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# sggqrf

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

SYNOPSIS
SUBROUTINE SGGQRF( N, M, P, 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
SGGQRF 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 assumes 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 an upper triangular matrix, 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 a P-by-P upper triangular matrix.

In particular, if B is square and nonsingular, the GQR fac-
torization 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, Z' denotes
the transpose of 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) REAL array, dimension (LDA,M)
On entry, the N-by-M matrix A.  On exit, the ele-
ments 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 orthog-
onal 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) REAL array, dimension (MIN(N,M))
The scalar factors of the elementary reflectors (see
Further Details).

B       (input/output) REAL 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 subdiag-
onal 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) REAL array, dimension (MIN(N,P))
The scalar factors of the elementary reflectors (see
Further Details).

WORK    (workspace) 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(1,N,M,P)*MAX(NB1,NB2,NB3), where NB1 is the
optimal blocksize for the QR factorization of an N-
by-M matrix A.  NB2 is the optimal blocksize for the
RQ factorization of an N-by-P matrix B.  NB3 is the
optimal blocksize for calling SORMQR.

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 SORGQR.
To use Q to update another matrix, use LAPACK subroutine
SORMQR.

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 SORGRQ.
To use Z to update another matrix, use LAPACK subroutine
SORMRQ.
```