nag_sum_invlaplace_weeks_eval (c06lc) evaluates an inverse Laplace transform at a given point, using the expansion coefficients computed by nag_sum_invlaplace_weeks (c06lb).

Syntax

[finv, ifail] = c06lc(t, sigma, b, acoef, errvec, 'm', m)

[finv, ifail] = nag_sum_invlaplace_weeks_eval(t, sigma, b, acoef, errvec, 'm', m)

Description

nag_sum_invlaplace_weeks_eval (c06lc) is designed to be used following a call to nag_sum_invlaplace_weeks (c06lb), which computes an inverse Laplace transform by representing it as a Laguerre expansion of the form:

\tilde{f} (t) = e^{σ t} \sum_{i = 0}^{m - 1} a_{i} e^{- b t / 2} L_{i} (b t), σ > σ_{O}, b > 0

where

L_{i} (x)

is the Laguerre polynomial of degree

i

This function simply evaluates the above expansion for a specified value of

t

nag_sum_invlaplace_weeks_eval (c06lc) is derived from the function MODUL2 in Garbow et al. (1988)

References

Garbow B S, Giunta G, Lyness J N and Murli A (1988) Algorithm 662: A Fortran software package for the numerical inversion of the Laplace transform based on Weeks' method ACM Trans. Math. Software 14 171–176

Parameters

Compulsory Input Parameters

1: $t$ – double scalar: The value $t$ for which the inverse Laplace transform $f (t)$ must be evaluated.
2: $sigma$ – double scalar
3: $b$ – double scalar
4: $acoef (m)$ – double array
5: $errvec (8)$ – double array: sigma, b, m, acoef and errvec must be unchanged from the previous call of nag_sum_invlaplace_weeks (c06lb).

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array acoef.
sigma, b, m, acoef and errvec must be unchanged from the previous call of nag_sum_invlaplace_weeks (c06lb).

Output Parameters

1: $finv$ – double scalar: The approximation to the inverse Laplace transform at $t$ .
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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: The approximation to $f (t)$ is too large to be representable: finv is set to $0.0$ .

W $ifail = 2$: The approximation to $f (t)$ is too small to be representable: finv is set to $0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The error estimate returned by nag_sum_invlaplace_weeks (c06lb) in

errvec (1)

has been found in practice to be a highly reliable bound on the pseudo-error

|f (t) - \tilde{f} (t)| e^{- σ t}

Further Comments

nag_sum_invlaplace_weeks_eval (c06lc) is primarily designed to evaluate

\tilde{f} (t)

when

t > 0

. When

t \leq 0

, the result approximates the analytic continuation of

f (t)

; the approximation becomes progressively poorer as

t

becomes more negative.

Example

See example for nag_sum_invlaplace_weeks (c06lb).

Open in the MATLAB editor: c06lc_example

function c06lc_example


fprintf('c06lc example results\n\n');

sigma0 = 3;
sigma = 0;
b = 0;
epstol = 1e-05;
[sigmaOut, bOut, m, acoef, errvec, ifail] = ...
    c06lb(@f, sigma0, sigma, b, epstol, 'mmax', int64(512));

fprintf('\nNo. of coefficients returned by c06lb = %d\n\n', m);
fprintf('                   Computed           Exact       Pseudo\n');
fprintf('        t              f(t)            f(t)        error\n');
for j = 0:5
  [finv, ifail] = c06lc(j, sigmaOut, bOut, acoef, errvec);
  exact = sinh(3*j);
  pserr = abs(finv-exact)/exp(sigmaOut*j);
  fprintf(' %10.2f %15.4f %15.4f %12.1g\n',j, finv, exact, pserr);
end



function [f] = f(s)
% Evaluate the Laplace transform function.
f=3.0/(s^2-9.0);
if isreal(f)
    f=complex(f);
end

c06lc example results


No. of coefficients returned by c06lb = 64

                   Computed           Exact       Pseudo
        t              f(t)            f(t)        error
       0.00          0.0000          0.0000        2e-09
       1.00         10.0179         10.0179        2e-09
       2.00        201.7132        201.7132        1e-10
       3.00       4051.5420       4051.5419        1e-09
       4.00      81377.3949      81377.3957        3e-10
       5.00    1634508.5023    1634508.6862        2e-09

PDF version (NAG web site, 64-bit version, 64-bit version)

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