Provided by: liblapack-doc_3.3.1-1_all #### NAME

```       LAPACK-3  - computes the generalized singular value decomposition (GSVD) of an M-by-N real
matrix A and P-by-N real matrix B

```

#### SYNOPSIS

```       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 )

CHARACTER      JOBQ, JOBU, JOBV

INTEGER        INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

INTEGER        IWORK( * )

DOUBLE         PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ),
U( LDU, * ), V( LDV, * ), WORK( * )

```

#### 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) ).

```

#### 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**T,B**T)**T.

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

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

ALPHA   (output) DOUBLE PRECISION array, dimension (N)
BETA    (output) 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       (output) 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     (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.

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

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

WORK    (workspace) DOUBLE PRECISION 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 DTGSJA.

```

#### 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.
Further Details
===============
2-96 Based on modifications by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA

LAPACK driver routine (version 3.3.1)      April 2011                            DGGSVD(3lapack)
```