NAG Library Routine Document

F06FJF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

1 Purpose

F06FJF updates the Euclidean norm of real vector in scaled form.

2 Specification

SUBROUTINE F06FJF (

N, X, INCX, SCAL, SUMSQ)

INTEGER	N, INCX
REAL (KIND=nag_wp)	X(*), SCAL, SUMSQ

3 Description

Given an

n

-element real vector

x

, and real scalars

α

and

ξ

, F06FJF returns updated values

\tilde{α}

and

\tilde{ξ}

such that

{\tilde{α}}^{2} \tilde{ξ} = x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2} + α^{2} ξ .

F06FJF is designed for use in the safe computation of the Euclidean norm of a real vector, without unnecessary overflow or destructive underflow. An initial call to F06FJF (with

ξ = 1

and

α = 0

) may be followed by further calls to F06FJF and finally a call to F06BMF to complete the computation. Multiple calls of F06FJF may be needed if the elements of the vector cannot all be accessed in a single array X.

4 References

None.

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of elements in $x$ .
2: X( $*$ ) – REAL (KIND=nag_wp) arrayInput: Note: the dimension of the array X must be at least $\max (1, 1 + (N - 1) \times INCX)$ .
On entry: the $n$ -element vector $x$ . $x_{i}$ must be stored in $X (1 + (i - 1) \times INCX)$ , for $i = 1, 2, \dots, N$ .
Intermediate elements of X are not referenced.
3: INCX – INTEGERInput: On entry: the increment in the subscripts of X between successive elements of $x$ .
Constraint: $INCX > 0$ .
4: SCAL – REAL (KIND=nag_wp)Input/Output: On entry: the scaling factor $α$ . On the first call to F06FJF $SCAL = 0.0$ .
Constraint: $SCAL \geq 0.0$ .

On exit: the updated scaling factor $\tilde{α} = \max_{i} (α, |x_{i}|)$ .
5: SUMSQ – REAL (KIND=nag_wp)Input/Output: On entry: the scaled sum of squares $ξ$ . On the first call to F06FJF $SUMSQ = 1.0$ .
Constraint: $SUMSQ \geq 1.0$ .

On exit: the updated scaled sum of squares $\tilde{ξ}$ , satisfying: $1 \leq \tilde{ξ} \leq ξ + n$ .