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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_real_trapez_rq (f01qg)

▸▿ 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_matop_real_trapez_rq (f01qg) reduces the

m

n

(

m \leq n

) real upper trapezoidal matrix

A

to upper triangular form by means of orthogonal transformations.

Syntax

[a, zeta, ifail] = f01qg(a, 'm', m, 'n', n)

[a, zeta, ifail] = nag_matop_real_trapez_rq(a, 'm', m, 'n', n)

Description

The

m

n

(

m \leq n

) real upper trapezoidal matrix

A

given by

A = (\begin{matrix} U & X \end{matrix}),

where

U

is an

m

m

upper triangular matrix, is factorized as

A = (\begin{matrix} R & 0 \end{matrix}) P^{T},

where

P

is an

n

n

orthogonal matrix and

R

is an

m

m

upper triangular matrix.

P

is given as a sequence of Householder transformation matrices

P = P_{m} \dots P_{2} P_{1},

the

(m - k + 1)

th transformation matrix,

P_{k}

, being used to introduce zeros into the

k

th row of

A

P_{k}

has the form

P_{k} = (\begin{array}{l} I & 0 \\ 0 & T_{k} \end{array}),

where

\begin{array}{c} T_{k} = I - u_{k} u_{k}^{T}, \\ u_{k} = (\begin{matrix} ζ_{k} \\ 0 \\ z_{k} \end{matrix}), \end{array}

ζ_{k}

is a scalar and

z_{k}

is an (

n - m

) element vector.

ζ_{k}

and

z_{k}

are chosen to annihilate the elements of the

k

th row of

X

The vector

u_{k}

is returned in the

k

th element of the array zeta and in the

k

th row of a, such that

ζ_{k}

is in

zeta (k)

and the elements of

z_{k}

are in

a (k, m + 1), \dots, a (k, n)

. The elements of

R

are returned in the upper triangular part of a.

For further information on this factorization and its use see Section 6.5 of Golub and Van Loan (1996).

References

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

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

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 leading $m$ by $n$ upper trapezoidal part of the array a must contain the matrix to be factorized.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of rows of the matrix $A$ .
When $m = 0$ then an immediate return is effected.

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: $n \geq m$ .

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 $\max (1, n)$ .

The $m$ by $m$ upper triangular part of a will contain the upper triangular matrix $R$ , and the $m$ by $(n - m)$ upper trapezoidal part of a will contain details of the factorization as described in Description.
2: $zeta (m)$ – double array: $zeta (k)$ contains the scalar $ζ_{k}$ for the $(m - k + 1)$ th transformation. If $T_{k} = I$ then $zeta (k) = 0.0$ , otherwise $zeta (k)$ contains $ζ_{k}$ as described in Description and $ζ_{k}$ is always in the range $(1.0, \sqrt{2.0})$ .
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = - 1$

On entry,	$m < 0$ ,
or	$n < m$ ,
or	$lda < m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computed factors

R

and

P

satisfy the relation

(R 0) P^{T} = A + E,

where

‖E‖ \leq c ε ‖A‖,

ε

is the machine precision (see nag_machine_precision (x02aj)),

c

is a modest function of

m

and

n

and

‖.‖

denotes the spectral (two) norm.

Further Comments

The approximate number of floating-point operations is given by

2 \times m^{2} (n - m)

Example

This example reduces the

3

5

matrix

A = (\begin{array}{r} 2.4 & 0.8 & - 1.4 & 3.0 & - 0.8 \\ 0.0 & 1.6 & 0.8 & 0.4 & - 0.8 \\ 0.0 & 0.0 & 1.0 & 2.0 & 2.0 \end{array})

to upper triangular form.

Open in the MATLAB editor: f01qg_example

function f01qg_example


fprintf('f01qg example results\n\n');

a = [2.4, 0.8, -1.4, 3,   -0.8;
     0,   1.6,  0.8, 0.4, -0.8;
     0,   0,    1,   2,    2];

[RQ, zeta, ifail] = f01qg(a);

disp('RQ Factorization of A');
disp('Vector zeta');
disp(zeta');
disp('Matrix A after factorization (R in left-hand upper triangle');
disp(RQ);

f01qg example results

RQ Factorization of A
Vector zeta
    1.2649    1.3416    1.1547

Matrix A after factorization (R in left-hand upper triangle
   -4.0000   -1.0000   -1.0000    0.6325   -0.0000
         0   -2.0000    0.0000    0.0000   -0.4472
         0         0   -3.0000    0.5774    0.5774

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Chapter Contents

Chapter Introduction

NAG Toolbox