Evaluate a changing function in loop

Question

I am writing a code that generates a function f in a loop. This function f changes in every loop, for example from f = x + 2x to f = 3x^2 + 1 (randomly), and I want to evaluate f at different points in every loop. I have tried using subs, eval, matlabFunction etc but it is still running slowly. How would you tackle a problem like this in the most efficient way?

This is as fast as I have been able to do it. ****matlabFunction and subs go slower than this. The code below is my solution and it is one loop. In my larger code the function f and point x0 change in every loop so you can imagine why I want this to go as fast as possible. I would greatly appreciate it if someone could go through this, and give me any pointers. If my coding is crap feel free to tell me :D

x = sym('x',[2,1]);
f = [x(1)-x(1)cos(x(2)), x(2)-3x(2)^2*cos(x(1))];
J = jacobian(f,x);
x0 = [2,1];
N=length(x0); % Number of equations

%% Transform into string
fstr = map2mat(char(f));
Jstr = map2mat(char(J));

% replace every occurence of 'xi' with 'x(i)'
Jstr = addPar(Jstr,N);
fstr = addPar(fstr,N);

x = x0;

phi0 = eval(fstr)
J = eval(Jstr)

function str = addPar(str,N)

% pstr = addPar(str,N)
% Transforms every occurence of xi in str into x(i)
% N is the maximum value of i
% replace every occurence of xi with x(i)
% note that we do this backwards to avoid x10 being
% replaced with x(1)0

for i=N:-1:1
is = num2str(i);
xis = ['x' is];
xpis = ['x(' is ')'];
str = strrep(str,xis,xpis);
end

function r = map2mat(r)

% MAP2MAT Maple to MATLAB string conversion.
% Lifted from the symbolic toolbox source code
% MAP2MAT(r) converts the Maple string r containing
% matrix, vector, or array to a valid MATLAB string.
%
% Examples: map2mat(matrix([[a,b], [c,d]]) returns
% [a,b;c,d]
% map2mat(array([[a,b], [c,d]]) returns
% [a,b;c,d]
% map2mat(vector([[a,b,c,d]]) returns
% [a,b,c,d]
% Deblank.

r(findstr(r,' ')) = [];

% Special case of the empty matrix or vector
if strcmp(r,'vector([])') | strcmp(r,'matrix([])') | ...
strcmp(r,'array([])')
r = [];
else
% Remove matrix, vector, or array from the string.
r = strrep(r,'matrix([[','['); r = strrep(r,'array([[','[');
r = strrep(r,'vector([','['); r = strrep(r,'],[',';');
r = strrep(r,']])',']'); r = strrep(r,'])',']');
end

Answer 1

There are several ways to get huge boosts in speed for this sort of problem:

• The java GUI front end slows everything down. Go back to version 2010a or earlier. Go back to when it was based on C or fortran. The MATLAB script runs as fast as if you had put it into the MATLAB "compiler".
• If you have MatLab compiler (or builder, I forget which) but not the coder, then you can process your code and have it run a few times faster without modifying the code.
• write it to a file, then call it as a function. I have done this for changing finite-element expressions, so large ugly math that makes $y = 3x^2 +1$ look simple. In that it gave me solid speed increase.
• vectorize, vectorize, vectorize. It used to reliably give 10x to 12x speed increase. Pull it out of loops. The java, I think, obscures this some by making everything slower.
• have you "profiled" your function to make sure that "eval" or such are the problem? If you fix "eval" and your bottleneck is elsewhere then you will have problems.
• If you have the choice between eval and subs, stick with eval. subs gives you a symbolic solution, not a numeric one.
• If there is a clean way to have multiple instances of MatLab running, especially if you have a decently core-rich cpu that MatLab does not fully utilize, then get several of them going. If you are at an educational institution you might try running several different versions (2010a, 2010b, 2009a,...) on the same system. I (fuzzily) recall they didn't collide when I did it. Running more than about 8 started slowing things down more than it improved them. Make sure they aren't colliding on file access if you are using files to share control.
• You could write your program in LabVIEW (not MathScript, not MatLab) and because it is a compiled language, there are times that code can run 1000x faster.
• You could go all numeric and make it a matrix activity. This depends on your code, but if you could randomly populate the columns in the matrix then matrix multiply it to a matrix $ \\left[ 1, x, x^{2}, ...\\right] $, that would likely be several hundreds or thousands of times faster than your current level of equation handling and still in MatLab.

About your coding:

• don't redeclare "x" as a symbol every loop, that is expensive.
• what is this "map2mat" then "addPar" stuff?
• the string handling functions are horrible for runtime. Stick to one language. The symbolic toolbox IS maple, and you don't have to get goofy hand-made parsing to make it work with the rest of MatLab.