what is the ilaplace function algorithm?

what is the algorithm used by the built-in ilaplace function?

Answer 1

[Hassaan]

1. Lookup Tables: Many inverse Laplace transforms can be computed using predefined tables of Laplace transform pairs. If the symbolic expression matches a known form, the corresponding time-domain function can be returned.
2. Partial Fraction Expansion: For rational expressions, the inverse Laplace transform can be found by performing a partial fraction expansion followed by applying the inverse Laplace transform to each term individually, which often corresponds to a known transform pair.
3. Complex Integration: The inverse Laplace transform can be theoretically computed by evaluating a Bromwich integral, which is a contour integral in the complex plane. However, this method is usually not used in symbolic computation software due to its complexity.
4. Algorithms for Symbolic Integration: Many times, the problem of finding an inverse Laplace transform reduces to a symbolic integration problem. MATLAB's symbolic engine includes algorithms to perform integration symbolically, which could be applied to solve inverse Laplace transform problems.
5. Numerical Inversion: When a symbolic solution is difficult to obtain, numerical techniques may be employed to approximate the inverse Laplace transform. However, this is less common in a purely symbolic context.

In practice, the ilaplace function likely uses a combination of these methods, with a heavy reliance on lookup tables and symbolic algebra techniques to simplify expressions and find corresponding time-domain functions. If you need the exact details of the algorithm, you might have to contact MathWorks support directly, although they may not provide specifics if it's proprietary information.

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[Euclides]

thank you Muhammad. I thought the ilaplace function relied solely on numerical inversion algorithms. I might need to compare the accuracy of ilaplace results with numerical inversion algorithms that are available on Matlab and other software products, to see whether the ilaplace function is good enough for my needs

[Torsten]

What is it good for to know the algorithm used ? Make test problems for your application, solve them using different software and compare results and efficiency. If you know which software is suited, ask for the algorithm used.

[Hassaan]

If you need to compare the accuracy of MATLAB's ilaplace function with numerical inversion algorithms, here's how you might do this:

1. Use a Known Function: Start with a function whose inverse Laplace transform is known. This will give you a benchmark to compare against.
2. Compare with ilaplace: Apply MATLAB's ilaplace function to the Laplace-transformed function and compare the result with the known inverse.
3. Apply Numerical Methods: Use numerical inversion algorithms available in MATLAB or other software. For MATLAB, functions like residue, deconv, or custom implementations based on numerical integration can be used.
4. Assess Accuracy: Evaluate the results from the symbolic and numerical methods. Compare these results to the known inverse, considering both the absolute error and the relative error.
5. Consider Computational Cost: Besides accuracy, consider the computational cost. Symbolic computation can be computationally intensive, especially for complex expressions or when high precision is required.
6. Real-world Data: If you are dealing with real-world data, compare how each method handles noise and discretization effects.
7. Software Comparison: If you are also comparing across different software products, ensure that you are using equivalent methods and precision settings for a fair comparison.

In MATLAB, the vpa function (Variable-Precision Arithmetic) can also be used to increase the precision of symbolic results if necessary, and this can be useful when you want to ensure that the symbolic results are as accurate as possible.

Remember that the most appropriate method will depend on the specific requirements of your application, such as the need for precision, computational efficiency, and the ability to handle real-world data.

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[Euclides]

@Torsten: I know pretty well which numerical inversion algorithms work for my application. I just wanted to know if ilaplace uses one of those algorithms or not.

[Torsten]

Since ilaplace is a symbolic function with symbolic F as input, my guess is that it tries to analytically integrate exp(s*t)*F(s) ds using MATLAB's "int". So you won't reach far for complicated applications by using this function.

It won't be far from what Octave's "ilaplace" does

https://octave.sourceforge.io/symbolic/function/@sym/ilaplace.html

[Dyuman Joshi]

That information is not available publicly.

You could ask TMW about it, however there's a good chance that they won't reveal details about it. But it's worth giving it a try.

Though, since it's a symbolic function, I'd doubt there would be any numerical algorithms involved per say.

[Walter Roberson]

@Euclides

ilaplace never does numeric approximation.

[Paul]

@Torsten

I'd be quite surprised if ilaplace used int, except possibly as a last resort when all else (lookup tables of transform pairs and rules) fails, and I'd be surprised even if that. Do you have reason to believe that the Octave implementation actually symbolically computes the integral, which also involves correct selection of C in the limits of integration?

[Torsten]

I couldn't think of another way to solve symbolic integrals apart from "int".

Do you really think it uses lookup tables ? Wouldn't this mean that it is quite limited in possible functions F ?

[Walter Roberson]

evalin(symengine, 'expose(ilaplace)')

ans = 

[Torsten]

Yes, Paul seems to be correct: it looks like applying rules for certain function types, doesn't it ?

[Euclides]

@Torsten: the fact that ilaplace takes a symbolic function as an argument does not mean that it cannot internally discretize it at given s points in the Laplace domain and use a numerical inversion algorithm to approximate the time domain function; or does it?

[Torsten]

"ilaplace" will definitly not use a numerical inversion method. See also @Walter Roberson 's comment above.

[Paul]

@Euclides

Keep in mind that ilaplace and laplace are only diretly applicable to/from causal signals.

[Paul]

@Torsten

I've always found the ilaplace integral to be deceptively not simple. Consider a simple function

syms t real
assumeAlso(t >= 0);
f(t) = exp(2*t)*heaviside(t);
syms s
F(s) = laplace(f(t),t,s)

F(s) = 

C = sym(3);
fr(t) = int(exp(s*t)*F(s),s,C-1i*inf,C+1i*inf)/(2*sym(pi)*1i)

fr(t) = 

We can try to evaluate the integral numerically for different values of C, and assuming that integrating over +-1000i is sufficient

fr = @(t,C) integral(@(s) exp(s.*t)./(s-2), C - 1i*1e3, C + 1i*1e3)/(2*pi*1i);

figure
tvals = 0:5;
for C = [2.1 3 3.5]
   for ii = 1:numel(tvals)
      fval(ii) = fr(tvals(ii),C);
   end
   plot(tvals,double(f(tvals)) - real(fval),'-o')
   hold on
end
legend('C = 2.1','C = 3','C = 3.5','Location','NorthWest')

The integral becomes less accurate as the line of integration defined by C moves further to the right away from the pole at s = 2, and the location of the rightmost pole might not be easy to find for higher order denominators.

[Torsten]

I must admit that I don't have any experience with Laplace or Inverse Laplace transforms.

I just saw its definition by the limit of an integral and naively assumed that for a symbolic solution, "int" would be the way to go.

[Paul]

If you're interested in seeing the potential scope of a table/rules approach, this link has a very extensive table of Laplace trasform rules and pairs, starting at page 613.