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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgeqrf (f08ae)

## Purpose

nag_lapack_dgeqrf (f08ae) computes the QR$QR$ factorization of a real m$m$ by n$n$ matrix.

## Syntax

[a, tau, info] = f08ae(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgeqrf(a, 'm', m, 'n', n)

## Description

nag_lapack_dgeqrf (f08ae) forms the QR$QR$ factorization of an arbitrary rectangular real m$m$ by n$n$ matrix. No pivoting is performed.
If mn$m\ge n$, the factorization is given by:
A = Q
 ( R ) 0
,
$A = Q R 0 ,$
where R$R$ is an n$n$ by n$n$ upper triangular matrix and Q$Q$ is an m$m$ by m$m$ orthogonal matrix. It is sometimes more convenient to write the factorization as
A =
 ( Q1 Q2 )
 ( R ) 0
,
$A = Q1 Q2 R 0 ,$
which reduces to
 A = Q1R , $A = Q1R ,$
where Q1${Q}_{1}$ consists of the first n$n$ columns of Q$Q$, and Q2${Q}_{2}$ the remaining mn$m-n$ columns.
If m < n$m, R$R$ is trapezoidal, and the factorization can be written
A = Q
 ( R1 R2 )
,
$A = Q R1 R2 ,$
where R1${R}_{1}$ is upper triangular and R2${R}_{2}$ is rectangular.
The matrix Q$Q$ is not formed explicitly but is represented as a product of min (m,n)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q$Q$ in this representation (see Section [Further Comments]).
Note also that for any k < n$k, the information returned in the first k$k$ columns of the array a represents a QR$QR$ factorization of the first k$k$ columns of the original matrix A$A$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The m$m$ by n$n$ matrix A$A$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,m)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
If mn$m\ge n$, the elements below the diagonal store details of the orthogonal matrix Q$Q$ and the upper triangle stores the corresponding elements of the n$n$ by n$n$ upper triangular matrix R$R$.
If m < n$m, the strictly lower triangular part stores details of the orthogonal matrix Q$Q$ and the remaining elements store the corresponding elements of the m$m$ by n$n$ upper trapezoidal matrix R$R$.
2:     tau( : $:$) – double array
Note: the dimension of the array tau must be at least max (1,min (m,n))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
Further details of the orthogonal matrix Q$Q$.
3:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

The computed factorization is the exact factorization of a nearby matrix (A + E)$\left(A+E\right)$, where
 ‖E‖2 = O(ε) ‖A‖2 , $‖E‖2 = O(ε) ‖A‖2 ,$
and ε$\epsilon$ is the machine precision.

The total number of floating point operations is approximately (2/3) n2 (3mn) $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if mn$m\ge n$ or (2/3) m2 (3nm) $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if m < n$m.
To form the orthogonal matrix Q$Q$ nag_lapack_dgeqrf (f08ae) may be followed by a call to nag_lapack_dorgqr (f08af):
```[a, info] = f08af(a, tau, 'k', min(m,n));
```
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_lapack_dgeqrf (f08ae).
When mn$m\ge n$, it is often only the first n$n$ columns of Q$Q$ that are required, and they may be formed by the call:
```[a, info] = f08af(a, tau);
```
To apply Q$Q$ to an arbitrary real rectangular matrix C$C$, nag_lapack_dgeqrf (f08ae) may be followed by a call to nag_lapack_dormqr (f08ag). For example,
```[c, info] = f08ag('Left', 'Transpose', a, tau, c, 'k', min(m,n));
```
forms C = QTC$C={Q}^{\mathrm{T}}C$, where C$C$ is m$m$ by p$p$.
To compute a QR$QR$ factorization with column pivoting, use nag_lapack_dgeqpf (f08be).
The complex analogue of this function is nag_lapack_zgeqrf (f08as).

## Example

```function nag_lapack_dgeqrf_example
a = [-0.57, -1.28, -0.39, 0.25;
-1.93, 1.08, -0.31, -2.14;
2.3, 0.24, 0.4, -0.35;
-1.93, 0.64, -0.66, 0.08;
0.15, 0.3, 0.15, -2.13;
-0.02, 1.03, -1.43, 0.5];
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];
% Compute the QR Factorisation of A
[a, tau, info] = nag_lapack_dgeqrf(a);
% Compute C = (C1) = (Q^T)*B, storing the result in B (C2)
[b, info] = nag_lapack_dormqr('Left', 'Transpose', a, tau, b);
% Compute least-squares solutions by backsubstitution in R*X = C1
[b, info] = nag_lapack_dtrtrs('Upper', 'No Transpose', 'Non-Unit', a, b, 'n', int64(4));
if (info >0)
fprintf('The upper triangular factor, R, of A is singular,\n');
fprintf('the least squares solution could not be computed.\n');
else
% Print least-squares solutions
[ifail] = nag_file_print_matrix_real_gen('General', ' ', b(1:4,:), 'Least-squares solution(s)');
% Compute and print estimates of the square roots of the residual
% sums of squares
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(b(5:6,j));
end
fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
fprintf('\t%11.2e    %11.2e\n', rnorm(1), rnorm(2));
end
```
```
Least-squares solution(s)
1          2
1      1.5339    -1.5753
2      1.8707     0.5559
3     -1.5241     1.3119
4      0.0392     2.9585

Square root(s) of the residual sum(s) of squares
2.22e-02       1.38e-02

```
```function f08ae_example
a = [-0.57, -1.28, -0.39, 0.25;
-1.93, 1.08, -0.31, -2.14;
2.3, 0.24, 0.4, -0.35;
-1.93, 0.64, -0.66, 0.08;
0.15, 0.3, 0.15, -2.13;
-0.02, 1.03, -1.43, 0.5];
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];
% Compute the QR Factorisation of A
[a, tau, info] = f08ae(a);
% Compute C = (C1) = (Q^T)*B, storing the result in B (C2)
[b, info] = f08ag('Left', 'Transpose', a, tau, b);
% Compute least-squares solutions by backsubstitution in R*X = C1
[b, info] = f07te('Upper', 'No Transpose', 'Non-Unit', a, b, 'n', int64(4));
if (info >0)
fprintf('The upper triangular factor, R, of A is singular,\n');
fprintf('the least squares solution could not be computed.\n');
else
% Print least-squares solutions
[ifail] = x04ca('General', ' ', b(1:4,:), 'Least-squares solution(s)');
% Compute and print estimates of the square roots of the residual
% sums of squares
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(b(5:6,j));
end
fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
fprintf('\t%11.2e    %11.2e\n', rnorm(1), rnorm(2));
end
```
```
Least-squares solution(s)
1          2
1      1.5339    -1.5753
2      1.8707     0.5559
3     -1.5241     1.3119
4      0.0392     2.9585

Square root(s) of the residual sum(s) of squares
2.22e-02       1.38e-02

```