What are some ways to find the numerical Laplace transform of an InterpolatingFunction? (I know that numerical Laplace transforms are rarely used but my application requires a numerical evaluation). Thanks in advance for any help -- I'm relatively new to Mathematica and would really appreciate it!

  • $\begingroup$ The problem is for each s you need to do an integral. You could use Fourier transforms and weight your interpolation function with Exp[-a t] . This should be fast. The kernel of the integration would be `Exp[-(a +I w )t]' which is what you want. $\endgroup$
    – Hugh
    Commented Jul 6, 2020 at 13:49

1 Answer 1


Taking the definition of LaplaceTransform and using NIntegrate is a possiblity:

(* 1d case *)
nlap[f_, s_?NumericQ] := NIntegrate[f[t] Exp[-s*t], {t, 0, ∞}]

The multi-dimensional case is a bit harder - I think this does the trick though:

(* multidimensional case *)
nlapnd[f_, s_?(VectorQ[#, NumericQ] &)] := 
 With[{vars = Array[t, Length@s]},
  With[{dots = vars.s, g = Apply[f, vars]},
    Evaluate[Sequence @@ ({#, 0, ∞} & /@ vars)]]]

This simple test case below for f[x]:=x^2 with LaplaceTransform 2/s^3 shows the numerical one matches the analytic one, so I think my implementation is correct, at least for 1D:

f[x_] := x^2
(* make some data and the interpolation function *)
data = Table[f[x], {x, 1, 5}];
intp = Interpolation[data];

(* show that they match up *)
 Plot[f[x], {x, 0, 5}]

(* get the laplace transform *)
lp = LaplaceTransform[f[x], x, s]
(* result: 2/s^3 *)

(* verify the error is very small between lp transform of f[x] and the numerical 
   transform of the interpolation function.  *)
Plot[nlap[intp, s] - lp, {s, 0, 3}]

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.