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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dgelqf (f08ah)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dgelqf (f08ah) computes the

L Q

factorization of a real

m

n

matrix.

Syntax

[a, tau, info] = f08ah(a, 'm', m, 'n', n)

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

Description

nag_lapack_dgelqf (f08ah) forms the

L Q

factorization of an arbitrary rectangular real

m

n

matrix. No pivoting is performed.

m \leq n

, the factorization is given by:

A = (\begin{array}{r} L & 0 \end{array}) Q

where

L

is an

m

m

lower triangular matrix and

Q

is an

n

n

orthogonal matrix. It is sometimes more convenient to write the factorization as

A = (\begin{array}{r} L & 0 \end{array}) (\begin{array}{r} Q_{1} \\ Q_{2} \end{array})

which reduces to

A = L Q_{1},

where

Q_{1}

consists of the first

m

rows of

Q

, and

Q_{2}

the remaining

n - m

rows.

m > n

L

is trapezoidal, and the factorization can be written

A = (\begin{array}{r} L_{1} \\ L_{2} \end{array}) Q

where

L_{1}

is lower triangular and

L_{2}

is rectangular.

The

L Q

factorization of

A

is essentially the same as the

Q R

factorization of

A^{T}

, since

A = (\begin{array}{r} L & 0 \end{array}) Q \Leftrightarrow A^{T} = Q^{T} (\begin{array}{r} L^{T} \\ 0 \end{array}) .

The matrix

Q

is not formed explicitly but is represented as a product of

\min (m, n)

elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with

Q

in this representation (see Further Comments).

Note also that for any

k < m

, the information returned in the first

k

rows of the array a represents an

L Q

factorization of the first

k

rows of the original matrix

A

References

None.

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – double array: The first dimension of the array a must be at least $\max (1, m)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The $m$ by $n$ matrix $A$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $\max (1, m)$ .
The second dimension of the array a will be $\max (1, n)$ .

If $m \leq n$ , the elements above the diagonal store details of the orthogonal matrix $Q$ and the lower triangle stores the corresponding elements of the $m$ by $m$ lower triangular matrix $L$ .
If $m > n$ , the strictly upper triangular part stores details of the orthogonal matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ lower trapezoidal matrix $L$ .
2: $tau (:)$ – double array: The dimension of the array tau will be $\max (1, \min (m, n))$

Further details of the orthogonal matrix $Q$ .
3: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ 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)

, where

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision.

Further Comments

The total number of floating-point operations is approximately

\frac{2}{3} m^{2} (3 n - m)

m \leq n

\frac{2}{3} n^{2} (3 m - n)

m > n

To form the orthogonal matrix

Q

nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dorglq (f08aj):

[a, info] = f08aj(a, tau, 'k', min(m,n));

but note that the first dimension of the array a, specified by the argument lda, must be at least n, which may be larger than was required by nag_lapack_dgelqf (f08ah).

When

m \leq n

, it is often only the first

m

rows of

Q

that are required, and they may be formed by the call:

[a, info] = f08aj(a, tau, 'k', m);

To apply

Q

to an arbitrary real rectangular matrix

C

, nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dormlq (f08ak). For example,

[c, info] = f08ak('Left', 'Transpose', a, tau, c, 'k', min(m, n));

forms the matrix product

C = Q^{T} C

, where

C

m

p

The complex analogue of this function is nag_lapack_zgelqf (f08av).

Example

This example finds the minimum norm solutions of the under-determined systems of linear equations

A x_{1} = b_{1} and A x_{2} = b_{2}

where

b_{1}

and

b_{2}

are the columns of the matrix

B

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) and B = (\begin{array}{r} - 2.87 & - 5.23 \\ 1.63 & 0.29 \\ - 3.52 & 4.76 \\ 0.45 & - 8.41 \end{array}) .

Open in the MATLAB editor: f08ah_example

function f08ah_example


fprintf('f08ah example results\n\n');

a = [-5.42, 3.28, -3.68, 0.27, 2.06, 0.46;
     -1.65, -3.4, -3.2, -1.03, -4.06, -0.01;
     -0.37, 2.35, 1.9, 4.31, -1.76, 1.13;
     -3.15, -0.11, 1.99, -2.7, 0.26, 4.5];
b = [-2.87, -5.23;
      1.63,  0.29;
     -3.52,  4.76;
      0.45, -8.41;
      0,     0;
      0,     0];
% Compute the LQ factorization of a
[a, tau, info] = f08ah(a);

% solve l*y=b
l = tril(a(:, 1:4));
b(1:4,:) = inv(l)*b(1:4,:);

% Compute minimum-norm solution x = (q^t)*b
[x, info] = f08ak( ...
		   'Left', 'Transpose', a, tau, b);

mtitle = 'Minimum-norm solution(s)';
[ifail] = x04ca( ...
                 'General', ' ', x, mtitle);

f08ah example results

 Minimum-norm solution(s)
             1          2
 1      0.2371     0.7383
 2     -0.4575     0.0158
 3     -0.0085    -0.0161
 4     -0.5192     1.0768
 5      0.0239    -0.6436
 6     -0.0543    -0.6613

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox