NAG Library Routine Document
F08BEF (DGEQPF) computes the factorization, with column pivoting, of a real by matrix.
||M, N, LDA, JPVT(*), INFO
||A(LDA,*), TAU(min(M,N)), WORK(3*N)
The routine may be called by its
F08BEF (DGEQPF) forms the factorization, with column pivoting, of an arbitrary rectangular real by matrix.
, the factorization is given by:
upper triangular matrix,
orthogonal matrix and
permutation matrix. It is sometimes more convenient to write the factorization as
which reduces to
consists of the first
is trapezoidal, and the factorization can be written
is upper triangular and
is not formed explicitly but is represented as a product of
elementary reflectors (see the F08 Chapter Introduction
for details). Routines are provided to work with
in this representation (see Section 8
Note also that for any
, the information returned in the first
columns of the array A
factorization of the first
columns of the permuted matrix
The routine allows specified columns of to be moved to the leading columns of at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the th stage the pivot column is chosen to be the column which maximizes the -norm of elements to over columns to .
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 1: M – INTEGERInput
On entry: , the number of rows of the matrix .
- 2: N – INTEGERInput
On entry: , the number of columns of the matrix .
- 3: A(LDA,) – REAL (KIND=nag_wp) arrayInput/Output
the second dimension of the array A
must be at least
On entry: the by matrix .
, the elements below the diagonal are overwritten by details of the orthogonal matrix
and the upper triangle is overwritten by the corresponding elements of the
upper triangular matrix
If , the strictly lower triangular part is overwritten by details of the orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the by upper trapezoidal matrix .
- 4: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F08BEF (DGEQPF) is called.
- 5: JPVT() – INTEGER arrayInput/Output
the dimension of the array JPVT
must be at least
On entry: if , then the th column of is moved to the beginning of before the decomposition is computed and is fixed in place during the computation. Otherwise, the th column of 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 . More precisely, if , then the th column of is moved to become the th column of ; in other words, the columns of are the columns of in the order .
- 6: TAU() – REAL (KIND=nag_wp) arrayOutput
On exit: further details of the orthogonal matrix .
- 7: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 8: INFO – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed factorization is the exact factorization of a nearby matrix
is the machine precision
The total number of floating point operations is approximately if or if .
To form the orthogonal matrix
F08BEF (DGEQPF) may be followed by a call to F08AFF (DORGQR)
but note that the second dimension of the array A
must be at least M
, which may be larger than was required by F08BEF (DGEQPF).
, it is often only the first
that are required, and they may be formed by the call:
to an arbitrary real rectangular matrix
, F08BEF (DGEQPF) may be followed by a call to F08AGF (DORMQR)
. For example,
CALL DORMQR('Left','Transpose',M,P,MIN(M,N),A,LDA,TAU,C,LDC,WORK, &
To compute a
factorization without column pivoting, use F08AEF (DGEQRF)
The complex analogue of this routine is F08BSF (ZGEQPF)
This example finds the basic solutions for the linear least squares problems
are the columns of the matrix
is approximately rank-deficient, and hence it is preferable to use F08BEF (DGEQPF) rather than F08AEF (DGEQRF)
9.1 Program Text
Program Text (f08befe.f90)
9.2 Program Data
Program Data (f08befe.d)
9.3 Program Results
Program Results (f08befe.r)