# NAG Library Routine Document

## 1Purpose

f08abf (dgeqrt) recursively computes, with explicit blocking, the $QR$ factorization of a real $m$ by $n$ matrix.

## 2Specification

Fortran Interface
 Subroutine f08abf ( m, n, nb, a, lda, t, ldt, work, info)
 Integer, Intent (In) :: m, n, nb, lda, ldt Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), t(ldt,*) Real (Kind=nag_wp), Intent (Out) :: work(nb*n)
#include nagmk26.h
 void f08abf_ (const Integer *m, const Integer *n, const Integer *nb, double a[], const Integer *lda, double t[], const Integer *ldt, double work[], Integer *info)
The routine may be called by its LAPACK name dgeqrt.

## 3Description

f08abf (dgeqrt) forms the $QR$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. No pivoting is performed.
It differs from f08aef (dgeqrf) in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the $QR$ factorization based on the algorithm of Elmroth and Gustavson (2000).
If $m\ge n$, the factorization is given by:
 $A = Q R 0 ,$
where $R$ is an $n$ by $n$ upper triangular matrix and $Q$ is an $m$ by $m$ orthogonal matrix. It is sometimes more convenient to write the factorization as
 $A = Q1 Q2 R 0 ,$
which reduces to
 $A = Q1 R ,$
where ${Q}_{1}$ consists of the first $n$ columns of $Q$, and ${Q}_{2}$ the remaining $m-n$ columns.
If $m, $R$ is upper trapezoidal, and the factorization can be written
 $A = Q R1 R2 ,$
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 represents a $QR$ factorization of the first $k$ columns of the original matrix $A$.

## 4References

Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel $QR$ Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44) 4 605–624
Golub G H and Van Loan C F (2012) Matrix Computations (4th 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:     $\mathbf{nb}$ – IntegerInput
On entry: the explicitly chosen block size to be used in computing the $QR$ factorization. See Section 9 for details.
Constraint: if $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)>0$, $1\le {\mathbf{nb}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 $R$.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08abf (dgeqrt) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{t}\left({\mathbf{ldt}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array t 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: further details of the orthogonal matrix $Q$. The number of blocks is $b=⌈\frac{k}{{\mathbf{nb}}}⌉$, where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)×{\mathbf{nb}}$. For each of the blocks, an upper triangular block reflector factor is computed: ${\mathbit{T}}_{1},{\mathbit{T}}_{2},\dots ,{\mathbit{T}}_{b}$. These are stored in the ${\mathbf{nb}}$ by $n$ matrix $T$ as $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
7:     $\mathbf{ldt}$ – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08abf (dgeqrt) is called.
Constraint: ${\mathbf{ldt}}\ge {\mathbf{nb}}$.
8:     $\mathbf{work}\left({\mathbf{nb}}×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
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 $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08abf (dgeqrt) 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 apply $Q$ to an arbitrary real rectangular matrix $C$, f08abf (dgeqrt) may be followed by a call to f08acf (dgemqrt). For example,
```Call dgemqrt('Left','Transpose',m,p,min(m,n),nb,a,lda,t,ldt,c,ldc, &
work,info)```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $m$ by $p$.
To form the orthogonal matrix $Q$ explicitly, simply initialize the $m$ by $m$ matrix $C$ to the identity matrix and form $C=QC$ using f08acf (dgemqrt) as above.
The block size, nb, used by f08abf (dgeqrt) is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of ${\mathbf{nb}}=64\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ is likely to achieve good efficiency and it is unlikely that an optimal value would exceed $340$.
To compute a $QR$ factorization with column pivoting, use f08bbf (dtpqrt) or f08bef (dgeqpf).
The complex analogue of this routine is f08apf (zgeqrt).

## 10Example

This example solves the linear least squares problems
 $minimize⁡ Axi - bi 2 , i=1,2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 and B= -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 .$

### 10.1Program Text

Program Text (f08abfe.f90)

### 10.2Program Data

Program Data (f08abfe.d)

### 10.3Program Results

Program Results (f08abfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017