NAG Library Routine Document

F01RJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F01RJF 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.

2 Specification

SUBROUTINE F01RJF (

M, N, A, LDA, THETA, IFAIL)

INTEGER	M, N, LDA, IFAIL
COMPLEX (KIND=nag_wp)	A(LDA,*), THETA(M)

3 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.

4 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

5 Parameters

1: M – INTEGERInput

On entry:

m

, the number of rows of the matrix

A

When

M = 0

then an immediate return is effected.

Constraint:

M \geq 0

2: N – INTEGERInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

N \geq M

3: A(LDA, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least

\max (1, N)

On entry: the leading

m

n

part of the array A must contain the matrix to be factorized.

On exit: 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 Section 3.

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F01RJF is called.

Constraint:

LDA \geq \max (1, M)

5: THETA(M) – COMPLEX (KIND=nag_wp) arrayOutput

On exit:

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 Section 3 and

θ_{k}

is always in the range

(1.0, \sqrt{2.0})

6: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = - 1$

On entry,	$M < 0$ ,
or	$N < M$ ,
or	$LDA < M$ .

7 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 X02AJF),

c

is a modest function of

m

and

n

, and

‖.‖

denotes the spectral (two) norm.

8 Further Comments

The approximate number of floating point operations is given by

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

The first

k

rows of the unitary matrix

P^{H}

can be obtained by calling F01RKF, which overwrites the

k

rows of

P^{H}

on the first

k

rows of the array A.

P^{H}

is obtained by the call:

IFAIL = 0
CALL F01RKF('Separate',M,N,K,A,LDA,THETA,WORK,IFAIL)

WORK

must be a

\max (m - 1, k - m, 1)

element array. If

K

is larger than

M

, then A must have been declared to have at least

K

rows.

Operations involving the matrix

R

can readily be performed by the Level 2 BLAS routines F06SFF (ZTRMV) and F06SJF (ZTRSV), (see Chapter F06), but note that no test for near singularity of

R

is incorporated into F06SFF (ZTRMV). If

R

is singular, or nearly singular then F02XUF can be used to determine the singular value decomposition of

R

9 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}) .

NAG Library Routine DocumentF01RJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F01RJF