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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: computes unitary matrices U, V and Q such that N-K-L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L &gt;= 0

```NAME
LAPACK-3  -  computes unitary matrices U, V and Q such that   N-K-L K L
U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0

SYNOPSIS
SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A,  LDA,  B,  LDB,  TOLA,
TOLB,  K,  L,  U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
TAU, WORK, INFO )

CHARACTER      JOBQ, JOBU, JOBV

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

DOUBLE         PRECISION TOLA, TOLB

INTEGER        IWORK( * )

DOUBLE         PRECISION RWORK( * )

COMPLEX*16     A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ),  U(
LDU, * ), V( LDV, * ), WORK( * )

PURPOSE
ZGGSVP computes unitary matrices U, V and Q such that
L ( 0     0   A23 )
M-K-L ( 0     0    0  )
N-K-L  K    L
=     K ( 0    A12  A13 )  if M-K-L < 0;
M-K ( 0     0   A23 )
N-K-L  K    L
V'*B*Q =   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.  K+L = the effective
numerical rank of the (M+P)-by-N matrix (A',B')'.  Z' denotes the
conjugate transpose of Z.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
ZGGSVD.

ARGUMENTS
JOBU    (input) CHARACTER*1
= 'U':  Unitary matrix U is computed;
= 'N':  U is not computed.

JOBV    (input) CHARACTER*1
= 'V':  Unitary matrix V is computed;
= 'N':  V is not computed.

JOBQ    (input) CHARACTER*1
= 'Q':  Unitary matrix Q is computed;
= 'N':  Q is not computed.

M       (input) INTEGER
The number of rows of the matrix A.  M >= 0.

P       (input) INTEGER
The number of rows of the matrix B.  P >= 0.

N       (input) INTEGER
The number of columns of the matrices A and B.  N >= 0.

A       (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section.

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

B       (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix described in
the Purpose section.

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

TOLA    (input) DOUBLE PRECISION
TOLB    (input) DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
numerical rank of matrix B and a subblock of A. 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.

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

U       (output) COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the unitary 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) COMPLEX*16 array, dimension (LDV,P)
If JOBV = 'V', V contains the unitary 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) COMPLEX*16 array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the unitary 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.

IWORK   (workspace) INTEGER array, dimension (N)

RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)

TAU     (workspace) COMPLEX*16 array, dimension (N)

WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P))

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS
The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
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