Symbolic differentiation for very long equation

Hi All,

I would like to do partial differentition for a relative long function. The function is give as following:

ni = 1;
nt = 1.4; 
R_eff = Eff_Reflection(ni,nt); %Reff represents the fraction of photons that is internally diffusely reflected at the boundary.

syms f(ua,us,g,R,rho);

D = 1 / 3 / (ua + (1-g) * us);

z0 = 1 / (ua + (1 - g) * us);

zb = (1 + R)/(1 - R) * 2 * D;

u_eff = sqrt(3 * ua * (ua + (1 - g) * us));

r1 = sqrt(z0^2 + rho^2);

r2 = sqrt((z0 + 2*zb)^2 + rho^2);

f(ua,us,g,R,rho) = z0 * (u_eff + 1/r1) * exp(-u_eff * r1) / r1^2 + (z0 + 2*zb) * (u_eff + 1/r2) * exp(-u_eff * r2) / r2^2;

Df = diff(f,ua);

The answer for running the code is:

(exp(-3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)*((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2))*(1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))*((3^(1/2)*(2*ua - us*(g - 1)))/(2*(ua*(ua - us*(g - 1)))^(1/2)) + ((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))*(1/(ua - us*(g - 1))^2 - (4*(R + 1))/(3*(ua - us*(g - 1))^2*(R - 1))))/(rho^2 + (1/(ua - us*(g - 1)) - (4*R + 4)/(3*(ua - us*(g - 1))*(R - 1)))^2)^(3/2)))/((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2) - (exp(-3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)*((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2))*(1/(ua - us*(g - 1))^2 - (4*(R + 1))/(3*(ua - us*(g - 1))^2*(R - 1)))*(1/((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2) + 3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)))/((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2) + (exp(-3^(1/2)*(1/(ua - us*(g - 1))^2 + rho^2)^(1/2)*(ua*(ua - us*(g - 1)))^(1/2))*(1/((1/(ua - us*(g - 1))^2 + rho^2)^(3/2)*(ua - us*(g - 1))^3) + (3^(1/2)*(2*ua - us*(g - 1)))/(2*(ua*(ua - us*(g - 1)))^(1/2))))/((1/(ua - us*(g - 1))^2 + rho^2)*(ua - us*(g - 1))) - (exp(-3^(1/2)*(1/(ua - us*(g - 1))^2 + rho^2)^(1/2)*(ua*(ua - us*(g - 1)))^(1/2))*(1/(1/(ua - us*(g - 1))^2 + rho^2)^(1/2) + 3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)))/((1/(ua - us*(g - 1))^2 + rho^2)*(ua - us*(g - 1))^2) + (2*exp(-3^(1/2)*(1/(ua - us*(g - 1))^2 + rho^2)^(1/2)*(ua*(ua - us*(g - 1)))^(1/2))*(1/(1/(ua - us*(g - 1))^2 + rho^2)^(1/2) + 3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)))/((1/(ua - us*(g - 1))^2 + rho^2)^2*(ua - us*(g - 1))^4) - (exp(-3^(1/2)*(1/(ua - us*(g - 1))^2 + rho^2)^(1/2)*(ua*(ua - us*(g - 1)))^(1/2))*(1/(1/(ua - us*(g - 1))^2 + rho^2)^(1/2) + 3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2))*((3^(1/2)*(1/(ua - us*(g - 1))^2 + rho^2)^(1/2)*(2*ua - us*(g - 1)))/(2*(ua*(ua - us*(g - 1)))^(1/2)) - (3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2))/((1/(ua - us*(g - 1))^2 + rho^2)^(1/2)*(ua - us*(g - 1))^3)))/((1/(ua - us*(g - 1))^2 + rho^2)*(ua - us*(g - 1))) - (exp(-3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)*((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2))*(1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))*((3^(1/2)*((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2)*(2*ua - us*(g - 1)))/(2*(ua*(ua - us*(g - 1)))^(1/2)) - (3^(1/2)*(1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))*(1/(ua - us*(g - 1))^2 - (4*(R + 1))/(3*(ua - us*(g - 1))^2*(R - 1)))*(ua*(ua - us*(g - 1)))^(1/2))/(rho^2 + (1/(ua - us*(g - 1)) - (4*R + 4)/(3*(ua - us*(g - 1))*(R - 1)))^2)^(1/2))*(1/((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2) + 3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)))/((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2) + (2*exp(-3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)*((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2))*(1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2*(1/(ua - us*(g - 1))^2 - (4*(R + 1))/(3*(ua - us*(g - 1))^2*(R - 1)))*(1/((1/(ua - us*(g - 1)) - (4*(R + 1))/(3*(ua - us*(g - 1))*(R - 1)))^2 + rho^2)^(1/2) + 3^(1/2)*(ua*(ua - us*(g - 1)))^(1/2)))/(rho^2 + (1/(ua - us*(g - 1)) - (4*R + 4)/(3*(ua - us*(g - 1))*(R - 1)))^2)^2

The answer that matlab give is very very long, it the results trustable?

Answer 1

[Torsten]

Here is a code to test MATLAB's answer:

syms ua us g R rho 
D = 1 / sym('3') / (ua + (1-g) * us); 
z0 = 1 / (ua + (1 - g) * us);
zb = (1 + R)/(1 - R) * sym('2') * D;   
u_eff = sqrt(sym('3') * ua * (ua + (1 - g) * us));
r1 = sqrt(z0^2 + rho^2);
r2 = sqrt((z0 + 2*zb)^2 + rho^2); 
f = z0 * (u_eff + 1/r1) * exp(-u_eff * r1) / r1^2 + (z0 + 2*zb) * (u_eff + 1/r2) * exp(-u_eff * r2) / r2^2;
Df = diff(f,ua)

Df = 

        
fnum = subs(f,[us,g,R,rho],[1 2 3 4]);
Dfnum = subs(Df,[us,g,R,rho],[1 2 3 4]);
        
ua = 2:0.1:10;     
        
fnum = matlabFunction(fnum);
dfnum = (fnum(ua+1e-6)-fnum(ua))*1e6;
        
Dfnum = matlabFunction(Dfnum);
Dfnum = Dfnum(ua);
        
plot(ua,[dfnum;Dfnum])

[Alex Lee]

Thanks a lot!