SumOfExpVPMR.jl

SumOfExpVPMR.jl is a Julia implentation the VPMR method by Zixuan Gao, Jiuyang Liang and Zhenli Xu in A Kernel-Independent Sum-of-Exponentials Method, which can be used to represent rapid decaying kernels via sum of exponentials.

Add this package in julia by typing ] in Julia REPL and then

pkg> add SumOfExpVPMR

to install the package.

The main function is

function VPMR_cal(
    f::Function, # function to be approximated, be like f(x::T) where{T<:Real}, and make sure it can produce highly accurate result for BigFloat
    nc::T, # width of soe
    n::Int, # terms of soe in VP
    N::Int, # order of Gaussian integral
    p::Int; # terms of MR
    T1::DataType = ComplexF64, # output type for s and w
    T2::DataType = Float64, # output type for \sigma
    digit::Int = 1024,
    print_info::Bool=false
    ) where{T}

for details please refer to the article.

Here is an example of using VPMR, where we find a 12 term approximation for Gaussian function:

julia> using SumOfExpVPMR

julia> f = x -> exp(-x^2)
#58 (generic function with 1 method)

julia> sw12, σ12 = VPMR_cal(f, 4.0, 40, 100, 12, print_info = true);
[ Info: VP error: 1.2159077902284768438357759665040000333168536446025393484114289668230574315365e-12
[ Info: MR error: 1.331145872202156879583878373566408054787581766492969993446545399873793531071339e-11
[ Info: Truncated MR error: 1.3310907931440852e-11

More benchmark is shown in the following figure, the code can be found in the expamle folder.

The SOE approach is quite useful since the $\exp(\cdot)$ function is easy to be handled. One of the most important usage is called the fast exponential transform (FET), which can be used to calculate the summation $\sum_{i,j} q_{1}^{i} q_{2}^{j} f \left( α^{-1} |x_i - x_j| \right)$ in $O(N)$ steps. For details, please see the implentation in src/fast_exp_transform.jl. Here we compare the cost of calculating $\sum_{i,j} q_{1}^{i} q_{2}^{j} \exp \left( -(x_i - x_j)^2 \right)$ directly and via FET, as shown in fig below.

If you find any bug or have any suggestion, please open an issue.

1. The VPMR article A Kernel-Independent Sum-of-Exponentials Method.
2. Matlab implementation of VPMR
3. C++ implementation of VPMR