NAG Library Routine Document

F01QJF

+− 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

F01QJF finds the

R Q

factorization of the real

m

n

(

m \leq n

) matrix

A

, so that

A

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

2 Specification

SUBROUTINE F01QJF (

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

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

3 Description

The

m

n

matrix

A

is factorized as

\begin{array}{l} A = (\begin{matrix} R & 0 \end{matrix}) P^{T} & when m < n, \\ A = R P^{T} & when m = n, \end{array}

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} = I - u_{k} u_{k}^{T},

where

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

ζ_{k}

is a scalar,

w_{k}

is an

(k - 1)

element vector and

z_{k}

is an

(n - m)

element vector.

u_{k}

is chosen to annihilate the elements in the

k

th row of

A

The vector

u_{k}

is returned in the

k

th element of ZETA and in the

k

th row of A, such that

ζ_{k}

is in

ZETA (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, $*$ ) – 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 $n$ 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 $m$ strictly lower triangular part of A and the $m$ by $(n - m)$ 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 F01QJF 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 $P_{k} = I$ 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

(\begin{matrix} R & 0 \end{matrix}) 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} (3 n - m) / 3

The first

k

rows of the orthogonal matrix

P^{T}

can be obtained by calling F01QKF, which overwrites the

k

rows of

P^{T}

on the first

k

rows of the array A.

P^{T}

is obtained by the call:

 
IFAIL = 0
CALL F01QKF('Separate',M,N,K,A,LDA,ZETA,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 F06PFF (DTRMV) and F06PJF (DTRSV) (see Chapter F06), but note that no test for near singularity of

R

is incorporated into F06PJF (DTRSV). If

R

is singular, or nearly singular then F02WUF 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} 2.0 & 2.0 & 1.6 & 2.0 & 1.2 \\ 2.5 & 2.5 & - 0.4 & - 0.5 & - 0.3 \\ 2.5 & 2.5 & 2.8 & 0.5 & - 2.9 \end{array}) .

NAG Library Routine DocumentF01QJF

+− 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

F01QJF