f08bef computes the $QR$ factorization, with column pivoting, of a real $m\times n$ matrix. f08bef is marked as deprecated by LAPACK; the replacement routine is f08bff which makes better use of Level 3 BLAS.
where $R$ is an $n\times n$ upper triangular matrix, $Q$ is an $m\times m$ orthogonal matrix and $P$ is an $n\times n$ permutation matrix. It is sometimes more convenient to write the factorization as
where ${R}_{1}$ is upper triangular and ${R}_{2}$ is rectangular.
The matrix $Q$ is not formed explicitly but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}(m,n)$ elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with $Q$ in this representation (see Section 9).
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 $i$th stage the pivot column is chosen to be the column which maximizes the $2$-norm of elements $i$ to $m$ over columns $i$ to $n$.
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{m}}\ge 0$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $m\times n$ matrix $A$.
On exit: if $m\ge n$, the elements below the diagonal are overwritten by details of the orthogonal matrix $Q$ and the upper triangle is overwritten by the corresponding elements of the $n\times n$ upper triangular matrix $R$.
If $m<n$, the strictly lower triangular part is overwritten by details of the orthogonal matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m\times n$ upper trapezoidal matrix $R$.
4: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bef is called.
Note: the dimension of the array jpvt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{jpvt}}\left(i\right)\ne 0$, 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 ${\mathbf{jpvt}}\left(i\right)=k$, the $k$th 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 ${\mathbf{jpvt}}\left(1\right),{\mathbf{jpvt}}\left(2\right),\dots ,{\mathbf{jpvt}}\left(n\right)$.
6: $\mathbf{tau}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{m}},{\mathbf{n}})\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: further details of the orthogonal matrix $Q$.
7: $\mathbf{work}\left(3\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
8: $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
The computed factorization is the exact factorization of a nearby matrix $(A+E)$, where
Background information to multithreading can be found in the Multithreading documentation.
f08bef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations is approximately $\frac{2}{3}{n}^{2}(3m-n)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}(3n-m)$ if $m<n$.
To form the orthogonal matrix $Q$f08bef may be followed by a call to f08aff
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