
SGGQRF(l) ) SGGQRF(l)
NAME
SGGQRF  compute a generalized QR factorization of an NbyM matrix A and an NbyP 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 NbyM matrix A and an NbyP matrix
B:
A = Q*R, B = Q*T*Z,
where Q is an NbyN orthogonal matrix, Z is a PbyP 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 ) NM N MN
M
where R11 is upper triangular, and
if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) NP,
PN 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) REAL array, dimension (LDA,M)
On entry, the NbyM matrix A. On exit, the elements on and above the diagonal of
the array contain the min(N,M)byM upper trapezoidal matrix R (R is upper trian
gular 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 Fur
ther 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 which represent the orthogonal
matrix Q (see Further Details). B (input/output) REAL array, dimension
(LDB,P) On entry, the NbyP matrix B. On exit, if N <= P, the upper triangle of
the subarray B(1:N,PN+1:P) contains the NbyN upper triangular matrix T; if N >
P, the elements on and above the (NP)th subdiagonal contain the NbyP 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 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
QR factorization of an NbyM matrix, NB2 is the optimal blocksize for the RQ fac
torization of an NbyP matrix, and NB3 is the optimal blocksize for a call of
SORMQR.
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 ith 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:i1) = 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(pk+i+1:p) = 0 and v(pk+i) = 1; v(1:pk+i1) is stored on exit in B(nk+i,1:pk+i1),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGRQ.
To use Z to update another matrix, use LAPACK subroutine SORMRQ.
LAPACK version 3.0 15 June 2000 SGGQRF(l) 
