Mathematica Tricks

This note records some useful tricks I learnt in the past years of using Mathematica (MMA). MMA provides a very powerful language system that realizes almost any functionality one would image. However, such a huge library means large learning cost.

Some exmaples are available in my MMA Repository and you can check there for more fun stuffs!

Cracking word puzzles

Recently, when I was playing some word puzzles, I suddenly came to realize that I can take advantages of using programs to look up the dictionary with some given patterns. MMA does support this feature with a built-in function DictionaryLookup[].

First, you can check if it supports a certain language by typing DictionaryLookup[All].

Now, to search for word with specific pattern, e.g., English words starting with ‘a’ and ending with ‘k’ and have only 5 letters, you can just type

Select[DictionaryLookup[{"English", "a" ~~ ___ ~~ "k"}], 
 StringLength[#] == 5 &]

Next, if you want to get a word using given letters and restrict its length to be 4, you could use

StringJoin /@ 
 Select[Tuples[{"e", "a", "r", "t", "h"}, 4], 
  DeleteDuplicates[#] == # && 
    Length[DictionaryLookup[StringJoin[#]]] != 0 &]

Besides these, this function can be used for a lot fun thins! You can count the number of the words starting with ‘a’ to ‘z’

Length[DictionaryLookup[# ~~ ___]] & /@ CharacterRange["a", "z"]

or find all the palindromic words

DictionaryLookup[x__ /; x === StringReverse[x]]

Reference

• Wolfram Documentation Center - “DictionaryLookup”

Numerical derivative of a list

In MMA, you may easily take partial derivative using D[f(x), x] or even the total derivative Dt[f(x), x]. However, most models I encountered in researches are not analytic so that the functions are usually stored as an array $f(x_i)$, where $x_i$ is an array of discrete points. I’ve searched on stackoverflow, stackExchange and Wolfram Community, but I never found a satisfactory solution. Thus, I have to write another subroutine myself to realize this functionality for the past years. Recently, I came across to the function DerivativeFilter and finally, it works! To take $n$/$m$-th order derivative with respect to $x$/$y$, you just type

DerivativeFilter[array, {n,m}],

where the array should be 2D here. The function applies to any array or even image!

Reference

• Wolfram Documentation Center - “DerivativeFilter”

fftshift in MMA

As I have to deal with wave functions, it is common for me to work between real/momentum spaces, which are connected through Fourier transform. I was so spoiled by fftshift in MatLab so I didn’t know what to do when I cannot find a similar function in MMA. In fact, this can be easily realized through

(*wfx is an array representing the wave function in real space*)
wfk = Fourier[wfx, FourierParameters->1];
wfk = RotateRight[wfx, Floor[(Length@wfk-1)/2]]

Now, the zero momentum component has been shifted to the center! Why this works? Check out DFT and FFT on wiki (or the last two reference if you read Chinese).

Reference

• Equivalent to Matlab fftshift in Mathematica
• MATALAB中的fft、fftshift相关原理说明
• FFT算法讲解——麻麻我终于会FFT了！