cggsvp(3)

NAME

CGGSVP - compute 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  CGGSVP(  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
    REAL           TOLA, TOLB
    INTEGER        IWORK( * )
    REAL           RWORK( * )
    COMPLEX        A( LDA, * ), B( LDB, * ), Q( LDQ, *  ),
TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )

PURPOSE

CGGSVP 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; 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 non

singular 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 subrou

tine CGGSVD.

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 array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, A con

tains 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 array, dimension (LDB,N)

On entry, the P-by-N matrix B. On exit, B con

tains the triangular matrix described in the Purpose section.

LDB (input) INTEGER

The leading dimension of the array B. LDB >=

max(1,P).

TOLA (input) REAL

TOLB (input) REAL TOLA and TOLB are the thresh

olds 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)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS. 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 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 array, dimension (LDV,M)

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 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) REAL array, dimension (2*N)

TAU (workspace) COMPLEX array, dimension (N)

WORK (workspace) COMPLEX array, dimension

(max(3*N,M,P))

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

The subroutine uses LAPACK subroutine CGEQPF for the QR

factorization with column pivoting to detect the effective numer

ical rank of the a matrix. It may be replaced by a better rank

determination strategy.

LAPACK version 3.0 15 June 2000

CGGSVP(3)