C06LBF (PDF version)
C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

C06LBF computes the inverse Laplace transform ft  of a user-supplied function Fs , defined for complex s. The routine uses a modification of Weeks' method which is suitable when ft  has continuous derivatives of all orders. The routine returns the coefficients of an expansion which approximates ft  and can be evaluated for given values of t by subsequent calls of C06LCF.

2  Specification

COMPLEX (KIND=nag_wp)  F

3  Description

Given a function f t  of a real variable t, its Laplace transform Fs  is a function of a complex variable s, defined by
F s = 0 e-st f t dt ,   Res > σ0 .
Then ft  is the inverse Laplace transform of Fs . The value σ0  is referred to as the abscissa of convergence of the Laplace transform; it is the rightmost real part of the singularities of Fs .
C06LBF, along with its companion C06LCF, attempts to solve the following problem:
The method is a modification of Weeks' method (see Garbow et al. (1988a)), which approximates ft  by a truncated Laguerre expansion:
f~ t = eσt i=0 m-1 ai e -bt / 2 Li bt ,   σ > σ0 ,   b > 0
where Li x  is the Laguerre polynomial of degree i. This routine computes the coefficients ai  of the above Laguerre expansion; the expansion can then be evaluated for specified t by calling C06LCF. You must supply the value of σ0 , and also suitable values for σ and b: see Section 8 for guidance.
The method is only suitable when ft  has continuous derivatives of all orders. For such functions the approximation f~ t  is usually good and inexpensive. The routine will fail with an error exit if the method is not suitable for the supplied function Fs .
The routine is designed to satisfy an accuracy criterion of the form:
ft- f~t e σt < ε tol ,   for all ​t
where εtol  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 σ>0  and t is large, the absolute error in f~ t  may be very large.
C06LBF is derived from the subroutine MODUL1 in Garbow et al. (1988a).

4  References

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

5  Parameters

1:     F – COMPLEX (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
F must return the value of the Laplace transform function Fs for a given complex value of s.
The specification of F is:
COMPLEX (KIND=nag_wp)  S
1:     S – COMPLEX (KIND=nag_wp)Input
On entry: the value of s for which Fs must be evaluated. The real part of S is greater than σ0.
F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C06LBF is called. Parameters denoted as Input must not be changed by this procedure.
2:     SIGMA0 – REAL (KIND=nag_wp)Input
On entry: the abscissa of convergence of the Laplace transform, σ0.
3:     SIGMA – REAL (KIND=nag_wp)Input/Output
On entry: the parameter σ of the Laguerre expansion. If on entry SIGMAσ0, SIGMA is reset to σ0+0.7.
On exit: the value actually used for σ, as just described.
4:     B – REAL (KIND=nag_wp)Input/Output
On entry: the parameter b of the Laguerre expansion. If on entry B < 2 σ - σ0 , B is reset to 2.5σ-σ0.
On exit: the value actually used for b, as just described.
5:     EPSTOL – REAL (KIND=nag_wp)Input
On entry: the required relative pseudo-accuracy, that is, an upper bound on f t - f~ t e-σt.
6:     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: MMAX=1024 is sufficient for all but a few exceptional cases.
Constraint: MMAX8.
7:     M – INTEGEROutput
On exit: the number of Laguerre expansion coefficients actually computed. The number of calls to F is M/2+2.
8:     ACOEF(MMAX) – REAL (KIND=nag_wp) arrayOutput
On exit: the first M elements contain the computed Laguerre expansion coefficients, ai.
9:     ERRVEC(8) – REAL (KIND=nag_wp) arrayOutput
On exit: an 8-component vector of diagnostic information.
Overall estimate of the pseudo-error f t - f~ t e-σt =ERRVEC2 + ERRVEC3 + ERRVEC4.
Estimate of the discretization pseudo-error.
Estimate of the truncation pseudo-error.
Estimate of the condition pseudo-error on the basis of minimal noise levels in function values.
K, coefficient of a heuristic decay function for the expansion coefficients.
R, base of the decay function for the expansion coefficients.
Logarithm of the largest expansion coefficient.
Logarithm of the smallest nonzero expansion coefficient.
The values K and R returned in ERRVEC5 and ERRVEC6 define a decay function KR-i constructed by the routine for the purposes of error estimation. It satisfies
ai < K R -i ,   ​ i= 1, 2, , m .
10:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ 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 parameters may be useful even if IFAIL0 on exit, the recommended value is -1. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry 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:
On entry,MMAX<8.
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.
Computation was terminated early because the estimate of rounding error was greater than EPSTOL. Increasing EPSTOL may help.
The decay rate of the coefficients is too small. Increasing MMAX may help.
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.
The behaviour of the coefficients does not enable reasonable prediction of error bounds. Check the value of SIGMA0. In this case, ERRVECi  is set to -1.0 , for i=1,2,,5.
When IFAIL3, changing SIGMA or B may help. If not, the method should be abandoned.

7  Accuracy

The error estimate returned in ERRVEC1  has been found in practice to be a highly reliable bound on the pseudo-error ft-f~t e-σt .

8  Further Comments

8.1  The Role of σ0

Nearly all techniques for inversion of the Laplace transform require you to supply the value of σ0 , the convergence abscissa, or else an upper bound on σ0 . For this routine, one of the reasons for having to supply σ0  is that the parameter σ must be greater than σ0 ; otherwise the series for f~t  will not converge.
If you do not know the value of σ0 , 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 Fs .
The value of σ0  is also relevant in defining a natural accuracy criterion. For large t, ft  is of uniform numerical order k e σ0t , so a natural measure of relative accuracy of the approximation f~ t  is:
εnat t = f~ t - f t / e σ0t .
C06LBF uses the supplied value of σ0  only in determining the values of σ and b (see Sections 8.2 and 8.3); thereafter it bases its computation entirely on σ and b.

8.2  Choice of σ

Even when the value of σ0  is known, choosing a value for σ is not easy. Briefly, the series for f~t  converges slowly when σ-σ0  is small, and faster when σ-σ0  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 σ-σ0 .
Hence, if you require meaningful results over a large range of values of t, you should choose σ-σ0  small, in which case the series for f~t  converges slowly; while for a smaller range of values of t, you can allow σ-σ0  to be larger and obtain faster convergence.
The default value for σ used by C06LBF is σ0+0.7 . There is no theoretical justification for this.

8.3  Choice of b

The simplest advice for choosing b is to set b/2 σ - σ0 . The default value used by the routine is 2.5 σ - σ0 . A more refined choice is to set
b/2 minj σ-sj
where sj  are the singularities of Fs .

9  Example

This example computes values of the inverse Laplace transform of the function
Fs = 3 s2-9 .
The exact answer is
ft = sinh3t .
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.

9.1  Program Text

Program Text (c06lbfe.f90)

9.2  Program Data

Program Data (c06lbfe.d)

9.3  Program Results

Program Results (c06lbfe.r)

Produced by GNUPLOT 4.4 patchlevel 0 1e-10 1e-08 1e-06 0.0001 0.01 1 100 10000 1e+06 1e+08 0 1 2 3 4 5 1e-10 1e-08 1e-06 0.0001 0.01 1 f(t) Pseudo Error t Example Program Inverse Laplace Transform of 3/(s2-9) f(t) pseudo error

C06LBF (PDF version)
C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012