NAG Library Routine Document

C06LBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

+− 8 Further Comments

8.1 The Role of σ0

8.2 Choice of σ

8.3 Choice of b

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C06LBF computes the inverse Laplace transform

f (t)

of a user-supplied function

F (s)

, defined for complex

s

. The routine uses a modification of Weeks' method which is suitable when

f (t)

has continuous derivatives of all orders. The routine returns the coefficients of an expansion which approximates

f (t)

and can be evaluated for given values of

t

by subsequent calls of C06LCF.

2 Specification

SUBROUTINE C06LBF (

F, SIGMA0, SIGMA, B, EPSTOL, MMAX, M, ACOEF, ERRVEC, IFAIL)

INTEGER	MMAX, M, IFAIL
REAL (KIND=nag_wp)	SIGMA0, SIGMA, B, EPSTOL, ACOEF(MMAX), ERRVEC(8)
COMPLEX (KIND=nag_wp)	F
EXTERNAL	F

3 Description

Given a function

f (t)

of a real variable

t

, its Laplace transform

F (s)

is a function of a complex variable

s

, defined by

F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t, Re (s) > σ_{0} .

Then

f (t)

is the inverse Laplace transform of

F (s)

. The value

σ_{0}

is referred to as the abscissa of convergence of the Laplace transform; it is the rightmost real part of the singularities of

F (s)

C06LBF, along with its companion C06LCF, attempts to solve the following problem:

given a function $F (s)$ , compute values of its inverse Laplace transform $f (t)$ for specified values of $t$ .

The method is a modification of Weeks' method (see Garbow et al. (1988a)), which approximates

f (t)

by a truncated Laguerre expansion:

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

where

L_{i} (x)

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

σ_{0}

, and also suitable values for

σ

and

b

: see Section 8 for guidance.

The method is only suitable when

f (t)

has continuous derivatives of all orders. For such functions the approximation

\tilde{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

F (s)

The routine is designed to satisfy an accuracy criterion of the form:

|\frac{f (t) - \tilde{f} (t)}{e^{σ t}}| < ε_{tol},   for all ​ t

where

ε_{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

σ > 0

and

t

is large, the absolute error in

\tilde{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

F (s)

for a given complex value of

s

The specification of F is:

FUNCTION F (

COMPLEX (KIND=nag_wp) F

COMPLEX (KIND=nag_wp)

1: S – COMPLEX (KIND=nag_wp)Input: On entry: the value of $s$ for which $F (s)$ must be evaluated. The real part of S is greater than $σ_{0}$ .

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,

σ_{0}

3: SIGMA – REAL (KIND=nag_wp)Input/Output

On entry: the parameter

σ

of the Laguerre expansion. If on entry

SIGMA \leq σ_{0}

, SIGMA is reset to

σ_{0} + 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 - σ_{0})

, B is reset to

2.5 (σ - σ_{0})

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) - \tilde{f} (t)| e^{- σ t}

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:

MMAX \geq 8

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,

a_{i}

9: ERRVEC( $8$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: an

8

-component vector of diagnostic information.

$ERRVEC (1)$: Overall estimate of the pseudo-error $|f (t) - \tilde{f} (t)| e^{- σ t} = ERRVEC (2) + ERRVEC (3) + ERRVEC (4)$ .
$ERRVEC (2)$: Estimate of the discretization pseudo-error.
$ERRVEC (3)$: Estimate of the truncation pseudo-error.
$ERRVEC (4)$: Estimate of the condition pseudo-error on the basis of minimal noise levels in function values.
$ERRVEC (5)$: $K$ , coefficient of a heuristic decay function for the expansion coefficients.
$ERRVEC (6)$: $R$ , base of the decay function for the expansion coefficients.
$ERRVEC (7)$: Logarithm of the largest expansion coefficient.
$ERRVEC (8)$: Logarithm of the smallest nonzero expansion coefficient.

The values

K

and

R

returned in

ERRVEC (5)

and

ERRVEC (6)

define a decay function

K R^{- i}

constructed by the routine for the purposes of error estimation. It satisfies

|a_{i}| < K R^{- i},   ​ i = 1, 2, \dots, 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

IFAIL \neq 0

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

- 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:

$IFAIL = 1$

On entry,

MMAX < 8

$IFAIL = 2$: 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.

$IFAIL = 3$: Computation was terminated early because the estimate of rounding error was greater than EPSTOL. Increasing EPSTOL may help.

$IFAIL = 4$: The decay rate of the coefficients is too small. Increasing MMAX may help.

$IFAIL = 5$: 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.

$IFAIL = 6$: The behaviour of the coefficients does not enable reasonable prediction of error bounds. Check the value of SIGMA0. In this case, $ERRVEC (i)$ is set to $- 1.0$ , for $i = 1, 2, \dots, 5$ .

When

IFAIL \geq 3

, changing SIGMA or B may help. If not, the method should be abandoned.

7 Accuracy

The error estimate returned 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}

8 Further Comments

8.1 The Role of $σ_{0}$

Nearly all techniques for inversion of the Laplace transform require you to supply the value of

σ_{0}

, the convergence abscissa, or else an upper bound on

σ_{0}

. For this routine, one of the reasons for having to supply

σ_{0}

is that the parameter

σ

must be greater than

σ_{0}

; otherwise the series for

\tilde{f} (t)

will not converge.

If you do not know the value of

σ_{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 (s)

The value of

σ_{0}

is also relevant in defining a natural accuracy criterion. For large

t

f (t)

is of uniform numerical order

k e^{σ_{0} t}

, so a natural measure of relative accuracy of the approximation

\tilde{f} (t)

is:

ε_{nat} (t) = (\tilde{f} (t) - f (t)) / e^{σ_{0} t} .

C06LBF uses the supplied value of

σ_{0}

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

σ_{0}

is known, choosing a value for

σ

is not easy. Briefly, the series for

\tilde{f} (t)

converges slowly when

σ - σ_{0}

is small, and faster when

σ - σ_{0}

is larger. However the natural accuracy measure satisfies

|ε_{nat} (t)| < ε_{tol} e^{(σ - σ_{0}) t}

and this degrades exponentially with

t

, the exponential constant being

σ - σ_{0}

Hence, if you require meaningful results over a large range of values of

t

, you should choose

σ - σ_{0}

small, in which case the series for

\tilde{f} (t)

converges slowly; while for a smaller range of values of

t

, you can allow

σ - σ_{0}

to be larger and obtain faster convergence.

The default value for

σ

used by C06LBF is

σ_{0} + 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 \geq σ - σ_{0}

. The default value used by the routine is

2.5 (σ - σ_{0})

. A more refined choice is to set

b / 2 \geq \min_{j} |σ - s_{j}|

where

s_{j}

are the singularities of

F (s)

9 Example

This example computes values of the inverse Laplace transform of the function

F (s) = \frac{3}{s^{2} - 9} .

The exact answer is

f (t) = \sinh 3 t .

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

NAG Library Routine DocumentC06LBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 The Role of σ0

8.2 Choice of σ

8.3 Choice of b

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

C06LBF

8.1 The Role of $σ_{0}$

8.2 Choice of $σ$

8.3 Choice of $b$