sggrqf(3)

NAME

SGGRQF - compute a generalized RQ factorization of an M

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

SYNOPSIS

SUBROUTINE  SGGRQF(  M,  P, N, A, LDA, TAUA, B, LDB, TAUB,
WORK, LWORK, INFO )
    INTEGER        INFO, LDA, LDB, LWORK, M, N, P
    REAL           A( LDA, * ), B( LDB, * ),  TAUA(  *  ),
TAUB( * ), WORK( * )

PURPOSE

SGGRQF 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 orthogonal matrix, Z is a P-by-P or

thogonal 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 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) REAL 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 orthogonal matrix Q as a product of elementary re

flectors (see Further Details).

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,M).

TAUA (output) REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors

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

(input/output) REAL 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 diagonal,

with the array TAUB, represent the orthogonal matrix Z as a prod

uct of elementary reflectors (see Further Details). LDB (in

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

max(1,P).

TAUB (output) REAL array, dimension (min(P,N))

The scalar factors of the elementary reflectors

which represent the orthogonal matrix Z (see Further Details).

WORK (workspace/output) REAL 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 SORMRQ.

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 INF0= -i, the i-th argument had an ille

gal 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 real scalar, and v is a real 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 SORGRQ.
To use Q to update another matrix, use LAPACK subroutine

SORMRQ.

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 real scalar, and v is a real 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 SORGQR.
To use Z to update another matrix, use LAPACK subroutine

SORMQR.

LAPACK version 3.0 15 June 2000

SGGRQF(3)