
dggqrf.f(3) LAPACK dggqrf.f(3)
NAME
dggqrf.f 
SYNOPSIS
Functions/Subroutines
subroutine dggqrf (N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
DGGQRF
Function/Subroutine Documentation
subroutine dggqrf (integerN, integerM, integerP, double precision, dimension( lda, * )A,
integerLDA, double precision, dimension( * )TAUA, double precision, dimension( ldb, * )B,
integerLDB, double precision, dimension( * )TAUB, double precision, dimension( * )WORK,
integerLWORK, integerINFO)
DGGQRF
Purpose:
DGGQRF 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**T*(inv(T)*R)
where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.
Parameters:
N
N is INTEGER
The number of rows of the matrices A and B. N >= 0.
M
M is INTEGER
The number of columns of the matrix A. M >= 0.
P
P is INTEGER
The number of columns of the matrix B. P >= 0.
A
A is DOUBLE PRECISION 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 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
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAUA
TAUA is DOUBLE PRECISION array, dimension (min(N,M))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q (see Further Details).
B
B is DOUBLE PRECISION 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
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
TAUB
TAUB is DOUBLE PRECISION array, dimension (min(N,P))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Z (see Further Details).
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is 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 factorization of an NbyP matrix, and NB3 is the optimal
blocksize for a call of DORMQR.
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
INFO is INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
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**T
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 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**T
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 DORGRQ.
To use Z to update another matrix, use LAPACK subroutine DORMRQ.
Definition at line 215 of file dggqrf.f.
Author
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Version 3.4.2 Tue Sep 25 2012 dggqrf.f(3) 
