Homepage > Man Pages > Category > Subroutines
Homepage > Man Pages > Name > L

# LAPACK-3

## man page of LAPACK-3

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

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

SYNOPSIS
SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L,  A,  LDA,  B,  LDB,
ALPHA,  BETA,  U,  LDU, V, LDV, Q, LDQ, WORK, RWORK,
IWORK, INFO )

CHARACTER      JOBQ, JOBU, JOBV

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

INTEGER        IWORK( * )

DOUBLE         PRECISION ALPHA( * ), BETA( * ), RWORK( * )

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

PURPOSE
ZGGSVD  computes the generalized singular value decomposition (GSVD) of
an M-by-N complex matrix A and P-by-N complex matrix B:
U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
where U, V and Q are unitary matrices, and Z' means the conjugate
transpose of Z.  Let K+L = the effective numerical rank of the
matrix (A',B')', 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 unitary
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'.
If ( A',B')' has orthnormal 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'*A x = lambda* B'*B x.
In some literature, the GSVD of A and B is presented in the form
U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and 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':  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.

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 Purpose.
K + L = effective numerical rank of (A',B')'.

A       (input/output) COMPLEX*16 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) COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of 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) COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M 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 P-by-P 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 N-by-N 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.

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

RWORK   (workspace) DOUBLE PRECISION array, dimension (2*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 ZTGSJA.

PARAMETERS
TOLA    DOUBLE PRECISION
TOLB    DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
rank of (A',B')'. 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.March 2011                       LAPACK-3(3)
```

Copyright © 2011–2018 by topics-of-interest.com . All rights reserved. Hosted by all-inkl.
Contact · Imprint · Privacy

Page generated in 24.24ms.

schuhefinden.de | tier-bedarf.com | autoresponder.name