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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_interp_1d_ratnl_eval (e01rb)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_interp_1d_ratnl_eval (e01rb) evaluates continued fractions of the form produced by nag_interp_1d_ratnl (e01ra).


[f, ifail] = e01rb(a, u, x, 'm', m)
[f, ifail] = nag_interp_1d_ratnl_eval(a, u, x, 'm', m)


nag_interp_1d_ratnl_eval (e01rb) evaluates the continued fraction
Rix=am-i+ 2x-um-i+ 1 1+Ri- 1x ,   for ​ i=m,m- 1,,2.  
for a prescribed value of x. nag_interp_1d_ratnl_eval (e01rb) is intended to be used to evaluate the continued fraction representation (of an interpolatory rational function) produced by nag_interp_1d_ratnl (e01ra).


Graves–Morris P R and Hopkins T R (1981) Reliable rational interpolation Numer. Math. 36 111–128


Compulsory Input Parameters

1:     am – double array
aj must be set to the value of the parameter aj in the continued fraction, for j=1,2,,m.
2:     um – double array
uj must be set to the value of the parameter uj in the continued fraction, for j=1,2,,m-1. (The element um is not used).
3:     x – double scalar
The value of x at which the continued fraction is to be evaluated.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the dimension of the arrays a, u. (An error is raised if these dimensions are not equal.)
m, the number of terms in the continued fraction.
Constraint: m1.

Output Parameters

1:     f – double scalar
The value of the continued fraction corresponding to the value of x.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
The value of x corresponds to a pole of Rx or is so close that an overflow is likely to ensue.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


See Accuracy in nag_interp_1d_ratnl (e01ra).

Further Comments

The time taken by nag_interp_1d_ratnl_eval (e01rb) is approximately proportional to m.


This example reads in the arguments aj and uj of a continued fraction (as determined by the example for nag_interp_1d_ratnl (e01ra)) and evaluates the continued fraction at a point x.
function e01rb_example

fprintf('e01rb example results\n\n');

% Calculate rational approximation coefficients
x = [0:4];
f = [4   2   4   7   10.4];

[m, a, u, ifail] = e01ra( ...
                          x, f);

% Evaluate at single point
x = 6;
[f, ifail] = e01rb( ...
                    a, u, x, 'm', m);

fprintf('x    = %12.4e\n',x);
fprintf('R(x) = %12.4e\n',f);

e01rb example results

x    =   6.0000e+00
R(x) =   1.7714e+01

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Chapter Introduction
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