MATLAB: Numerical values of integrals

Question

I am dealing with a problem of finding accurate numerical values of integrals. Specifically, the integral is introduced by using the best approximation scheme (Legendre Polynomials) to approximate a vector valued function whose indefinite integral is not easy to be explicitly written down. The code is provided as follows:

 r1 = 0.7;  r2 = 1; r3 = r2 - r1; d = 8;
 syms y real 
 le = [];
 for i = 0:d
    l = [coeffs(legendreP(i,(2/r1)*y+1))   zeros(1,d-i)];
 le = [le; l];
 end
 leg = [];
 for i = 0:d
    l = [coeffs(legendreP(i,(2*y + r1 + r2)/r3))   zeros(1,d-i)];
 leg = [leg; l];
 end
 syms x
 t = [];
 for i = 0 : d
 t = [t ; x^i];     
 end
 xp = t;
 lp = le*xp;  la = leg*xp; 
 fi1 = [exp(sin(x)); exp(cos(x))]; fi2 = [sin(x^2); cos(x^2)];
 ny1 = size(fi1,1); ny2 = size(fi2,1); ny = ny1 + ny2; 
 ga1 = fi1*lp';   ga2 = fi2*la';
 Ga1 = double(int(ga1,x,-r1,0)*diag(2.*(0:d)+1)*(1/r1));
 Ga2 = double(int(ga2,x,-r2,-r1)*diag(2.*(0:d)+1)*(1/r3));
 ep1 = fi1 - Ga1*lp;   ep2 = fi2 - Ga2*la;
 E1  = double(int(ep1*ep1',x,-r1,0)); E2  = double(int(ep2*ep2',x,-r2,-r1));

The code works fine until d = 8 when an error is returned to state that DOUBLE cannot convert the input expression into a double array. If the input expression contains a symbolic variable, use VPA.

I have tried vpa function but the same problem happens still.

One may suggest to use integral numerical integration instead of int. However, the numerical integration seems produce inaccurate result compared to the symbolic representation. Note that the error matrix E1 and E2 become extremely small when the approximation degree d becomes large.

To summarize, the problem here is how to extract, or accurately calculate if anyone has suggestions, the numerical values of E1 and E2.

Thanks a lot!

Answer 1

If you change your first line to

 syms n
 r1 = 0.7;  r2 = 1; r3 = r2 - r1; d = 8; di = n*(d+1);

then you can complete down through E1. After that, unfortunately MATLAB does not know how to do the integral, even though there is a closed-form solution for it. You will need to switch to numeric:

FF = ep2*ep2';
FF1 = matlabFunction(simplify(FF(1));
FF2 = matlabFunciton(simplify(FF(2));
E2(1) = integral(FF1, -r2, -r1);
E2(2) = integral(FF2, -r2, -r1);