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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_matop_complex_trapez_rq (f01rg)

▸▿ 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_trapez_rq (f01rg) reduces the complex

m

n

(

m \leq n

) upper trapezoidal matrix

A

to upper triangular form by means of unitary transformations.

Syntax

[a, theta, ifail] = f01rg(a, 'm', m, 'n', n)

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

Description

The

m

n (m \leq n)

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^{H},

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

where

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

γ_{k}

is a scalar for which

Re (γ_{k}) = 1.0

ζ_{k}

is a real scalar and

z_{k}

is an

(n - m)

element vector.

γ_{k}

ζ_{k}

and

z_{k}

are chosen to annihilate the elements of the

k

th row of

X

and to make the diagonal elements of

R

real.

The scalar

γ_{k}

and the vector

u_{k}

are returned in the

k

th element of the array 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)

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: $a (lda :)$ – 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 $n$ 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: $a (lda :)$ – 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

(n - m)

upper trapezoidal part of a 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

T_{k} = I

then

theta (k) = 0.0

; if

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

then

theta (k) = α

, otherwise

theta (k)

contains

θ_{k}

as described in Description and

Re (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

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

Example

This example reduces the

3

4

matrix

(\begin{array}{r} 2.4 & 0.8 + 0.8 i & - 1.4 + 0.6 i & 3.0 - 1.0 i \\ 0 & 1.6 & 0.8 + 0.3 i & 0.4 + 0.5 i \\ 0 & 0 & 1.0 & 2.0 - 1.0 i \end{array})

to upper triangular form.

Open in the MATLAB editor: f01rg_example

function f01rg_example


fprintf('f01rg example results\n\n');

a = [ 2.4,     0.8 + 0.8i, -1.4 + 0.6i,  3   - i;
      0 + 0i,  1.6 + 0i,    0.8 + 0.3i,  0.4 + 0.5i;
      0 + 0i,  0   + 0i,    1   + 0i,    2   - i];

[RQ, theta, ifail] = f01rg(a);

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

f01rg example results

RQ Factorization of A
Vector theta
   1.2924 + 0.0000i   1.3861 + 0.0000i   1.1867 + 0.0000i

Matrix A after factorization (R in left-hand upper triangle
  -3.5808 + 0.0000i   0.2533 - 0.9059i  -2.2862 - 0.6532i   0.5120 + 0.2601i
   0.0000 + 0.0000i  -1.7369 + 0.0000i  -0.4491 - 0.6940i  -0.2544 - 0.1187i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -2.4495 + 0.0000i   0.6880 + 0.3440i

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Chapter Contents

Chapter Introduction

NAG Toolbox