Hello,
I'm currently trying to use fSolve to solve a (relatively) simple nonlinear equation to represent the effects of drag on an airplane. The equation is the integral of x/(A*x^3 + B), where A and B are constants. I used wolfram to find the integral, and used it as the objective in fsolve. It worked just fine, but I wanted to try to speed it up, as I have to solve the equation repeatedly under a large number of conditions.
For this reason, I was going to use a jacobian to speed up calculations. I used the jacobian J = x/(A*x^3 + B). I was under the impression that since this equation has only one variable that would be all I needed to do. However, after implementing this jacobian fsolve stopped working entirely.
Setup for fsolve:
options = optimoptions('fsolve','Jacobian','on');
[finalSpeed fval] = fsolve(@(finalSpeed)straightLineDrag(distance,currentSpeed,finalSpeed,A,B),finalSpeedEstimate,options);
My function for F and J:
[F,J] = straightLineDrag(dist,V_0,V_f,A,B)
signA = 1;
signB = 1;
if A < 0
signA = -1;
end
if B < 0
signB = -1;
end
A_1_3 = signA*norm(A^(1/3));
A_2_3 = norm(A^(2/3));
B_1_3 = signB*norm(B^(1/3));
B_2_3 = norm(B^(2/3));
term_1_1 = -2*sqrt(3)*atan((1-(2*A_1_3*V_f)/B_1_3)/sqrt(3));
term_1_2 = -2*sqrt(3)*atan((1-(2*A_1_3*V_0)/B_1_3)/sqrt(3));
term_2_1 = -2*log(B_1_3 + A_1_3*V_f);
term_2_2 = -2*log(B_1_3 + A_1_3*V_0);
term_3_1 = log(B_2_3 - A_1_3*B_1_3*V_f + A_2_3*(V_f^2));
term_3_2 = log(B_2_3 - A_1_3*B_1_3*V_0 + A_2_3*(V_0^2));
term_4 = dist*(6*A_2_3*B_1_3);
F = term_1_1 - term_1_2 + term_2_1 - term_2_2 + term_3_1 - term_3_2 + term_4;
J = V_f/(A*(V_f^3) + B);
end
A and B values: (I don't think it matters, but I'm including it anyways)
A_prime = rho*A_S/(2*m);
A = (C_D_0 + C_D_alpha*abs(alpha))*A_prime;
B = g*sin(alpha);
Where everything is a constant except for alpha, which is determined from the angle of a straight line path (varies +/- 15 degrees)
