NAG Library Routine Document
F08BSF (ZGEQPF)
1 Purpose
F08BSF (ZGEQPF) computes the QR factorization, with column pivoting, of a complex m by n matrix.
2 Specification
INTEGER |
M, N, LDA, JPVT(*), INFO |
REAL (KIND=nag_wp) |
RWORK(2*N) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), TAU(min(M,N)), WORK(N) |
|
The routine may be called by its
LAPACK
name zgeqpf.
3 Description
F08BSF (ZGEQPF) forms the QR factorization, with column pivoting, of an arbitrary rectangular complex m by n matrix.
If
m≥n, the factorization is given by:
where
R is an
n by
n upper triangular matrix (with real diagonal elements),
Q is an
m by
m unitary matrix and
P is an
n by
n permutation matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
Q1 consists of the first
n columns of
Q, and
Q2 the remaining
m-n columns.
If
m<n,
R is trapezoidal, and the factorization can be written
where
R1 is upper triangular and
R2 is rectangular.
The matrix
Q is not formed explicitly but is represented as a product of
minm,n elementary reflectors (see 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 first
k columns of the array
A represents a
QR factorization of the first
k columns of the permuted matrix
AP.
The routine allows specified columns of A to be moved to the leading columns of AP at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the ith stage the pivot column is chosen to be the column which maximizes the 2-norm of elements i to m over columns i to n.
4 References
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 elements below the diagonal are overwritten by details of the unitary matrix
Q and the upper triangle is overwritten by the corresponding elements of the
n by
n upper triangular matrix
R.
If m<n, the strictly lower triangular part is overwritten by details of the unitary matrix Q and the remaining elements are overwritten by the corresponding elements of the m by n upper trapezoidal matrix R.
The diagonal elements of R are real.
- 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08BSF (ZGEQPF) is called.
Constraint:
LDA≥max1,M.
- 5: JPVT(*) – INTEGER arrayInput/Output
-
Note: the dimension of the array
JPVT
must be at least
max1,N.
On entry: if JPVTi≠0, then the i th column of A is moved to the beginning of AP before the decomposition is computed and is fixed in place during the computation. Otherwise, the i th column of A is a free column (i.e., one which may be interchanged during the computation with any other free column).
On exit: details of the permutation matrix P. More precisely, if JPVTi=k, then the kth column of A is moved to become the i th column of AP; in other words, the columns of AP are the columns of A in the order JPVT1,JPVT2,…,JPVTn.
- 6: TAU(minM,N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: further details of the unitary matrix Q.
- 7: WORK(N) – COMPLEX (KIND=nag_wp) arrayWorkspace
- 8: RWORK(2×N) – REAL (KIND=nag_wp) arrayWorkspace
- 9: 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 F08BSF (ZGEQPF) may be followed by a call to
F08ATF (ZUNGQR):
CALL ZUNGQR(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 F08BSF (ZGEQPF).
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 ZUNGQR(M,N,N,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an arbitrary complex rectangular matrix
C, F08BSF (ZGEQPF) may be followed by a call to
F08AUF (ZUNMQR). For example,
CALL ZUNMQR('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.
To compute a
QR factorization without column pivoting, use
F08ASF (ZGEQRF).
The real analogue of this routine is
F08BEF (DGEQPF).
9 Example
This example solves the linear least squares problems
where
b1 and
b2 are the columns of the matrix
B,
and
Here
A is approximately rank-deficient, and hence it is preferable to use F08BSF (ZGEQPF) rather than
F08ASF (ZGEQRF).
9.1 Program Text
Program Text (f08bsfe.f90)
9.2 Program Data
Program Data (f08bsfe.d)
9.3 Program Results
Program Results (f08bsfe.r)