NAG Library Routine Document
F08CSF (ZGEQLF)
1 Purpose
F08CSF (ZGEQLF) computes a QL factorization of a complex m by n matrix A.
2 Specification
INTEGER |
M, N, LDA, LWORK, INFO |
COMPLEX (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name zgeqlf.
3 Description
F08CSF (ZGEQLF) forms the QL factorization of an arbitrary rectangular complex m by n matrix.
If
m≥n, the factorization is given by:
where
L is an
n by
n lower triangular matrix and
Q is an
m by
m unitary matrix. If
m<n the factorization is given by
where
L is an
m by
n lower trapezoidal matrix and
Q is again an
m by
m unitary matrix. In the case where
m>n the factorization can be expressed as
where
Q1 consists of the first
m-n columns of
Q, and
Q2 the remaining
n columns.
The matrix
Q is not formed explicitly but is represented as a product of
minm,n elementary reflectors (see
Section 3.3.6 in the F08 Chapter Introduction for details). Routines are provided to work with
Q in this representation (see
Section 8).
Note also that for any
k<n, the information returned in the last
k columns of the array
A represents a
QL factorization of the last
k columns of the original matrix
A.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint:
M≥0.
- 2: N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the m by n matrix A.
On exit: if
m≥n, the lower triangle of the subarray
Am-n+1:m1:n contains the
n by
n lower triangular matrix
L.
If
m≤n, the elements on and below the
n-mth superdiagonal contain the
m by
n lower trapezoidal matrix
L. The remaining elements, with the array
TAU, represent the unitary matrix
Q as a product of elementary reflectors (see
Section 3.3.6 in the F08 Chapter Introduction).
- 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08CSF (ZGEQLF) is called.
Constraint:
LDA≥max1,M.
- 5: TAU(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAU
must be at least
max1,minM,N.
On exit: the scalar factors of the elementary reflectors (see
Section 8).
- 6: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 7: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08CSF (ZGEQLF) is called.
If
LWORK=-1, 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.
Suggested value:
for optimal performance, LWORK≥N×nb, where nb is the optimal block size.
Constraint:
LWORK≥max1,N.
- 8: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
The computed factorization is the exact factorization of a nearby matrix
A+E, where
and
ε is the
machine precision.
8 Further Comments
The total number of real floating point operations is approximately
83
n2
3m-n
if m≥n or
83
m2
3n-m
if m<n.
To form the unitary matrix
Q F08CSF (ZGEQLF) may be followed by a call to
F08CTF (ZUNGQL):
CALL ZUNGQL(M,M,MIN(M,N),A,LDA,TAU,WORK,LWORK,INFO)
but note that the second dimension of the array
A must be at least
M, which may be larger than was required by F08CSF (ZGEQLF).
When
m≥n, it is often only the first
n columns of
Q that are required, and they may be formed by the call:
CALL ZUNGQL(M,N,N,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an arbitrary complex rectangular matrix
C, F08CSF (ZGEQLF) may be followed by a call to
F08CUF (ZUNMQL). For example,
CALL ZUNMQL('Left','Conjugate Transpose',M,P,MIN(M,N),A,LDA,TAU, &
C,LDC,WORK,LWORK,INFO)
forms
C=QHC, where
C is
m by
p.
The real analogue of this routine is
F08CEF (DGEQLF).
9 Example
This example solves the linear least squares problems
for
x1 and
x2, where
bj is the
jth column of the matrix
B,
and
The solution is obtained by first obtaining a QL factorization of the matrix A.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
9.1 Program Text
Program Text (f08csfe.f90)
9.2 Program Data
Program Data (f08csfe.d)
9.3 Program Results
Program Results (f08csfe.r)