# dtgsja.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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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 (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision TOLA, double precision TOLB, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK, integer NCYCLE, integer INFO)

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

NAG Ltd.

Date:
December 2016

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 380 of file dtgsja.f.

## Author

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