hide long namesshow long names

Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int

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

Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_real_gen_rq (f01qj)

▸▿ 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_gen_rq (f01qj) finds the

R Q

factorization of the real

m

n

(

m \leq n

) matrix

A

, so that

A

is reduced to upper triangular form by means of orthogonal transformations from the right.

Syntax

[a, zeta, ifail] = f01qj(a, 'm', m, 'n', n)

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

Description

The

m

n

matrix

A

is factorized as

\begin{array}{l} A = (\begin{matrix} R & 0 \end{matrix}) P^{T} & when m < n, \\ A = R P^{T} & when m = n, \end{array}

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} = I - u_{k} u_{k}^{T},

where

u_{k} = (\begin{array}{l} w_{k} \\ ζ_{k} \\ 0 \\ z_{k} \end{array}),

ζ_{k}

is a scalar,

w_{k}

is an

(k - 1)

element vector and

z_{k}

is an

(n - m)

element vector.

u_{k}

is chosen to annihilate the elements in the

k

th row of

A

The vector

u_{k}

is returned in the

k

th element of zeta and in the

k

th row of a, such that

ζ_{k}

is in

zeta (k)

, the elements of

w_{k}

are in

a (k, 1), \dots, a (k, k - 1)

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.

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 $n$ 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 $m$ strictly lower triangular part of a and the $m$ by $(n - m)$ rectangular part of a to the right of the upper triangular part 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 $P_{k} = I$ 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

(\begin{matrix} R & 0 \end{matrix}) 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} (3 n - m) / 3

The first

k

rows of the orthogonal matrix

P^{T}

can be obtained by calling nag_matop_real_gen_rq_formq (f01qk), which overwrites the

k

rows of

P^{T}

on the first

k

rows of the array a.

P^{T}

is obtained by the call:

[a, ifail] = f01qk('Separate', m, k, a, zeta);

Example

This example obtains the

R Q

factorization of the

3

5

matrix

A = (\begin{array}{r} 2.0 & 2.0 & 1.6 & 2.0 & 1.2 \\ 2.5 & 2.5 & - 0.4 & - 0.5 & - 0.3 \\ 2.5 & 2.5 & 2.8 & 0.5 & - 2.9 \end{array}) .

Open in the MATLAB editor: f01qj_example

function f01qj_example


fprintf('f01qj example results\n\n');

a = [2,   2,    1.6,  2,    1.2;
     2.5, 2.5, -0.4, -0.5, -0.3;
     2.5, 2.5,  2.8,  0.5, -2.9];

[RQ, zeta, ifail] = f01qj(a);

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

f01qj example results

RQ Factorization of A
Vector zeta
    1.0092    1.2981    1.2329

Matrix A after factorization (R in left-hand upper triangle
   -3.1446   -1.0705   -2.2283    0.6333    0.7619
    0.5277   -2.8345   -2.2283   -0.1662    0.0945
    0.3766    0.3766   -5.3852    0.0753   -0.4368

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

Chapter Contents

Chapter Introduction

NAG Toolbox