# NAG Library Routine Document

## 1Purpose

c06lcf evaluates an inverse Laplace transform at a given point, using the expansion coefficients computed by c06lbf.

## 2Specification

Fortran Interface
 Subroutine c06lcf ( t, b, m, finv,
 Integer, Intent (In) :: m Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: t, sigma, b, acoef(m), errvec(8) Real (Kind=nag_wp), Intent (Out) :: finv
#include nagmk26.h
 void c06lcf_ (const double *t, const double *sigma, const double *b, const Integer *m, const double acoef[], const double errvec[], double *finv, Integer *ifail)

## 3Description

c06lcf is designed to be used following a call to c06lbf, which computes an inverse Laplace transform by representing it as a Laguerre expansion of the form:
 $f~ t = eσt ∑ i=0 m-1 ai e -bt/2 Li bt , σ > σO , b > 0$
where ${L}_{i}\left(x\right)$ is the Laguerre polynomial of degree $i$.
This routine simply evaluates the above expansion for a specified value of $t$.
c06lcf is derived from the subroutine MODUL2 in Garbow et al. (1988)

## 4References

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

## 5Arguments

1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: the value $t$ for which the inverse Laplace transform $f\left(t\right)$ must be evaluated.
2:     $\mathbf{sigma}$ – Real (Kind=nag_wp)Input
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
4:     $\mathbf{m}$ – IntegerInput
5:     $\mathbf{acoef}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
6:     $\mathbf{errvec}\left(8\right)$ – Real (Kind=nag_wp) arrayInput
On entry: sigma, b, m, acoef and errvec must be unchanged from the previous call of c06lbf.
7:     $\mathbf{finv}$ – Real (Kind=nag_wp)Output
On exit: the approximation to the inverse Laplace transform at $t$.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
The approximation to $f\left(t\right)$ is too large to be representable: finv is set to $0.0$.
${\mathbf{ifail}}=2$
The approximation to $f\left(t\right)$ is too small to be representable: finv is set to $0.0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The error estimate returned by c06lbf in ${\mathbf{errvec}}\left(1\right)$ has been found in practice to be a highly reliable bound on the pseudo-error $\left|f\left(t\right)-\stackrel{~}{f}\left(t\right)\right|{e}^{-\sigma t}$.

## 8Parallelism and Performance

c06lcf is not threaded in any implementation.

c06lcf is primarily designed to evaluate $\stackrel{~}{f}\left(t\right)$ when $t>0$. When $t\le 0$, the result approximates the analytic continuation of $f\left(t\right)$; the approximation becomes progressively poorer as $t$ becomes more negative.