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Numerical methods challenge: Summary

Nicolás Guarín-Zapata

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During October (2017) I wrote a program per day for some well-known numerical methods in both Python and Julia. It was intended to be an exercise, then don't expect the code to be good enough for real use. Also, I should mention that I had almost no experience with Julia, so it probably is not idiomatic Julia but more Python-like Julia.

Summary

This post is a summary of that challenge. For the source code you can check the repository.

The verdict

Since the challenge is with myself, and the main purpose was to learn some Julia the verdict is: success. Nevertheless, I failed during day 26 with the Boundary Element Method.

The list of methods

Day

Numerical method

01

Bisection

02

Regula falsi

03

Newton

04

Newton multivariate

05

Broyden

06

Gradient descent

07

Nelder-Mead

08

Newton for optimization

09

Lagrange interpolation

10

Lagrange interpolation with Lobatto sampling

11

Lagrange interpolation using Vandermonde matrix

12

Hermite interpolation

13

Spline interpolation

14

Trapezoid quadrature

15

Simpson quadrature

16

Clenshaw-Curtis quadrature

17

Euler integration

18

Runge-Kutta integration

19

Verlet integration

20

Shooting method

21

Finite differences with Jacobi method

22

Finite differences for eigenvalues

23

Ritz method

24

Finite element method in 1D

25

Finite element method in 2D

26

Boundary element method

27

Monte-Carlo integration

28

LU factorization

29

Cholesky factorization

30

Conjugate gradient

31

Finite element method with solver

Conclusions

  • This was an exercise of code-kata to learn some of the details of Julia for scientific computing. As such, it was really useful for me to get my hands dirty with Julia.

  • Implementing the Boundary Element Method in one day seems to be rough. I knew this beforehand, but I gave it a try anyways … without succcess.

  • Julia syntax is somewhat in a sweetspot between Python and MATLAB. This makes it really easy to use, although the documentation of some packages is at an arcane stage right now.

  • I won't take a challenge like this in a while. It takes a lot of atttention to get it done.

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