zggrqf(3)

NAME

ZGGRQF - compute a generalized RQ factorization of an M

by-N matrix A and a P-by-N matrix B

SYNOPSIS

SUBROUTINE  ZGGRQF(  M,  P, N, A, LDA, TAUA, B, LDB, TAUB,
WORK, LWORK, INFO )
    INTEGER        INFO, LDA, LDB, LWORK, M, N, P
    COMPLEX*16     A( LDA, * ), B( LDB, * ),  TAUA(  *  ),
TAUB( * ), WORK( * )

PURPOSE

ZGGRQF computes a generalized RQ factorization of an M-by

N matrix A and a P-by-N matrix B:

A = R*Q, B = Z*T*Q,

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary

matrix, and R and T assume one of the forms:

if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 )

M-N,

N-M M ( R21 )

N

N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11

T12 ) P,

( 0 ) P-N P N

P

N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ

factorization of A and B implicitly gives the RQ factorization of

A*inv(B):

A*inv(B) = (R*inv(T))*Z'

where inv(B) denotes the inverse of the matrix B, and Z'

denotes the conjugate transpose of the matrix Z.

ARGUMENTS

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, if M <=

N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the

M-by-M upper triangular matrix R; if M > N, the elements on and

above the (M-N)-th subdiagonal contain the M-by-N upper trape

zoidal matrix R; the remaining elements, with the array TAUA,

represent the unitary matrix Q as a product of elementary reflec

tors (see Further Details).

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,M).

TAUA (output) COMPLEX*16 array, dimension (min(M,N))

The scalar factors of the elementary reflectors

which represent the unitary matrix Q (see Further Details). B

(input/output) COMPLEX*16 array, dimension (LDB,N) On entry, the

P-by-N matrix B. On exit, the elements on and above the diagonal

of the array contain the min(P,N)-by-N upper trapezoidal matrix T

(T is upper triangular if P >= N); the elements below the diago

nal, with the array TAUB, represent the unitary matrix Z as a

product of elementary reflectors (see Further Details). LDB

(input) INTEGER The leading dimension of the array B. LDB >=

max(1,P).

TAUB (output) COMPLEX*16 array, dimension (min(P,N))

The scalar factors of the elementary reflectors

which represent the unitary matrix Z (see Further Details). WORK

(workspace/output) COMPLEX*16 array, dimension (LWORK) On exit,

if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >=

max(1,N,M,P). For optimum performance LWORK >=

max(N,M,P)*max(NB1,NB2,NB3), where NB1 is the optimal blocksize

for the RQ factorization of an M-by-N matrix, NB2 is the optimal

blocksize for the QR factorization of a P-by-N matrix, and NB3 is

the optimal blocksize for a call of ZUNMRQ.

If LWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the WORK array,

returns this value as the first entry of the WORK array, and no

error message related to LWORK is issued by XERBLA.

INFO (output) INTEGER

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

value.

FURTHER DETAILS

The matrix Q is represented as a product of elementary re

flectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - taua * v * v'

where taua is a complex scalar, and v is a complex vector

with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on

exit in A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine ZUNGRQ.
To use Q to update another matrix, use LAPACK subroutine

ZUNMRQ.

The matrix Z is represented as a product of elementary re

flectors

Z = H(1) H(2) . . . H(k), where k = min(p,n).

Each H(i) has the form

H(i) = I - taub * v * v'

where taub is a complex scalar, and v is a complex vector

with v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in

B(i+1:p,i), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine ZUNGQR.
To use Z to update another matrix, use LAPACK subroutine

ZUNMQR.

LAPACK version 3.0 15 June 2000

ZGGRQF(3)