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
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 https://www.netlib.org/lapack/lug
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 f08bff 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 ${\mathbf{jpvt}}\left(j\right)\ne 0$, the $j$ 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 $j$ 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 ${\mathbf{jpvt}}\left(j\right)=k$, the $k$th column of $A$ is moved to become the $j$ 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: $\mathbf{tau}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{m}},{\mathbf{n}}))$.
On exit: the scalar factors of the elementary reflectors.
7: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{lwork}})\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
8: $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08bff is called.
If ${\mathbf{lwork}}=\mathrm{-1}$, 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, ${\mathbf{lwork}}\ge 2\times {\mathbf{n}}+({\mathbf{n}}+1)\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint:
${\mathbf{lwork}}\ge 3\times {\mathbf{n}}+1$ or ${\mathbf{lwork}}=\mathrm{-1}$.
9: $\mathbf{info}$ – 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
f08bff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bff 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 $Q$f08bff may be followed by a call to f08aff
:
and ${b}_{j}$ is the $j$th column of the matrix $B$. The solution is obtained by first obtaining a $QR$ factorization with column pivoting of the matrix $A$. A tolerance of $0.01$ is used to estimate the rank of $A$ from the upper triangular factor, $R$.
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.