Crthuky

I typically write my programs in Matlab and then port the picture into LaTeX. At times it appears that one has better efficiency or flexibility to have the program within LaTeX itself, either you have access to more control over graphing or that the work-flow is easier.
I am looking for an intermediate article that explains elements of mathematical programming in LaTeX. Typical math programs are items such as Newton method for root finding, Runge-Kutta solution of differential equations, basic Monte Carlo, etc. If no such article exists links to your sample math-within-LaTeX would be appreciated.

Where Matlab fails is in "annotation". It is good so long as you are plotting some graph. But if you are putting pieces together and adding text, arrows, etc to a plot then it becomes a bit of problem to use Matlab. Matlab has a "annotation" function to help you do this but it is geared to "normalized windows coordinate", which somehow takes your window to be $[0,1]*[0,1]$ and asks you specify the coordinates with respect to this window, as opposed to your data coordinates. The switch over is confusing enough to have created a side industry of third party contributed functions. Most of which do not work because of one issue or another.

You can integrate python code into your LaTeX document using pythontex.

Here is a simple example:

documentclassarticle

usepackage[gobble=auto]pythontex
usepackagepgfplots

begindocument

beginpycode
 from sympy import *
 x = symbols('x')
 f = integrate(cos(x)*sin(x), x)
endpycode


beginpysub
begintikzpicture
 beginaxis[xlabel=$x$,ylabel=$y$,samples=200,no markers,title=!latex(f)]
 addplot[black] gnuplot !f;
 endaxis 
endtikzpicture
endpysub

enddocument

Here is another example:

documentclassarticle

usepackage[gobble=auto]pythontex
usepackagepgfplots
usepackagesiunitx

sisetup
 round-mode=places,
 round-precision=3
 

DeclareDocumentCommandpyNum m O
%
py'\num[#2]' + str(#1).replace('(','').replace(')','') + r''%



begindocument

beginpycode
import numpy as np
from scipy import optimize as op
def f(x):
 return x**2 + 3*x -3
x = np.arange(-5,5,0.1)
np.savetxt('file.dat',zip(x,f(x)),fmt='%0.5f')
endpycode

A root of $f$ is pyNumop.newton(f,-2).


begincenter
begintikzpicture
 beginaxis[xlabel=$x$,ylabel=$y$,samples=200,no markers,axis lines=center]
 addplot[black] table file.dat;
 endaxis 
endtikzpicture
endcenter

enddocument

Here is a further example solving an ODE for a driven oscillator:

documentclassarticle

usepackage[gobble=auto]pythontex
usepackagepgfplots

pgfplotssetcompat=1.15

begindocument

beginpycode
import numpy as np
from scipy.integrate import odeint

omega = 3

omega_ext = 2
c = 0.1
d = 0.5
m = 1
e = 1
k = omega**2*m

def Force(t,x,v):
 return -k*x + np.sin(omega_ext*t) - d*v

def dgl(xv, t):
 x, v = xv 
 return [v, 1/m*Force(t,x,v)]

xv0 = [1, 0] 

tmax = 30
t_out = np.arange(0, tmax, 0.05) 

xv_res = odeint(dgl, xv0, t_out)

x,v = xv_res.T

tv = list(zip(t_out,v))
np.savetxt('osciTV.dat',tv)
endpycode


beginpysub
begintikzpicture
 beginaxis[xlabel=$t$,ylabel=$v$,samples=200,no markers]
 addplot[black] table osciTV.dat;
 addplot[dashed,variable=t,domain=0:!tmax] gnuplot sin(!omega_ext*t); 
 endaxis 
endtikzpicture
endpysub


enddocument

See also the examples from the pythontex-gallery.

Python provides many libraries for scientific computing.

Another option would to use sagetex which let's you include sage-code into your document.

Note that it makes sense to think about choosing an editor which supports switching between two languages in one document. Emacs can do this for example with polymode.

For LuaLaTeX, and using Lua, but other than that:

• "Numerical methods with LuaLaTeX", by Juan Montijano, Mario Pérez, Luis Rández and Juan Luis Varona. TUGboat issue 35.1: https://www.tug.org/TUGboat/tb35-1/tb109montijano.pdf
  

pweave was mentioned in the answer by jonaslb, so it would make sense to also mention sweave (which was the inspiration for pweave) and knitr. These are frameworks for similar concepts, but for the R language.

