# NAG Library Routine Document

## 1Purpose

c06lbf computes the inverse Laplace transform $f\left(t\right)$ of a user-supplied function $F\left(s\right)$, defined for complex $s$. The routine uses a modification of Weeks' method which is suitable when $f\left(t\right)$ has continuous derivatives of all orders. The routine returns the coefficients of an expansion which approximates $f\left(t\right)$ and can be evaluated for given values of $t$ by subsequent calls of c06lcf.

## 2Specification

Fortran Interface
 Subroutine c06lbf ( f, b, mmax, m,
 Integer, Intent (In) :: mmax Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: m Real (Kind=nag_wp), Intent (In) :: sigma0, epstol Real (Kind=nag_wp), Intent (Inout) :: sigma, b Real (Kind=nag_wp), Intent (Out) :: acoef(mmax), errvec(8) Complex (Kind=nag_wp), External :: f
#include nagmk26.h
 void c06lbf_ (Complex (NAG_CALL *f)(const Complex *s),const double *sigma0, double *sigma, double *b, const double *epstol, const Integer *mmax, Integer *m, double acoef[], double errvec[], Integer *ifail)

## 3Description

Given a function $f\left(t\right)$ of a real variable $t$, its Laplace transform $F\left(s\right)$ is a function of a complex variable $s$, defined by
 $F s = ∫0∞ e-st f t dt , Res > σ0 .$
Then $f\left(t\right)$ is the inverse Laplace transform of $F\left(s\right)$. The value ${\sigma }_{0}$ is referred to as the abscissa of convergence of the Laplace transform; it is the rightmost real part of the singularities of $F\left(s\right)$.
c06lbf, along with its companion c06lcf, attempts to solve the following problem:
• given a function $F\left(s\right)$, compute values of its inverse Laplace transform $f\left(t\right)$ for specified values of $t$.
The method is a modification of Weeks' method (see Garbow et al. (1988a)), which approximates $f\left(t\right)$ by a truncated Laguerre expansion:
 $f~ t = eσt ∑ i=0 m-1 ai e -bt / 2 Li bt , σ > σ0 , b > 0$
where ${L}_{i}\left(x\right)$ is the Laguerre polynomial of degree $i$. This routine computes the coefficients ${a}_{i}$ of the above Laguerre expansion; the expansion can then be evaluated for specified $t$ by calling c06lcf. You must supply the value of ${\sigma }_{0}$, and also suitable values for $\sigma$ and $b$: see Section 9 for guidance.
The method is only suitable when $f\left(t\right)$ has continuous derivatives of all orders. For such functions the approximation $\stackrel{~}{f}\left(t\right)$ is usually good and inexpensive. The routine will fail with an error exit if the method is not suitable for the supplied function $F\left(s\right)$.
The routine is designed to satisfy an accuracy criterion of the form:
 $ft- f~t e σt < ε tol , for all ​t$
where ${\epsilon }_{\mathit{tol}}$ is a user-supplied bound. The error measure on the left-hand side is referred to as the pseudo-relative error, or pseudo-error for short. Note that if $\sigma >0$ and $t$ is large, the absolute error in $\stackrel{~}{f}\left(t\right)$ may be very large.
c06lbf is derived from the subroutine MODUL1 in Garbow et al. (1988a).

