ctgsja.f man page

ctgsja.f —

Synopsis

Functions/Subroutines

subroutine ctgsja (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)
CTGSJA

Function/Subroutine Documentation

subroutine ctgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN, integerK, integerL, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA, complex, dimension( ldu, * )U, integerLDU, complex, dimension( ldv, * )V, integerLDV, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( * )WORK, integerNCYCLE, integerINFO)

CTGSJA

Purpose:

CTGSJA computes the generalized singular value decomposition (GSVD)
of two complex 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 CGGSVP
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**H *A*Q = D1*( 0 R ),    V**H *B*Q = D2*( 0 R ),

where U, V and Q are unitary 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 unitary 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 a unitary matrix U1 on entry, and
        the product U1*U is returned;
= 'I':  U is initialized to the unit matrix, and the
        unitary matrix U is returned;
= 'N':  U is not computed.

JOBV

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

JOBQ

JOBQ is CHARACTER*1
= 'Q':  Q must contain a unitary matrix Q1 on entry, and
        the product Q1*Q is returned;
= 'I':  Q is initialized to the unit matrix, and the
        unitary 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 CTGSJA.
See Further Details.

A

A is COMPLEX 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 COMPLEX 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 REAL

TOLB

TOLB is REAL

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)*MACHEPS,
    TOLB = MAX(P,N)*norm(B)*MACHEPS.

ALPHA

ALPHA is REAL array, dimension (N)

BETA

BETA is 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) = 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
  BETA(K+L+1:N)  = 0.

U

U is COMPLEX array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the unitary matrix returned by CGGSVP).
On exit,
if JOBU = 'I', U contains the unitary 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 COMPLEX array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the unitary matrix returned by CGGSVP).
On exit,
if JOBV = 'I', V contains the unitary 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 COMPLEX array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the unitary matrix returned by CGGSVP).
On exit,
if JOBQ = 'I', Q contains the unitary 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 COMPLEX 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

CTGSJA 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**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,

where U1, V1 and Q1 are unitary matrix.
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 378 of file ctgsja.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

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

Sat Nov 16 2013 Version 3.4.2 LAPACK