MetaPost is also integrated in LuaTeX. As a programming language it allows the implementation of some numerical methods. See this tutorial for an implementation of the Newton iterative method (p. 34).

As a graphic language it also allows some geometric computations, like finding the intersection of two curves, building a box plot out of a stats diagram, etc.

Edit: as an example, here is a slightly modified implementation of the Newton method I mentioned above, applied to the function f(x)=x^2-2. It is a geometric version of this method, that is to say that it is based upon the given curve and its tangents, not upon the function itself and its derivative. (It could have been done that way, of course.)

documentclassscrartcl
usepackageluamplib
 mplibtextextlabelenable
 mplibsetformatmetafun
 mplibnumbersystemdouble
begindocument
 beginmplibcode
 vardef f(expr x) = x**2 - 2 enddef;
 u = 3cm; v = 1.5cm; xmax = 2.75; ymax = 6;
 path curve; numeric t[]; dx = 1E-4;
 curve = (0, f(0)) 
 for i = dx step dx until xmax: .. (i, f(i)) endfor;
 beginfig(1);
 draw curve xyscaled (u, v);
 x0 = 2.5; i := 0;
 forever:
 (t[i],whatever) = curve intersectiontimes 
 ((x[i], -infinity) -- (x[i],infinity));
 y[i] = ypart (point t[i] of curve); 
 (x[i+1],0) = z[i] + whatever*direction t[i] of curve; 
 draw ((x[i], 0) -- z[i] -- (x[i+1], 0)) xyscaled (u, v);
 drawdot (z[i] xyscaled (u, v)) withpen pencircle scaled 4bp;
 i := i+1; 
 exitif abs(x[i]-x[i-1]) < dx;
 endfor;
 label.bot(btex $x_0$ etex, (x0*u, 0)); 
 label.bot(btex $x_1$ etex, (x1*u, 0)); 
 label.bot(btex $x_2$ etex, (x2*u, 0));
 label.lrt("$x_" & decimal i & "=" & decimal x[i] & "$", 
 (x[i]*u, 0) shifted (0, -.75cm)); 
 drawarrow origin -- (xmax*u, 0);
 for i = 0 upto xmax:
 draw (i*u, -2bp) -- (i*u, 2bp);
 label.bot("$" & decimal i & "$", (i*u, -2bp));
 endfor;
 label.bot("$x$", (xmax*u, 0));
 endfig;
 endmplibcode
enddocument

Along the lines of percusse's comment, the sagetex package allows you to access to a computer algebra system, called Sage, which has math programming built in as well as Python. Searching for sagetex on this site will get you some quick examples. For example, if you want to plot the Riemann zeta function you can use the fact that Sage knows the Reimann zeta function and then you can print it out with tikz/pgfplots as is done in my answer here. Not having to program the zeta function saves you from having to waste time "reinventing the wheel". If the function isn't defined, such as the Cantor function, then it can be programmed in Python with the output nicely typeset in tikz/pgfplots as I showed in my answer here. Whether your math requires permutations, graph theory, statistics, matrices, randomly generated polynomial problems, blurring an image, or the various topics mentioned in the substantial documentation here, Sage plus Python is built to handle math. You can find documentation on Runge-Kutta here with Monte Carlo and various root finding methods. Your intermediate document is the sagetex link above to get you started and the sagemath links. Some basics of Python are discussed here.

The Sage CAS is not part of a LaTeX distribution. It can be download and installed locally on your machine or, if you have an internet connection, you can access it through the free CoCalc.

What I usually do when I want to typeset something based on results of some programming, is to include jinja-generated latex code. It's particularly great for tables, I think. There are some tips on how to do that here.

Another method, that I have not tried, but which looks extremely promising, is pweave, which lets you write Python code inside your latex document!

And as some of the other answers here write, you can use Lua with LuaLatex. Lua is a little more obscure though, so if you have to learn the language and is coming from Matlab, Python should be easier.