## 4References

Garbow B S, Giunta G, Lyness J N and Murli A (1988a) Software for an implementation of Weeks' method for the inverse laplace transform problem ACM Trans. Math. Software 14 163–170
Garbow B S, Giunta G, Lyness J N and Murli A (1988b) 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:     $\mathbf{f}$ – Complex (Kind=nag_wp) Function, supplied by the user.External Procedure
f must return the value of the Laplace transform function $F\left(s\right)$ for a given complex value of $s$.
The specification of f is:
Fortran Interface
 Function f ( s)
 Complex (Kind=nag_wp) :: f Complex (Kind=nag_wp), Intent (In) :: s
#include nagmk26.h
 Complex f (const Complex *s)
1:     $\mathbf{s}$ – Complex (Kind=nag_wp)Input
On entry: the value of $s$ for which $F\left(s\right)$ must be evaluated. The real part of s is greater than ${\sigma }_{0}$.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c06lbf is called. Arguments denoted as Input must not be changed by this procedure.
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c06lbf. If your code inadvertently does return any NaNs or infinities, c06lbf is likely to produce unexpected results.
2:     $\mathbf{sigma0}$ – Real (Kind=nag_wp)Input
On entry: the abscissa of convergence of the Laplace transform, ${\sigma }_{0}$.
3:     $\mathbf{sigma}$ – Real (Kind=nag_wp)Input/Output
On entry: the parameter $\sigma$ of the Laguerre expansion. If on entry ${\mathbf{sigma}}\le {\sigma }_{0}$, sigma is reset to ${\sigma }_{0}+0.7$.
On exit: the value actually used for $\sigma$, as just described.
4:     $\mathbf{b}$ – Real (Kind=nag_wp)Input/Output
On entry: the parameter $b$ of the Laguerre expansion. If on entry ${\mathbf{b}}<2\left(\sigma -{\sigma }_{0}\right)$, b is reset to $2.5\left(\sigma -{\sigma }_{0}\right)$.
On exit: the value actually used for $b$, as just described.
5:     $\mathbf{epstol}$ – Real (Kind=nag_wp)Input
On entry: the required relative pseudo-accuracy, that is, an upper bound on $\left|f\left(t\right)-\stackrel{~}{f}\left(t\right)\right|{e}^{-\sigma t}$.
6:     $\mathbf{mmax}$ – IntegerInput
On entry: an upper bound on the number of Laguerre expansion coefficients to be computed. The number of coefficients actually computed is always a power of $2$, so mmax should be a power of $2$; if mmax is not a power of $2$ then the maximum number of coefficients calculated will be the largest power of $2$ less than mmax.
Suggested value: ${\mathbf{mmax}}=1024$ is sufficient for all but a few exceptional cases.
Constraint: ${\mathbf{mmax}}\ge 8$.
7:     $\mathbf{m}$ – IntegerOutput
On exit: the number of Laguerre expansion coefficients actually computed. The number of calls to f is ${\mathbf{m}}/2+2$.
8:     $\mathbf{acoef}\left({\mathbf{mmax}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the first m elements contain the computed Laguerre expansion coefficients, ${a}_{i}$.
9:     $\mathbf{errvec}\left(8\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: an $8$-component vector of diagnostic information.
${\mathbf{errvec}}\left(1\right)$
Overall estimate of the pseudo-error $\left|f\left(t\right)-\stackrel{~}{f}\left(t\right)\right|{e}^{-\sigma t}={\mathbf{errvec}}\left(2\right)+{\mathbf{errvec}}\left(3\right)+{\mathbf{errvec}}\left(4\right)$.
${\mathbf{errvec}}\left(2\right)$
Estimate of the discretization pseudo-error.
${\mathbf{errvec}}\left(3\right)$
Estimate of the truncation pseudo-error.
${\mathbf{errvec}}\left(4\right)$
Estimate of the condition pseudo-error on the basis of minimal noise levels in function values.
${\mathbf{errvec}}\left(5\right)$
$K$, coefficient of a heuristic decay function for the expansion coefficients.
${\mathbf{errvec}}\left(6\right)$
$R$, base of the decay function for the expansion coefficients.
${\mathbf{errvec}}\left(7\right)$
Logarithm of the largest expansion coefficient.
${\mathbf{errvec}}\left(8\right)$
Logarithm of the smallest nonzero expansion coefficient.
The values $K$ and $R$ returned in ${\mathbf{errvec}}\left(5\right)$ and ${\mathbf{errvec}}\left(6\right)$ define a decay function $K{R}^{-i}$ constructed by the routine for the purposes of error estimation. It satisfies
 $ai < K R -i , ​ i= 1, 2, …, m .$
10:   $\mathbf{ifail}$ – 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, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ 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).
Note: c06lbf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{mmax}}<8$.
${\mathbf{ifail}}=2$
The estimated pseudo-error bounds are slightly larger than epstol. Note, however, that the actual errors in the final results may be smaller than epstol as bounds independent of the value of $t$ are pessimistic.
${\mathbf{ifail}}=3$
Computation was terminated early because the estimate of rounding error was greater than epstol. Increasing epstol may help.
${\mathbf{ifail}}=4$
The decay rate of the coefficients is too small. Increasing mmax may help.
${\mathbf{ifail}}=5$
The decay rate of the coefficients is too small. In addition the rounding error is such that the required accuracy cannot be obtained. Increasing mmax or epstol may help.
${\mathbf{ifail}}=6$
The behaviour of the coefficients does not enable reasonable prediction of error bounds. Check the value of sigma0. In this case, ${\mathbf{errvec}}\left(\mathit{i}\right)$ is set to $-1.0$, for $\mathit{i}=1,2,\dots ,5$.
${\mathbf{ifail}}=-99$
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.
When ${\mathbf{ifail}}\ge {\mathbf{3}}$, changing sigma or b may help. If not, the method should be abandoned.

