juliamath/inverselaplace.jl

Inverse Laplace transform

InverseLaplace

Numerical inverse Laplace transform

This package implements some numerical methods for computing inverse Laplace transforms in Julia.

See the documentation https://JuliaMath.github.io/InverseLaplace.jl/latest .

inverselaplace.jl's Issues

talbotarr yields NaN with single element array argument [0.0]

Apologies if this is expected behavior (if so I will try and put in a PR to the docs), but it was unexpected to me:

julia> using InverseLaplace

julia> Jbar(s) = (1/s^2)*(1 + s^(0.5))/(1 + 1/s+ s^(-0.5));

julia> InverseLaplace.talbotarr(s -> Jbar(s), [0.0, 1.0])
2-element Array{Float64,1}:
 0.02312633959753949
 0.7837570956002141

julia> InverseLaplace.talbotarr(s -> Jbar(s), [0.0])
1-element Array{Float64,1}:
 NaN

However,

julia> InverseLaplace.talbotarr(s -> Jbar(s), [1.0])
1-element Array{Float64,1}:
 0.7837570956002141

So the 0.0 argument works when the array size > 1, but fails when 0.0 is the only element.

And single-element arrays work fine when the single array element > 0.0.

I've reproduced this behavior on the most up to date versions compatible with Julia 0.6 and Julia 1.0.1. Using Windows 7, 64 bit.

v0.3.1

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SpecialFunctions dependency version limit out of date

SpecialFunctions.jl is currently at v2.1.4, compat bounds at v0.10.3.

Excessive memory allocation when evaluating w(t)::Complex compared to w(t)::Float64

The following code solves a linear system of equations in the Laplace domain. Then it uses Weeks' method to determine the inverse Laplace transform of the first element of the solution.

using InverseLaplace
using Base.LinAlg.LAPACK: gesv!

######  Function definitions

function randmatvec(N::Int64)
    # Initialize the matrix A  and the vector b for the equation (sI - A)x = b
    A = complex.(rand(N^2,N^2),rand(N^2,N^2))
    b = zeros(Complex{Float64},N^2)
    b[N*(N-1)] = complex.(1.0,0.0)
    return (A, b)
end

function matrixequation!(A::AbstractArray{T},s::T) where T
    # Returns the l.h.s. of the equation (sI - A)x = b
    A .= s.*eye(A) .- A
    return A
end

function solvematrixequation!(A::AbstractArray{T},b::AbstractVector{T},s::T) where T
    # Return the first element of the solution of (sI - A)x = b
    gesv!(matrixequation!(A,s),b)
    return b[1]
end

function laplaceelement(s::Complex{Float64}, N::Int64)
    # Wrapper for first element of x(s) = (sI -A)⁻¹b
    (A,b) = randmatvec(N)
    return solvematrixequation!(A,b,s)
end
######


N = 8
laplaceelement(s::Complex) = laplaceelement(s, N)

trange = collect(linspace(0.0,3.0,1E5))

begin
    println("Float64")
    @time w1 = Weeks.(laplaceelement)
    @time w1.(trange)
    println("")

    println("Complex{Float64}")
    @time w2 = Weeks.(laplaceelement,datatype=Complex)
    @time w2.(trange)
    println("")
end

After running the begin block twice, I get the following output:

Float64
  0.044574 seconds (2.05 k allocations: 24.277 MiB, 16.83% gc time)
  0.267619 seconds (6 allocations: 781.484 KiB)

Complex{Float64}
  0.042009 seconds (2.32 k allocations: 24.293 MiB, 10.89% gc time)
  0.869980 seconds (37.80 M allocations: 1.030 GiB, 18.26% gc time)

The evaluation of the complex w(t) takes three times as long as the real one.
I would have expected it to take only twice as long.
What causes the increase in memory allocation?

Also do the Laplace Transform

Currently your package provides the inverse laplace transform. Would it be possible to add an Laplace transform too? Or is that functionality best contained in a another package?

