C06LBF computes the inverse Laplace transform of a user-supplied function , defined for complex . The routine uses a modification of Weeks' method which is suitable when has continuous derivatives of all orders. The routine returns the coefficients of an expansion which approximates and can be evaluated for given values of by subsequent calls of C06LCF.
Given a function of a real variable , its Laplace transform is a function of a complex variable , defined by
Then is the inverse Laplace transform of . The value is referred to as the abscissa of convergence of the Laplace transform; it is the rightmost real part of the singularities of .
C06LBF, along with its companion C06LCF, attempts to solve the following problem:
given a function , compute values of its inverse Laplace transform for specified values of .
The method is a modification of Weeks' method (see Garbow et al. (1988a)), which approximates by a truncated Laguerre expansion:
where is the Laguerre polynomial of degree . This routine computes the coefficients of the above Laguerre expansion; the expansion can then be evaluated for specified by calling C06LCF. You must supply the value of , and also suitable values for and : see Section 8 for guidance.
The method is only suitable when has continuous derivatives of all orders. For such functions the approximation is usually good and inexpensive. The routine will fail with an error exit if the method is not suitable for the supplied function .
The routine is designed to satisfy an accuracy criterion of the form:
where is a user-supplied bound. The error measure on the left-hand side is referred to as the pseudo-relativeerror, or pseudo-error for short. Note that if and is large, the absolute error in may be very large.
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. Software14 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. Software14 171–176
1: F – COMPLEX (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
F must return the value of the Laplace transform function for a given complex value of .
On entry: the value of for which must be evaluated. The real part of S is greater than .
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, .
3: SIGMA – REAL (KIND=nag_wp)Input/Output
On entry: the parameter of the Laguerre expansion. If on entry , SIGMA is reset to .
On exit: the value actually used for , as just described.
4: B – REAL (KIND=nag_wp)Input/Output
On entry: the parameter of the Laguerre expansion. If on entry , B is reset to .
On exit: the value actually used for , as just described.
5: EPSTOL – REAL (KIND=nag_wp)Input
On entry: the required relative pseudo-accuracy, that is, an upper bound on .
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 , so MMAX should be a power of ; if MMAX is not a power of then the maximum number of coefficients calculated will be the largest power of less than MMAX.
is sufficient for all but a few exceptional cases.
7: M – INTEGEROutput
On exit: the number of Laguerre expansion coefficients actually computed. The number of calls to F is .
On exit: the first M elements contain the computed Laguerre expansion coefficients, .
9: ERRVEC() – REAL (KIND=nag_wp) arrayOutput
On exit: an -component vector of diagnostic information.
Overall estimate of the pseudo-error .
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.
, coefficient of a heuristic decay function for the expansion coefficients.
, base of the decay function for the expansion coefficients.
Logarithm of the largest expansion coefficient.
Logarithm of the smallest nonzero expansion coefficient.
The values and returned in and define a decay function constructed by the routine for the purposes of error estimation. It satisfies
10: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to , . 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 is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if on exit, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , 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:
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 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, is set to , for .
When , changing SIGMA or B may help. If not, the method should be abandoned.
The error estimate returned in has been found in practice to be a highly reliable bound on the pseudo-error .
8 Further Comments
8.1 The Role of
Nearly all techniques for inversion of the Laplace transform require you to supply the value of , the convergence abscissa, or else an upper bound on . For this routine, one of the reasons for having to supply is that the parameter must be greater than ; otherwise the series for will not converge.
If you do not know the value of , 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 .
The value of is also relevant in defining a natural accuracy criterion. For large , is of uniform numerical order , so a natural measure of relative accuracy of the approximation is:
C06LBF uses the supplied value of only in determining the values of and (see Sections 8.2 and 8.3); thereafter it bases its computation entirely on and .
8.2 Choice of
Even when the value of is known, choosing a value for is not easy. Briefly, the series for converges slowly when is small, and faster when is larger. However the natural accuracy measure satisfies
and this degrades exponentially with , the exponential constant being .
Hence, if you require meaningful results over a large range of values of , you should choose small, in which case the series for converges slowly; while for a smaller range of values of , you can allow to be larger and obtain faster convergence.
The default value for used by C06LBF is . There is no theoretical justification for this.
8.3 Choice of
The simplest advice for choosing is to set . The default value used by the routine is . A more refined choice is to set
where are the singularities of .
This example computes values of the inverse Laplace transform of the function
The exact answer is
The program first calls C06LBF to compute the coefficients of the Laguerre expansion, and then calls C06LCF to evaluate the expansion at , , , , , .