MATLAB: What is the best way to numerically solve a system of polynomial multivariate equations

optimization polynomial multivariate solution

I have a system of polynomial equations that I need to solve a whole bunch of times. What do you think is the most efficient way to solve them using some numerical solver in matlab? I have the optimization toolbox, etc.

Thanks,

Chris

Best Answer

If they really are multivariate polynomials (e.g., no sin() or gamma() or integrals, no sqrt() or x^n when n is not 0 or a positive integer), then the \ operator might be appropriate, especially if it is only the right-hand sides that are varying (in which case they can all be done at the same time.)

If the coefficients are changing each time, I would consider using symbolic algebra to solve() for the variables, and then turn the resulting formula into MATLAB code (matlabFunction() might help.) Any RootOf() symbolic polynomials can be converted in to roots() calls.

Related Solutions

MATLAB: Dsolve very slow for system of 2 second order ODEs

Hi,

One way to speed-up the execution is to add a limit on the maximum degree of radicals. The following shows you an example,

S = dsolve(..,'MaxDegree',2);

This will force dsolve to assume implicit formulas for polynomials of degree greater than the specified value.

Refer to the dsolve documentation for more information.

If the exact solution for the input set of ordinary differential equations is relatively complex, then this might be another reason the DSOLVE function takes long time.

Hope this helps!

Thanks!

DISCLAIMER: These are my own views and in no way depict those of MathWorks.

MATLAB: Multivariate Guassian Distribution

You're really trying to do two things here. The first is, you have some random data and you want to fit it to a multivariate normal distribution. Your approach to this part works, although it can be streamlined:

n = 1000; d=2;
X = randn(n,2);

Get mean and covariance:

mumat=mean(X);
cov_mat=cov(X);

The second part is plotting the resulting distribution. Here you need a regular grid for your variables, not the random values you generated above:

x = -3:.2:3; y = -3:.2:3;
[X,Y] = meshgrid(x,y);
X = X-mumat(1); Y = Y-mumat(2);

Combine X and Y in a way that each row represents one 2D variable.

Z = [X(:) Y(:)];

Now calculate the probabilities.

scale=((2*pi)^(d/2))*sqrt(abs(det(cov_mat))); 
p = zeros(length(Z),1);
for ii=1:length(Z)
  p(ii) = exp(-Z(ii,:)/cov_mat*Z(ii,:)'/2)/scale;
end

Reshape and plot.

p = reshape(p,length(x),length(y));
surf(x,y,p)
xlabel('x'), ylabel('y')

Best Answer

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MATLAB: Dsolve very slow for system of 2 second order ODEs

MATLAB: Multivariate Guassian Distribution

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