NAG FL Interface
f08bff (dgeqp3)
1
Purpose
f08bff computes the $QR$ factorization, with column pivoting, of a real $m$ by $n$ matrix.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
m, n, lda, lwork 
Integer, Intent (Inout) 
:: 
jpvt(*) 
Integer, Intent (Out) 
:: 
info 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), tau(*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 

C Header Interface
#include <nag.h>
void 
f08bff_ (const Integer *m, const Integer *n, double a[], const Integer *lda, Integer jpvt[], double tau[], double work[], const Integer *lwork, Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08bff_ (const Integer &m, const Integer &n, double a[], const Integer &lda, Integer jpvt[], double tau[], double work[], const Integer &lwork, Integer &info) 
}

The routine may be called by the names f08bff, nagf_lapackeig_dgeqp3 or its LAPACK name dgeqp3.
3
Description
f08bff forms the $QR$ factorization, with column pivoting, of an arbitrary rectangular real $m$ by $n$ matrix.
If
$m\ge n$, the factorization is given by:
where
$R$ is an
$n$ by
$n$ upper triangular matrix,
$Q$ is an
$m$ by
$m$ orthogonal matrix and
$P$ is an
$n$ by
$n$ permutation matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
${Q}_{1}$ consists of the first
$n$ columns of
$Q$, and
${Q}_{2}$ the remaining
$mn$ columns.
If
$m<n$,
$R$ is trapezoidal, and the factorization can be written
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
$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ 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 $n$.
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
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments

1:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of columns of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ 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
$n$ by
$n$ 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 $m$ by $n$ upper trapezoidal matrix $R$.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08bff is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.

5:
$\mathbf{jpvt}\left(*\right)$ – Integer array
Input/Output

Note: the dimension of the array
jpvt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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) array
Output

Note: the dimension of the array
tau
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the scalar factors of the elementary reflectors.

7:
$\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) array
Workspace

On exit: if
${\mathbf{info}}={\mathbf{0}}$,
${\mathbf{work}}\left(1\right)$ contains the minimum value of
lwork required for optimal performance.

8:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f08bff is called.
If
${\mathbf{lwork}}=1$, 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, ${\mathbf{lwork}}\ge 2\times {\mathbf{n}}+\left({\mathbf{n}}+1\right)\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint:
${\mathbf{lwork}}\ge 3\times {\mathbf{n}}+1$ or ${\mathbf{lwork}}=1$.

9:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error 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.
7
Accuracy
The computed factorization is the exact factorization of a nearby matrix
$\left(A+E\right)$, where
and
$\epsilon $ is the
machine precision.
8
Parallelism and Performance
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 implementationspecific information.
The total number of floatingpoint operations is approximately $\frac{2}{3}{n}^{2}\left(3mn\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3nm\right)$ if $m<n$.
To form the orthogonal matrix
$Q$ f08bff may be followed by a call to
f08aff:
Call dorgqr(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
f08bff.
When
$m\ge n$, it is often only the first
$n$ columns of
$Q$ that are required, and they may be formed by the call:
Call dorgqr(m,n,n,a,lda,tau,work,lwork,info)
To apply
$Q$ to an arbitrary real rectangular matrix
$C$,
f08bff may be followed by a call to
f08agf. For example,
Call dormqr('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc,work, &
lwork,info)
forms
$C={Q}^{\mathrm{T}}C$, where
$C$ is
$m$ by
$p$.
To compute a
$QR$ factorization without column pivoting, use
f08aef.
The complex analogue of this routine is
f08btf.
10
Example
This example solves the linear least squares problems
for the basic solutions
${x}_{1}$ and
${x}_{2}$, where
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results