Overcoming floating point issues when Inverse does not exist but determinant provides nonzero result

Question

I have an expression (let's say determinant of matrix A) expressed in symbolic form in terms of 2 decision variables x1, x2 and 2 parameters q1 and q2. I'm minimizing this using fmincon for different start point and parameter value combinations.

For certain combinations of x1, x2, q1, q2 the matrix A is not invertible so I would want determinant evaluation to be zero but Matlab just gives a very small number like 9.7166e-91.

If I had created the matrix A with numeric values for which inverse doesnt exist and then take the inverse it does give me the usual warning "Warning: Matrix is close to singular or badly scaled. Results may be inaccurate." This warning is very good as I know that the output can be discarded.

But I need to work with symbolic expression as this is what needs to be fed to fmincon.

So what approach should I use in order to know that certain outputs from the numeric evaluation of a symbolic expression should be discarded?Like am happy to have "NaN" when inverse doesn't exist. Hope there is a trick which doesnt slow down my program

Note, the Digits setting is 32 so should I use this and infer that if the determinant result is between -1E-32 and 1E-32 then result should be discarded ? (this approach does not seem correct)

Any suggestions are welcome.

%Providing code below in order to replicate easily;

syms x1 x2 q1 q2;% x1 and x2 are the decision variables while q1 and q2 are the parameters. For different values of parameters I'm trying to minimize a function using fmincon


%matrix A is DesignMatrixMult. Example provided below for illustrative purposes but its form can change depending on user input;
 DesignMatrixMult = [                    ((q1*(exp(-q1*x1) - exp(-q2*x1)))/(q1 - q2)^2 - (exp(-q1*x1) - exp(-q2*x1))/(q1 - q2) + (q1*x1*exp(-q1*x1))/(q1 - q2))^2 + ((q1*(exp(-q1*x2) - exp(-q2*x2)))/(q1 - q2)^2 - (exp(-q1*x2) - exp(-q2*x2))/(q1 - q2) + (q1*x2*exp(-q1*x2))/(q1 - q2))^2, - ((q1*(exp(-q1*x1) - exp(-q2*x1)))/(q1 - q2)^2 + (q1*x1*exp(-q2*x1))/(q1 - q2))*((q1*(exp(-q1*x1) - exp(-q2*x1)))/(q1 - q2)^2 - (exp(-q1*x1) - exp(-q2*x1))/(q1 - q2) + (q1*x1*exp(-q1*x1))/(q1 - q2)) - ((q1*(exp(-q1*x2) - exp(-q2*x2)))/(q1
    - q2)^2 + (q1*x2*exp(-q2*x2))/(q1 - q2))*((q1*(exp(-q1*x2) - exp(-q2*x2)))/(q1 - q2)^2 - (exp(-q1*x2) - exp(-q2*x2))/(q1 - q2) + (q1*x2*exp(-q1*x2))/(q1 - q2));
     - ((q1*(exp(-q1*x1) - exp(-q2*x1)))/(q1 - q2)^2 + (q1*x1*exp(-q2*x1))/(q1 - q2))*((q1*(exp(-q1*x1) - exp(-q2*x1)))/(q1 - q2)^2 - (exp(-q1*x1) - exp(-q2*x1))/(q1 - q2) + (q1*x1*exp(-q1*x1))/(q1 - q2)) - ((q1*(exp(-q1*x2) - exp(-q2*x2)))/(q1
    - q2)^2 + (q1*x2*exp(-q2*x2))/(q1 - q2))*((q1*(exp(-q1*x2) - exp(-q2*x2)))/(q1 - q2)^2 - (exp(-q1*x2) - exp(-q2*x2))/(q1 - q2) + (q1*x2*exp(-q1*x2))/(q1 - q2)),                                        ((q1*(exp(-q1*x1) - exp(-q2*x1)))/(q1 - q2)^2 + (q1*x1*exp(-q2*x1))/(q1 - q2))^2 + ((q1*(exp(-q1*x2) - exp(-q2*x2)))/(q1 - q2)^2 + (q1*x2*exp(-q2*x2))/(q1 - q2))^2];


% determinant is calculated below;  
DCritn = simplify(simplify(det(DesignMatrixMult),'IgnoreAnalyticConstraints', true,'steps',500),'full');
%creating an anonymous function handle to DCritn;
DCritnF = matlabFunction(DCritn,'vars',{x1,x2,q1,q2},'file','');



% few lines of code for generating different possible values of the parameter q1 and q2;
% As an example, one set of values is provided below;
q1=0.9287; q2 = 0.83;

For the above parameter values, the function DCritnF is minimized using fmincon;

The challenge I have is that for certain combinations of x1 and x2, DCritnF gives non-zero result even though I know that the inverse of DesignMatrixMult (matrix A) does not exist. For example, DCritnF is called with value of x1 = 42.5 and x2 = 95, it returns 9.7166e-91

On the other hand, if I do the following,

TrialCheck = double(subs(DesignMatrixMult));
TrialCheckInverse = inv(TrialCheck);

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 5.404300e-18. ans =

1.0e+42 *

5.4788 -1.3938

-1.3938 0.3546

Also one smaller question;if I had done

TrialCheck_Alternate = double(subs(DesignMatrixMult))
TrialCheckDeterm_Alternate = det(TrialCheck_Alternate)

Then I get no warning? If for inverse of matrix it issues a warning then shouldn't it also issue warning while finding determinant of that matrix?

TrialCheck_Alternate =

1.0e-25 *

0.0139 0.0546 0.0546 0.2146

TrialCheckDeterm_Alternate =

3.9175e-69

Answer 1

I'm guessing what you want is some way to tell that your square matrix $A$ is either singular or non-singular. For that purpose, I'd use the rank function in MATLAB, which is internally using the SVD with some thresholding to determine the number of nonzero singular values.