 'U':  Orthogonal matrix U is computed;
= 'N':  U is not computed.

JOBV

JOBV is CHARACTER*1
= 'V':  Orthogonal matrix V is computed;
= 'N':  V is not computed.

JOBQ

JOBQ is CHARACTER*1
= 'Q':  Orthogonal matrix Q is computed;
= 'N':  Q is not computed.

M

M is INTEGER
The number of rows of the matrix A.  M >= 0.

N

N is INTEGER
The number of columns of the matrices A and B.  N >= 0.

P

P is INTEGER
The number of rows of the matrix B.  P >= 0.

K

K is INTEGER

L

L is INTEGER

On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**T,B**T)**T.

A

A is 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

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B

B is 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

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).

ALPHA

ALPHA is DOUBLE PRECISION array, dimension (N)

BETA

BETA is 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

U is 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

LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

V

V is 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

LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.

Q

Q is 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

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK

WORK is DOUBLE PRECISION array,
            dimension (max(3*N,M,P)+N)

IWORK

IWORK is 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

INFO is 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 331 of file dggsvd.f.

Author

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Referenced By

dggsvd(3) is an alias of dggsvd.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK