Finding all solutions to a non-linear equation system with MuPAD

Question

My question is if there is a good way to use MuPAD functions in a Matlab script. The background is that I have a problem where I need to find all solutions to a set of non-linear equations. The previous solution was to use solve in Matlab, which works for some of my simulations (i.e., some of the sets of input T) but not always. So instead I'm using MuPAD in the following way:

function ut1 = testMupadSolver(T)
% # Input T should be a vector of 15 elements

mupadCommand = ['numeric::polysysroots({' eq1(T) ' = 0,' ...
    eq2(T) '= 0},[u, v])'];
allSolutions = evalin(symengine, mupadCommand);
ut1 = allSolutions;

end

function strEq = eq1(T)
sT = @(x) ['(' num2str(T(x)) ')'];

strEq = [ '-' sT(13) '*u^4 + (4*' sT(15) '-2*' sT(10) '-' sT(11) '*v)*u^3 + (3*' ...
    sT(13) '-3*' sT(6) '+v*(3*' sT(14) '-2*' sT(7) ')-' sT(8) '*v^2)*u^2 + (2*' ...
    sT(10) '-4*' sT(1) '+v*(2*' sT(11) '-3*' sT(2) ')+v^2*(2*' sT(12) ' - 2*' ...
    sT(3) ')-' sT(4) '*v^3)*u + v*(' sT(7) '+' sT(8) '*v+' sT(9) '*v^2)+' sT(6)];

end

function strEq = eq2(T)
sT = @(x) ['(' num2str(T(x)) ')'];
strEq = ['(' sT(14) '-' sT(13) '*v)*u^3 + u^2*' '(' sT(11) '+(2*' sT(12) '-2*' sT(10) ...
    ')*v-' sT(11) '*v^2) + u*(' sT(7) '+v*(2*' sT(8) '-3*' sT(6) ')+v^2*(3*' sT(9) ...
    '-2*' sT(7) ') - ' sT(8) '*v^3) + v*(2*' sT(3) '-4*' sT(1) '+v*(3*' sT(4) ...
    '-3*' sT(2) ')+v^2*(4*' sT(5) ' - 2*' sT(3) ')-' sT(4) '*v^3)+' sT(2)];

end

I have two queries:

1) In order to use MuPAD I need to rewrite my two equations for the equation-system as strings, as you can see above. Is there a better way to do this, preferably without the string step?

2) And regarding the format output; when

T = [0 0 0 0 0 0 0 0 0 0 1 0 1 0 1];

the output is:

testMupadSolver(T)

ans =

matrix([[u], [v]]) in {matrix([[4.4780323328249527319374854327354], [0.21316518769990291263811232040432]]), matrix([[- 0.31088044854742790561428736573347 - 0.67937835289645431373983117422178*i], [1.1103383836576028262792542770062 + 0.39498445715599777249947213893789*i]]), matrix([[- 0.31088044854742790561428736573347 + 0.67937835289645431373983117422178*i], [1.1103383836576028262792542770062 - 0.39498445715599777249947213893789*i]]), matrix([[0.47897094942962218512261248590261], [-1.26776233072168360314707025141]]), matrix([[-0.83524238515971910583152318717102], [-0.66607962429342496204955062300669]])} union solvelib::VectorImageSet(matrix([[0], [z]]), z, C_)

Can MuPAD give the solutions as a set of vectors or similarly? In order to use the answer above I need to sort out the solutions from that string-set of solutions. Is there a clever way to do this? My solution so far is to find the signs I know will be present in the solution, such as '([[' and pick the numbers following, which is really ugly, and if the solution for some reason looks a little bit different than the cases I've covered it doesn't work.

EDIT

When I'm using the solution suggested in the answer below by @horchler, I get the same solution as with my previous implementation. But for some cases (not all) it takes much longer time. Eg. for the T below the solution suggested below takes more than a minute whilst using evalin (my previous implementation) takes one second.

T = [2.4336 1.4309 0.5471 0.0934 9.5838 -0.1013 -0.2573 2.4830 ...
     36.5464 0.4898 -0.5383 61.5723 1.7637 36.0816 11.8262]

The new function:

function ut1 = testMupadSolver(T)
% # Input T should be a vector of 15 elements
allSolutions = feval(symengine,'numeric::polysysroots', ...
    [eq1(T),eq2(T)],'[u,v]');
end
function eq = eq1(T)
syms u v
eq = -T(13)*u^4 + (4*T(15) - 2*T(10) - T(11)*v)*u^3 + (3*T(13) - 3*T(6) ...
    + v*(3*T(14) -2*T(7)) - T(8)*v^2)*u^2 + (2*T(10) - 4*T(1) + v*(2*T(11) ...
    - 3*T(2)) + v^2*(2*T(12) - 2*T(3)) - T(4)*v^3)*u + v*(T(7) + T(8)*v ...
    + T(9)*v^2) + T(6);
end


function eq = eq2(T)
syms u v
eq = (T(14) - T(13)*v)*u^3 + u^2*(T(11) + (2*T(12) - 2*T(10))*v ...
    - T(11)*v^2) + u*(T(7) + v*(2*T(8) - 3*T(6) ) + v^2*(3*T(9) - 2*T(7)) ...
    - T(8)*v^3) + v*(2*T(3) - 4*T(1) + v*(3*T(4) - 3*T(2)) + v^2*(4*T(5) ...
    - 2*T(3)) - T(4)*v^3) + T(2);
end

Is there a good reason to why it takes so much longer time?

Answer 1

Firstly, Matlab communicates with MuPAD via string commands so ultimately there is no way of getting around the use of strings. And because it's the native format, if you're passing large amounts of data into MuPAD, the best approach will be to convert everything to strings fast and efficiently (sprintf is usually best). However, in your case, I think that you can use feval instead of evalin which allows you to pass in regular Matlab datatypes (under the hood sym/feval does the string conversion and calls evalin). This method is discussed in this MathWorks article. The following code could be used:

T = [0 0 0 0 0 0 0 0 0 0 1 0 1 0 1];
syms u v;
eq1 = -T(13)*u^4 + (4*T(15) - 2*T(10) - T(11)*v)*u^3 + (3*T(13) - 3*T(6) ...
    + v*(3*T(14) -2*T(7)) - T(8)*v^2)*u^2 + (2*T(10) - 4*T(1) + v*(2*T(11) ...
    - 3*T(2)) + v^2*(2*T(12) - 2*T(3)) - T(4)*v^3)*u + v*(T(7) + T(8)*v ...
    + T(9)*v^2) + T(6);
eq2 = (T(14) - T(13)*v)*u^3 + u^2*(T(11) + (2*T(12) - 2*T(10))*v ...
    - T(11)*v^2) + u*(T(7) + v*(2*T(8) - 3*T(6) ) + v^2*(3*T(9) - 2*T(7)) ...
    - T(8)*v^3) + v*(2*T(3) - 4*T(1) + v*(3*T(4) - 3*T(2)) + v^2*(4*T(5) ...
    - 2*T(3)) - T(4)*v^3) + T(2);
allSolutions = feval(symengine, 'numeric::polysysroots',[eq1,eq2],'[u,v]');

The last argument still needed to be a string (or omitted) and adding ==0 to the equations also doesn't work, but the zero is implicit anyways.

For the second question, the result returned by numeric::polysysroots is very inconvenient and not easy to work with. It's a set (DOM_SET) of matrices. I tried using coerce to convert the result to something else to no avail. I think you best bet it to convert the output to a string (using char) and parse the result. I do this for simpler output formats. I'm not sure if it will be helpful, but feel free to look at my sym2float which just handles symbolic matrices (the 'matrix([[ ... ]])' part go your output) using a few optimizations.

A last thing. Is there a reason your helper function includes superfluous parentheses? This seems sufficient

sT = @(x)num2str(T(x),17);

or

sT = @(x)sprintf('%.17g',T(x));

Note that num2str only converts to four decimal places by default. int2str (or %d should be used if T(x) is always an integer).