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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_rq (f01rj)

▸▿ 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_complex_gen_rq (f01rj) finds the

R Q

factorization of the complex

m

n

(

m \leq n

), matrix

A

, so that

A

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

Syntax

[a, theta, ifail] = f01rj(a, 'm', m, 'n', n)

[a, theta, ifail] = nag_matop_complex_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^{H} & when ​ m < n, \\ A = R P^{H} & when ​ m = n, \end{array}

where

P

is an

n

n

unitary 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 - γ_{k} u_{k} u_{k}^{H},

where

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

γ_{k}

is a scalar for which

Re (γ_{k}) = 1.0

ζ_{k}

is a real scalar,

w_{k}

is a

(k - 1)

element vector and

z_{k}

is an

(n - m)

element vector.

γ_{k}

and

u_{k}

are chosen to annihilate the elements in the

k

th row of

A

The scalar

γ_{k}

and the vector

u_{k}

are returned in the

k

th element of theta and in the

k

th row of a, such that

θ_{k}

, given by

θ_{k} = (ζ_{k}, Im (γ_{k})) .

is in

theta (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: $a (lda :)$ – complex 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: $a (lda :)$ – complex 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

m

upper triangular part of a will contain the upper triangular matrix

R

, and the

m

m

strictly lower triangular part of a and the

m

(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: $theta (m)$ – complex array

theta (k)

contains the scalar

θ_{k}

for the

(m - k + 1)

th transformation. If

P_{k} = I

then

theta (k) = 0.0

; if

T_{k} = (\begin{array}{l} I & 0 & 0 \\ 0 & α & 0 \\ 0 & 0 & I \end{array}), Re (α) < 0.0

then

theta (k) = α

, otherwise

theta (k)

contains

θ_{k}

as described in Description and

θ_{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^{H} = 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

8 \times m^{2} (3 n - m) / 3

The first

k

rows of the unitary matrix

P^{H}

can be obtained by calling nag_matop_complex_gen_rq_formq (f01rk), which overwrites the

k

rows of

P^{H}

on the first

k

rows of the array a.

P^{H}

is obtained by the call:

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

Example

This example obtains the

R Q

factorization of the

3

5

matrix

A = (\begin{array}{r} - 0.5 i & 0.4 - 0.3 i & 0.4 & 0.3 - 0.4 i & 0.3 i \\ - 0.5 - 1.5 i & 0.9 - 1.3 i & - 0.4 - 0.4 i & 0.1 - 0.7 i & 0.3 - 0.3 i \\ - 1.0 - 1.0 i & 0.2 - 1.4 i & 1.8 & 0.0 & - 2.4 i \end{array}) .

Open in the MATLAB editor: f01rj_example

function f01rj_example


fprintf('f01rj example results\n\n');

a = [  0   - 0.5i,  0.4 - 0.3i,  0.4 + 0i,    0.3 + 0.4i,  0   + 0.3i;
      -0.5 - 1.5i,  0.9 - 1.3i, -0.4 - 0.4i,  0.1 - 0.7i,  0.3 - 0.3i;
      -1   - i,     0.2 - 1.4i,  1.8 + 0i,    0   + 0i,    0   - 2.4i];

[RQ, theta, ifail] = f01rj(a);

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

f01rj example results

RQ Factorization of A
Vector theta
   1.0387 + 0.1006i   1.1810 - 0.3809i   1.2244 + 0.0000i

Matrix A after factorization (R in left-hand upper triangle
   0.7878 + 0.0000i  -0.2549 - 0.4006i  -0.2774 - 0.2774i  -0.2850 + 0.5586i   0.1154 + 0.7031i
   0.0396 + 0.5222i  -2.1122 + 0.0000i  -1.1094 - 0.5547i   0.1283 + 0.2317i   0.0790 - 0.0361i
  -0.2265 + 0.2265i   0.0453 + 0.3171i  -3.6056 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.5436i

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

Chapter Contents

Chapter Introduction

NAG Toolbox