CODEWITHSUNDEEP
matlab
user1869583
user1869583

Reputation:

Why is Matlab saving values as symbolic variables with massive fractions instead of decimal approximations?

I'm currently working on a rudimentary optimization algorithm in Matlab, and I'm running into issues with Matlab saving variables at ridiculous precision. Within a few iterations the variables are so massive that it's actually triggering some kind of infinite loop in sym.m.

Here's the line of code that's starting it all:

SLine = (m * (X - P(1))) + P(2);

Where P = [2,2] and m = 1.2595. When I type this line of code into the command line manually, SLine is saved as the symbolic expression (2519*X)/2000 - 519/1000. I'm not sure why it isn't using a decimal approximation, but at least these fractions have the correct value. When this line of code runs in my program, however, it saves SLine as the expression (2836078626493975*X)/2251799813685248 - 584278812808727/1125899906842624, which when divided out isn't even precise to four decimals. These massive fractions are getting carried through my program, growing with each new line of code, and causing it to grind to a halt.

Does anyone have any idea why Matlab is behaving in this way? Is there a way to specify what precision it should use while performing calculations? Thanks for any help you can provide.

Upvotes: 2

Views: 1475

Answers (3)

Cassio
Cassio

Reputation: 3102

So, from MATLAB reference documentation on Symbolic computations, the symbolic representation will always be in exact rational form, as opposed to decimal approximation of a floating-point number [1]. The reason this is done, apparently, is to "to help avoid rounding errors and representation errors" [2].

The exact representation is something that cannot be overcome by just doing symbolic arithmetic. However, you can use Variable-Precision Arithmetic (vpa) in Matlab to get the same precision [3].

For example

>> sym(pi)
ans = 
0

>> vpa(sym(pi))
ans = 
3.1415926535897932384626433832795

References

[1] http://www.mathworks.com/help/symbolic/create-symbolic-numbers-variables-and-expressions.html

[2]https://en.wikibooks.org/wiki/MATLAB_Programming/Advanced_Topics/Toolboxes_and_Extensions/Symbolic_Toolbox

[3]http://www.mathworks.com/help/symbolic/vpa.html

Upvotes: 0

Derek O'Connor
Derek O'Connor

Reputation: 325

Welcome to the Joys of Symbolic Computing!

Most Symbolic Algebra systems represent numbers as rationals, $(p,q) = \frac{p}{q}$, and perform rational arithmetic operations (+,-,*,/) on these numbers, which produce rational results. Generally, these results are exact (also called infinite precision).

It is well-known that the sizes of the rationals generated by rationals operations on rationals grow exponentially. Hence, if you try to solve a realistic problem with any Symbolic Algebra system, you eventually run out of space or time.

Here is the last word on this topic, where Nick Trefethen FRS shows why floating point arithmetic is absolutely vital for solving realistic numeric problems.

http://people.maths.ox.ac.uk/trefethen/publication/PDF/2007_123.pdf

Try this in Matlab:

function xnew = NewtonSym(xstart,niters);
% Symbolic Newton on simple polynomial
% Derek O'Connor 2 Dec 2012. [email protected]
x = sym(xstart,'f');

for iter = 1:niters
    xnew = x - (x^5-2*x^4-3*x^3+3*x^2-2*x-1)/...
               (5*x^4-8*x^3-9*x^2+6*x-2);
    x    = xnew;
end

function xnew = TestNewtonSym(maxits);
% Test the running time of Symbolic Newton
% Derek O'Connor 2 Dec 2012.

time=zeros(maxits,1);
for niters=1:maxits
    xstart=0;
    tic;
    xnew = NewtonSym(xstart,niters);
    time(niters,1)=toc;
end;
semilogy((1:maxits)',time)

Symbolic Newton

Upvotes: 1

user85109
user85109

Reputation:

You've told us what m and P are, but what is X? X is apparently a symbolic variable. So further computations are all done symbolically.

Upvotes: 1

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