`talbot` yields NaN where it shouldn't

This is related, but different to, issue #5. Apologies if I have misunderstood something from your answer in that issue.

I have a function defined in the time domain as J(t) = 1 - exp(-t) which yields 0.0 at t=0.0. The Laplace transform of this function is Jbar(s) = 1/s - 1/(s + 1).

For various reasons, the inputs and outputs should be of type Float64. However, recalling what you said about using BigFloat, I defined a few different versions of this:

J_i(s) = 1/s - 1/(s + 1);

J_f64(s) = 1.0/s - 1.0/(s + 1.0);

For all numerically computed values I get NaN:

julia> talbot(s -> J_i(s), big(0.0))
NaN

julia> talbot(s -> J_f64(s), 0)
NaN

julia> talbot(s -> J_f64(s), 0.0)
NaN

julia> talbot(s -> J_f64(s), big(0))
NaN

Is this just a corner case that causes numerical instabilities? It is strange because it happens for all the creep moduli (https://en.wikipedia.org/wiki/Creep_%28deformation%29) in my program that are defined in terms of their Laplace transform, and I am sure that they all have well defined values at t=0.0 in the time domain.

Note that it gets closer to the correct value for extremely small values of t, and this is the approximation I am currently using to get round the issue:

julia> talbot(s -> J_f64(s), 1e-5)
9.999950291077498e-6

julia> talbot(s -> J_f64(s), 1e-15)
2.4006857918196764e-12

GaverStehfest Algorithm

Hi,

Do you think the Gaver-Stehfest algorithm would be a good feature for this package? (https://arxiv.org/pdf/1305.0715.pdf)
I could submit a PR if you are interested

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Query: are there any plans on extending this to be symbolic?

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Check status of Optim.jl

Use of Optim.jl was disabled pre julia 1.0. Check if the brokenness has been resolved. Consider reenabling.

Post's inversion formula

I'm interested in implementing Post's inversion formula which is used in neurocomputational models of memory and learning. I imagine that it would be dependent on ForwardDiff. Is this something you would like to add to this package? Do you have any suggestions?

Scalar method requires element-wise evaluation of array-valued functions

I have a function definition that solves a system of equations by calling gesv!.
If the system is large, gesv! is called multiple times for each solution element.
This is because the current FFT call in '_wcoeff' only handles scalar valued functions.

I can make a seperate method that deals with array valued functions by storing the Laguerre-coefficients in higher dimensional arrays and applying a FFT along the first dimension of those arrays.

For simplicity, I'd start with vector valued functions.

Registering current changes as v0.3.3

Is there anything preventing this repo from releasing/registering changes since v0.3.2? It looks like the current commit (7811acf) is passing all the checks and there hasn't been a commit for about a year. I found myself needing to use the Weeks method parameter optimizer, which is not functional in v0.3.2 due to a removed Optim dependency, but seems to work fine on the current commit (dependency reinstated with 6205482). Installing directly via GitHub is certainly a solution, but I'm just curious if there are lingering issues that would prevent releasing current changes as a v0.3.3

Many thanks!

add Module docstring

Fix documentation deployment

There are lots of check marks and green things in Actions when building docs and deploying. But this is not reflected in the gh pages. Instead, the last pages built before transferring from jlapeyre to JuliaMath appear.

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Update ReadMe

The ReadMe seems very out of date.

Use @testset

Also export `gaverstehfest` as alias to `postwilder`.

Hi,

Would it be possible to also export a gaverstehfest method that would be an alias for the current postwilder ? I was looking for this algorithm and was about to start a fork to implement it because I did not saw it.

In a large part of the literature, the name of this algorithm is 'Gaver-Stehfest', in the names of Gaver, who found the approximating series, and Stehfest, who found the accelerated convergence tricks. Post-Wilder are the ones that found the continuous formula that Gaver's work derives from.

Also, the principal cited source in postwilder.jl is written by Stehfest, not Post-Wilder.

I do not ask for a change of name -- would probably be breaking --, an alias would be enough.