# NAG FL Interfacef08bff (dgeqp3)

## 1Purpose

f08bff computes the $QR$ factorization, with column pivoting, of a real $m$ by $n$ matrix.

## 2Specification

Fortran Interface
 Subroutine f08bff ( m, n, a, lda, jpvt, tau, work, info)
 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))
#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)
The routine may be called by the names f08bff, nagf_lapackeig_dgeqp3 or its LAPACK name dgeqp3.

## 3Description

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:
 $AP= Q R 0 ,$
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
 $AP= Q1 Q2 R 0 ,$
which reduces to
 $AP= Q1 R ,$
where ${Q}_{1}$ consists of the first $n$ columns of $Q$, and ${Q}_{2}$ the remaining $m-n$ columns.
If $m, $R$ is trapezoidal, and the factorization can be written
 $AP= Q R1 R2 ,$
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, 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: $\mathbf{m}$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, 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, ${\mathbf{lwork}}\ge 2×{\mathbf{n}}+\left({\mathbf{n}}+1\right)×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge 3×{\mathbf{n}}+1$ or ${\mathbf{lwork}}=-1$.
9: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 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 $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

## 8Parallelism 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 implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m.
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.

## 10Example

This example solves the linear least squares problems
 $minx bj - Axj 2 , j=1,2$
for the basic solutions ${x}_{1}$ and ${x}_{2}$, where
 $A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 and B= 7.4 2.7 4.2 -3.0 -8.3 -9.6 1.8 1.1 8.6 4.0 2.1 -5.7$
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.

### 10.1Program Text

Program Text (f08bffe.f90)

### 10.2Program Data

Program Data (f08bffe.d)

### 10.3Program Results

Program Results (f08bffe.r)