## 7Accuracy

The error estimate returned 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

c06lbf is not threaded in any implementation.

### 9.1The Role of ${\sigma }_{0}$

Nearly all techniques for inversion of the Laplace transform require you to supply the value of ${\sigma }_{0}$, the convergence abscissa, or else an upper bound on ${\sigma }_{0}$. For this routine, one of the reasons for having to supply ${\sigma }_{0}$ is that the argument $\sigma$ must be greater than ${\sigma }_{0}$; otherwise the series for $\stackrel{~}{f}\left(t\right)$ will not converge.
If you do not know the value of ${\sigma }_{0}$, you must be prepared for significant preliminary effort, either in experimenting with the method and obtaining chaotic results, or in attempting to locate the rightmost singularity of $F\left(s\right)$.
The value of ${\sigma }_{0}$ is also relevant in defining a natural accuracy criterion. For large $t$, $f\left(t\right)$ is of uniform numerical order $k{e}^{{\sigma }_{0}t}$, so a natural measure of relative accuracy of the approximation $\stackrel{~}{f}\left(t\right)$ is:
 $εnat t = f~ t - f t / e σ0t .$
c06lbf uses the supplied value of ${\sigma }_{0}$ only in determining the values of $\sigma$ and $b$ (see Sections 9.2 and 9.3); thereafter it bases its computation entirely on $\sigma$ and $b$.

### 9.2Choice of $\sigma$

Even when the value of ${\sigma }_{0}$ is known, choosing a value for $\sigma$ is not easy. Briefly, the series for $\stackrel{~}{f}\left(t\right)$ converges slowly when $\sigma -{\sigma }_{0}$ is small, and faster when $\sigma -{\sigma }_{0}$ is larger. However the natural accuracy measure satisfies
 $εnat t < εtol e σ - σ0 t$
and this degrades exponentially with $t$, the exponential constant being $\sigma -{\sigma }_{0}$.
Hence, if you require meaningful results over a large range of values of $t$, you should choose $\sigma -{\sigma }_{0}$ small, in which case the series for $\stackrel{~}{f}\left(t\right)$ converges slowly; while for a smaller range of values of $t$, you can allow $\sigma -{\sigma }_{0}$ to be larger and obtain faster convergence.
The default value for $\sigma$ used by c06lbf is ${\sigma }_{0}+0.7$. There is no theoretical justification for this.

### 9.3Choice of $b$

The simplest advice for choosing $b$ is to set $b/2\ge \sigma -{\sigma }_{0}$. The default value used by the routine is $2.5\left(\sigma -{\sigma }_{0}\right)$. A more refined choice is to set
 $b/2 ≥ minj σ-sj$
where ${s}_{j}$ are the singularities of $F\left(s\right)$.

## 10Example

This example computes values of the inverse Laplace transform of the function
 $Fs = 3 s2-9 .$
 $ft = sinh⁡3t .$
The program first calls c06lbf to compute the coefficients of the Laguerre expansion, and then calls c06lcf to evaluate the expansion at $t=0$, $1$, $2$, $3$, $4$, $5$.

### 10.1Program Text

Program Text (c06lbfe.f90)

### 10.2Program Data

Program Data (c06lbfe.d)

### 10.3Program Results

Program Results (c06lbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017