F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08BEF (DGEQPF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08BEF (DGEQPF) computes the $QR$ factorization, with column pivoting, of a real $m$ by $n$ matrix.

## 2  Specification

 SUBROUTINE F08BEF ( M, N, A, LDA, JPVT, TAU, WORK, INFO)
 INTEGER M, N, LDA, JPVT(*), INFO REAL (KIND=nag_wp) A(LDA,*), TAU(min(M,N)), WORK(3*N)
The routine may be called by its LAPACK name dgeqpf.

## 3  Description

F08BEF (DGEQPF) 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 8).
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 $n$.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     N – INTEGERInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
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:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08BEF (DGEQPF) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
5:     JPVT($*$) – INTEGER arrayInput/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 ${\mathbf{JPVT}}\left(i\right)\ne 0$, then 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 ${\mathbf{JPVT}}\left(i\right)=k$, then 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:     TAU($\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)$) – REAL (KIND=nag_wp) arrayOutput
On exit: further details of the orthogonal matrix $Q$.
7:     WORK($3×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
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
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

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 $Q$ F08BEF (DGEQPF) may be followed by a call to F08AFF (DORGQR):
```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 F08BEF (DGEQPF).
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 $C$, 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, &
LWORK,INFO)
```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $m$ by $p$.
To compute a $QR$ factorization without column pivoting, use F08AEF (DGEQRF).
The complex analogue of this routine is F08BSF (ZGEQPF).

## 9  Example

This example finds the basic solutions for the linear least squares problems
 $minimize⁡ Axi - bi 2 , i=1,2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 and B= -0.01 -0.04 0.04 -0.03 0.05 0.01 -0.03 -0.02 0.02 0.05 -0.06 0.07 .$
Here $A$ 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)