New answers tagged nonlinear-equations
1
I recommend using a global optimization solver.
You should be able to solve this, or at least try, using the BMIBNB global optimization solver https://yalmip.github.io/solver/bmibnb/ under YALMIP https://yalmip.github.io/tutorial/basics/ under MATLAB.
In particular, note that YALMIP supports the exponential integral, expint, https://yalmip.github.io/command/...
1
I would also like to point at MatlabAutoDiff, which supports sparse Jacobians. Have tried it myself: it is possible to compute large Jacobians (tried with N=1e5) in a small amount of time.
7
Julia has a whole ecosystem for generating sparsity patterns and doing sparse automatic differentiation in a way that mixes with scientific computing and machine learning (or scientific machine learning). Tools like SparseDiffTools.jl, ModelingToolkit.jl, and SparsityDetection.jl will do things like:
Automatically find sparsity patterns from code
Generate ...
2
There are finite difference stencils specifically designed to have rotational symmetry. For example, instead of the standard second order stencil
$$
\frac{1}{h^2} \begin{bmatrix}
& 1 & \\
1 & -4 & 1\\
& 1 &
\end{bmatrix}
$$
you can use
$$
\frac{1}{6 h^2} \begin{bmatrix}
1 & 4 & 1 \\
4 & -20 & 4 \\
1 & 4 & 1
\...
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