Loop over initial guesses with fsolve

Hi everyone!

I have a system of 7 non-linear equations with both real solutions and complex solutions, depending on my initial guess. In order to find the real solutions, I want to loop over initial guesses.

I have 7 variables, I want my initial guess vector to take the values [1, 1, 1, 1, 0.1, 0.1, 0.1], my second initial guess vector to increment over the first four guesses by 0.1 and over the last 3 by 0.01 so that it takes the values [1.1, 1.1, 1.1, 1.1, 0.11, 0.11, 0.11, 0.11] until the guess [1.9, 1.9, 1.9, 1.9, 0.1, 0.1, 0.1] after ten steps.

Finally, I want to save all the guesses in a matrix 10x7 (or 7x10).

I tried the following code, and I have the error

Index exceeds the number of array elements (1).

Caused by:

Failure in initial objective function evaluation. FSOLVE cannot continue.

Any idea? Thanks in advance!

func=@f;
x0=[1:0.1:1.9,1:0.1:1.9,1:0.1:1.9,1:0.1:1.9,0.01:0.01:0.1,0.01:0.01:0.1,0.01:0.01:0.1]
x = zeros(10,7)
for i=1:10
    x(i)=fsolve(func,x0(i))
end

function my_func=f(x)
rho = 1/2;
c1=1;
c2=11/10;
c3=12/10;
c4=7/10;
y = 20;
pall = 2;
sigma = 10;
eta = 2;
alpha=3/9;
my_func(1)= (sigma*(1 - (x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c1 -x(1);
my_func(2)= (sigma*(1 - (x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c2 -x(2);
my_func(3)= (sigma*(1 - (x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c3 -x(3);
my_func(4)= (sigma*(1 - (x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma)) + eta*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma))/(sigma*(1 - (x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma)) + eta*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma) - 1)*c4 -x(4);
my_func(5)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(1)^(-sigma)*(x(1) - c1)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(5)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(5)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
my_func(6)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(2)^(-sigma)*(x(2) - c2)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(6)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(6)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
my_func(7)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(3)^(-sigma)*(x(3) - c3)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(7)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(7)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
end

[Alex Sha]

The unique real solution should be:

x1: 1.13597072587292
x2: 1.23385304815548
x3: 1.33911918335085
x4: 0.783110223199337
x5: 0.017879890424952
x6: 0.000114204714425553
x7: 1.05898294440764E-6

[Olimpia]

Thank you @Alex Sha, but how did you obtain the real solution? When I tried the loop over the guesses I still got only complex solutions...

Answer 1

[Davide Masiello]

Try with

func=@f;
x0=[1:0.1:1.9;1:0.1:1.9;1:0.1:1.9;1:0.1:1.9;0.01:0.01:0.1;0.01:0.01:0.1;0.01:0.01:0.1];

x = zeros(10,7);

for i=1:10

      x(i,:)=fsolve(func,x0(:,i))

end

Please note the use of ";" instead of "," in the x0 matrix and the correct recursive indexing in the loop (x0(:,i)).

[Torsten]

x(i,:)=fsolve(func,x0(:,i))

instead of

x(i)=fsolve(func,x0(:,i))

[Davide Masiello]

Yes, thanks @Torsten!

[Olimpia]

Thanks a lot!

[Davide Masiello]

No problem. If it helped, please consider to Accept the answer!