ZGGQRF(1) LAPACK routine (version 3.2) ZGGQRF(1)NAME
ZGGQRF - computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B
SYNOPSIS
SUBROUTINE ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, P
COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
WORK( * )
PURPOSE
ZGGQRF 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 unitary matrix, Z is a P-by-P unitary 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
conjugate 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) COMPLEX*16 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
unitary 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) COMPLEX*16 array, dimension (min(N,M))
The scalar factors of the elementary reflectors which represent
the unitary matrix Q (see Further Details). B
(input/output) COMPLEX*16 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 triangu‐
lar 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 unitary
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) COMPLEX*16 array, dimension (min(N,P))
The scalar factors of the elementary reflectors which represent
the unitary matrix Z (see Further Details). WORK
(workspace/output) COMPLEX*16 array, dimension (MAX(1,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 factoriza‐
tion of an N-by-P matrix, and NB3 is the optimal blocksize for
a call of ZUNMQR. 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 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 complex scalar, and v is a complex 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 ZUNGQR.
To use Q to update another matrix, use LAPACK subroutine ZUNMQR. 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 complex scalar, and v is a complex 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 ZUNGRQ.
To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.
LAPACK routine (version 3.2) November 2008 ZGGQRF(1)