DGGQRF(l) — LAPACK routine (version 2.0)

NAME

DGGQRF - compute a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B

SYNOPSIS

SUBROUTINE DGGQRF(
N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO )

INTEGER INFO, LDA, LDB, LWORK, M, N, P

DOUBLE PRECISION A( LDA, ∗ ), B( LDB, ∗ ), TAUA( ∗ ), TAUB( ∗ ), WORK( ∗ )

PURPOSE

DGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B:

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

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms:

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

where R11 is upper triangular, and

if N <= P, T = ( 0 T12 ) N,   or if N > P, T = ( T11 ) N-P,
                 P-N N                           ( T21 ) P
                                                     P

where T12 or T21 is upper triangular.

In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)∗A:

             inv(B)∗A = Z’∗(inv(T)∗R)

where inv(B) denotes the inverse of the matrix B, and Z’ denotes the transpose of the matrix Z.

ARGUMENTS

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

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

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

A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, the N-by-M matrix A. On exit, the elements on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the orthogonal matrix Q as a product of min(N,M) elementary reflectors (see Further Details).

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

TAUA (output) DOUBLE PRECISION array, dimension (min(N,M))
The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details). B (input/output) DOUBLE PRECISION array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details).

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

TAUB (output) DOUBLE PRECISION array, dimension (min(N,P))
The scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details). WORK (workspace/output) DOUBLE PRECISION 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 QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of DORMQR.

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 reflectors

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

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(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine DORGQR.
To use Q to update another matrix, use LAPACK subroutine DORMQR.

The matrix Z is represented as a product of elementary reflectors

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

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(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine DORGRQ.
To use Z to update another matrix, use LAPACK subroutine DORMRQ.