NAG Library Routine Document

F01QGF

+− 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

F01QGF reduces the

m

n

(

m \leq n

) real upper trapezoidal matrix

A

to upper triangular form by means of orthogonal transformations.

2 Specification

SUBROUTINE F01QGF (

M, N, A, LDA, ZETA, IFAIL)

INTEGER	M, N, LDA, IFAIL
REAL (KIND=nag_wp)	A(LDA,*), ZETA(M)

3 Description

The

m

n

(

m \leq n

) real 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^{T},

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

where

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

ζ_{k}

is a scalar and

z_{k}

is an (

n - m

) element vector.

ζ_{k}

and

z_{k}

are chosen to annihilate the elements of the

k

th row of

X

The vector

u_{k}

is returned in the

k

th element of the array ZETA and in the

k

th row of A, such that

ζ_{k}

is in

ZETA (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, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .
On entry: the leading $m$ by $n$ upper trapezoidal part of the array A must contain the matrix to be factorized.

On exit: the $m$ by $m$ upper triangular part of A will contain the upper triangular matrix $R$ , and the $m$ by $(n - m)$ 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 F01QGF is called.
Constraint: $LDA \geq \max (1, M)$ .
5: ZETA(M) – REAL (KIND=nag_wp) arrayOutput: On exit: $ZETA (k)$ contains the scalar $ζ_{k}$ for the $(m - k + 1)$ th transformation. If $T_{k} = I$ then $ZETA (k) = 0.0$ , otherwise $ZETA (k)$ contains $ζ_{k}$ as described in Section 3 and $ζ_{k}$ 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^{T} = 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

2 m^{2} (n - m)

9 Example

This example reduces the

3

5

matrix

A = (\begin{array}{r} 2.4 & 0.8 & - 1.4 & 3.0 & - 0.8 \\ 0.0 & 1.6 & 0.8 & 0.4 & - 0.8 \\ 0.0 & 0.0 & 1.0 & 2.0 & 2.0 \end{array})

to upper triangular form.

NAG Library Routine DocumentF01QGF

+− 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

F01QGF