dtgsja.f man page

dtgsja.f —

Synopsis

Functions/Subroutines

subroutine 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)
DTGSJA

Function/Subroutine Documentation

subroutine dtgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN, integerK, integerL, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precisionTOLA, double precisionTOLB, 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, integerNCYCLE, integerINFO)

DTGSJA

Purpose:

DTGSJA computes the generalized singular value decomposition (GSVD)
of two real upper triangular (or trapezoidal) matrices A and B.

On entry, it is assumed that matrices A and B have the following
forms, which may be obtained by the preprocessing subroutine DGGSVP
from a general M-by-N matrix A and P-by-N matrix B:

             N-K-L  K    L
   A =    K ( 0    A12  A13 ) if M-K-L >= 0;
          L ( 0     0   A23 )
      M-K-L ( 0     0    0  )

           N-K-L  K    L
   A =  K ( 0    A12  A13 ) if M-K-L < 0;
      M-K ( 0     0   A23 )

           N-K-L  K    L
   B =  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.

On exit,

       U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),

where U, V and Q are orthogonal matrices.
R is a nonsingular upper triangular matrix, and D1 and D2 are
“diagonal” matrices, which are of the following structures:

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 ) K
            L (  0    0   R22 ) L

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.

R = ( 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 computation of the orthogonal transformation matrices U, V or Q
is optional.  These matrices may either be formed explicitly, or they
may be postmultiplied into input matrices U1, V1, or Q1.

Parameters:

JOBU

JOBU is CHARACTER*1
= 'U':  U must contain an orthogonal matrix U1 on entry, and
        the product U1*U is returned;
= 'I':  U is initialized to the unit matrix, and the
        orthogonal matrix U is returned;
= 'N':  U is not computed.

JOBV

JOBV is CHARACTER*1
= 'V':  V must contain an orthogonal matrix V1 on entry, and
        the product V1*V is returned;
= 'I':  V is initialized to the unit matrix, and the
        orthogonal matrix V is returned;
= 'N':  V is not computed.

JOBQ

JOBQ is CHARACTER*1
= 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
        the product Q1*Q is returned;
= 'I':  Q is initialized to the unit matrix, and the
        orthogonal matrix Q is returned;
= 'N':  Q is not computed.

M

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

P

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

N

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

K

K is INTEGER

L

L is INTEGER

K and L specify the subblocks in the input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
of A and B, whose GSVD is going to be computed by DTGSJA.
See Further Details.

A

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
a part of R.  See Purpose for details.

LDB

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

TOLA

TOLA is DOUBLE PRECISION

TOLB

TOLB is DOUBLE PRECISION

TOLA and TOLB are the convergence criteria for the Jacobi-
Kogbetliantz iteration procedure. Generally, they are the
same as used in the preprocessing step, say
    TOLA = max(M,N)*norm(A)*MAZHEPS,
    TOLB = max(P,N)*norm(B)*MAZHEPS.

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) = diag(C),
  BETA(K+1:K+L)  = diag(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.
Furthermore, if K+L < N,
  ALPHA(K+L+1:N) = 0 and
  BETA(K+L+1:N)  = 0.

U

U is DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBU = 'I', U contains the orthogonal matrix U;
if JOBU = 'U', U contains the product U1*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)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBV = 'I', V contains the orthogonal matrix V;
if JOBV = 'V', V contains the product V1*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)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBQ = 'I', Q contains the orthogonal matrix Q;
if JOBQ = 'Q', Q contains the product Q1*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 (2*N)

NCYCLE

NCYCLE is INTEGER
The number of cycles required for convergence.

INFO

INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
= 1:  the procedure does not converge after MAXIT cycles.
  Internal Parameters
  ===================

  MAXIT   INTEGER
          MAXIT specifies the total loops that the iterative procedure
          may take. If after MAXIT cycles, the routine fails to
          converge, we return INFO = 1..fi

 

Author:
Univ. of Tennessee 

Univ. of California Berkeley 

Univ. of Colorado Denver 

NAG Ltd. 

Date:
November 2011 

Further Details: 

  DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
  matrix B13 to the form:

           U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

  where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
  of Z.  C1 and S1 are diagonal matrices satisfying

                C1**2 + S1**2 = I,

  and R1 is an L-by-L nonsingular upper triangular matrix.

Definition at line 377 of file dtgsja.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

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

Sat Nov 16 2013 Version 3.4.2 LAPACK