NAG Library Routine Document

F01RGF

+− 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

F01RGF reduces the complex

m

n

(

m \leq n

) upper trapezoidal matrix

A

to upper triangular form by means of unitary transformations.

2 Specification

SUBROUTINE F01RGF (

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

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

3 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).

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

upper trapezoidal 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

(n - m)

upper trapezoidal part of A 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 F01RGF 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

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 Section 3 and

Re (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

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

9 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.

NAG Library Routine DocumentF01RGF

+− 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

F01RGF