# Problems with integral in fsolve

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Oeer on 9 Apr 2019
Commented: Walter Roberson on 9 Apr 2019
Hey
I have three functions with three unknown variables, but I can't find a solution for them. the features look like this:
I've tried to do the following code:
m = 0.31;
r = 0.05;
delta = 0.02;
y_bar = 2;
w_1 = 5;
w_2_bar = 5;
sigma = 1;
b_bar = 3;
z_0=[0.5,3,4];
sol=fsolve(@(z)([
w_1*(1-z(1))-z(2)-((1+delta)/((1-m)*(1+r)))*int((b_bar+(1-z(1))*(1-m)*x+(1-m)*(1+r)*z(2))*(1/(2*pi*sigma^2))*exp(-((x-w_2_bar)^2)/(2*sigma^2)),x,0,z(3))
+((1+delta)/(1+r))*int(((1-z(1))*k+(1+r)*z(2))*(1/(2*pi*sigma^2))*exp(-((k-w_2_bar)^2)/(2*sigma^2)),k,z(3),inf);
z(1)*w_1+int(z(1)*h*(1/(2*pi*sigma^2))*exp(-((h-w_2_bar)^2)/(2*sigma^2)),h,0,z(3))
-int((b_bar-m*((1-z(1))*y+(1+r)*z(2)))*(1/(2*pi*sigma^2))*exp(-((y-w_2_bar)^2)/(2*sigma^2)),y,0,z(3));
z(3)-(1/(1-z(1))*y_bar+((1+r)/(1-z(1)))*z(2))]),z_0);
Hope there is one who can correct my code so it works, thank you very much if it is you. :)

Walter Roberson on 9 Apr 2019
Code attached.
It is not fast code. You could improve the performance by changing the symbolic integrations into numeric integrations.
Or, since you are using symbolic integration, you could process your function once with symbolic z variables, and simplify. vpasolve() does a good job on what is left, or you can matlabFunction() and fsolve()

#### 1 Comment

Walter Roberson on 9 Apr 2019
Having the int() inside the fsolve() is slow, and the int() have closed form representations if you use symbolic z. The performance gains from executing once symbolically and using the result with vpasolve() or fsolve() is substantial.

Torsten on 9 Apr 2019
Edited: Torsten on 9 Apr 2019
function main
z_0 = [0.5,3,4];
options = optimset('TolFun',1e-8,'TolX',1e-8);
sol = fminsearch(@fun,z_0,options)
end
function res = fun(z)
m = 0.31;
r = 0.05;
delta = 0.02;
y_bar = 2;
w_1 = 5;
w_2_bar = 5;
sigma = 1;
b_bar = 3;
re(1) = w_1*(1-z(1))-z(2)-((1+delta)/((1-m)*(1+r)))*integral(@(x)(b_bar+(1-z(1))*(1-m)*x+(1-m)*(1+r)*z(2))*(1/(2*pi*sigma^2))*exp(-((x-w_2_bar).^2)/(2*sigma^2)),0,z(3),'ArrayValued',1)...
+((1+delta)/(1+r))*integral(@(k)((1-z(1))*k+(1+r)*z(2))*(1/(2*pi*sigma^2)).*exp(-((k-w_2_bar).^2)/(2*sigma^2)),z(3),inf,'ArrayValued',1);
re(2) = z(1)*w_1+integral(@(h)z(1)*h*(1/(2*pi*sigma^2)).*exp(-((h-w_2_bar).^2)/(2*sigma^2)),0,z(3),'ArrayValued',1)...
-integral(@(y)(b_bar-m*((1-z(1))*y+(1+r)*z(2)))*(1/(2*pi*sigma^2)).*exp(-((y-w_2_bar).^2)/(2*sigma^2)),0,z(3),'ArrayValued',1);
re(3) = z(3)-(1/(1-z(1))*y_bar+((1+r)/(1-z(1)))*z(2));
res = norm(re)
end