fminsearch with an integral function

Hello!

I want to minimize a function with an integral using fminsearch. The question is that I dont know how to define it properly.

The function i want to minimize is:

I want to minimize it respective to θ being A, given by the user before any calculation.

How can i define the function and the fminsearch procedure?

Thanks!

Answer 1

[Ameer Hamza]

See this code

A = 1;
sigma0 = 2;

int_term = @(theta) integral(@(phi) 1/(sigma0*sqrt(2*pi))*exp(-1/2*(phi/sigma0).^2).*sin(theta-phi), 0, 2*pi);
obj_fun = @(theta) -A*cos(theta)+int_term(theta);

theta_sol = fsolve(obj_fun, 0)

[Pedro Araújo]

I could solve some examples with this but when i try to take a step further I get some errors.

    int_term = @(theta,Sigma0) integral(@(phi) 1/(Sigma0*sqrt(2*pi))*exp(-1/2*(phi/Sigma0).^2).**(sin(theta-phi)).^2,0,2*pi);
    Func = @(theta,Sigma0,A) -A*cos(theta) + @(theta,Sigma0);

where then i define my variables

A=120;
Sigma0= 1;

and try to solve it

FuncAux = @(theta) Func(theta,Sigma0,A)
theta_sol = fminsearch(Func, 0)

The following error appears

Error using *

Incorrect dimensions for matrix multiplication. Check that the number of columns in the first matrix matches the number of rows in the second matrix. To perform elementwise multiplication, use '.*'.

I dont understand why it appears in this case.

The final goal is to fit this function to experimental data and extract the value of Sigma0 but first i want to try to plot the minimization with several given A values for constant Sigma 0.

[Ameer Hamza]

There were some mistakes in the pasted code, I corrected it and i got an asnwer

A=120;
Sigma0= 1;

int_term = @(theta,Sigma0) integral(@(phi) 1/(Sigma0*sqrt(2*pi))*exp(-1/2*(phi/Sigma0).^2).*(sin(theta-phi)).^2,0,2*pi);
Func = @(theta,Sigma0,A) -A*cos(theta) + int_term(theta,Sigma0);

FuncAux = @(theta) Func(theta,Sigma0,A);
theta_sol = fminsearch(FuncAux, 0.1);

Can you exactly paste the code which cause error?

[Pedro Araújo]

The code I used is this one: (the variables are all with different names as i tied to simplify to post here)

MultAmtoOe=    10^(4) * 4*pi /(10^7);
MultAmtoemucc = 1/(10^3);
mu0=4*pi/(10^7);
MS_AMR = 1320                / MultAmtoemucc;
t_AMR = 20                   /(10^10);
KuAVG_AMR=(150                  * 10 ^(-4) / mu0) * mu0 * MS_AMR /2;
Sigma_Angle = 55       * pi /180 ;
Sigma_Ku = 0.1* KuAVG_AMR ;

EnergyFunc1 = @(theta,SigmaKu,SigmaTheta0,H,thetaH) -mu0*H*MS_AMR*t_AMR*cos(thetaH-theta) + integral(@(phi) 1/(SigmaTheta0*sqrt(2*pi))*exp(-1/2*(phi/SigmaTheta0).^2).*KuAVG_AMR*t_AMR*(sin(theta-phi)).^2,0,2*pi);

Hteste=120/MultAmtoOe;
thetaH=90 * pi/180;

fundecoy = @(thetaPL)EnergyFunc1(thetaPL,Sigma_Ku,Sigma_Angle,Hteste,thetaH);
[thetaval,energyval]=fminsearch(fundecoy,0);

[Ameer Hamza]

The multipilication should be done with .* operator. Simply using * implies matrix multipilcation. Try the following code

MultAmtoOe=    10^(4) * 4*pi /(10^7);
MultAmtoemucc = 1/(10^3);
mu0=4*pi/(10^7);
MS_AMR = 1320                / MultAmtoemucc;
t_AMR = 20                   /(10^10);
KuAVG_AMR=(150                  * 10 ^(-4) / mu0) * mu0 * MS_AMR /2;
Sigma_Angle = 55       * pi /180 ;
Sigma_Ku = 0.1* KuAVG_AMR ;

EnergyFunc1 = @(theta,SigmaKu,SigmaTheta0,H,thetaH) -mu0*H*MS_AMR*t_AMR*cos(thetaH-theta) + integral(@(phi) 1/(SigmaTheta0*sqrt(2*pi))*exp(-1/2*(phi/SigmaTheta0).^2).*KuAVG_AMR*t_AMR.*(sin(theta-phi)).^2,0,2*pi);
                                                                                                                                                                                     %^ added this dot  

Hteste=120/MultAmtoOe;
thetaH=90 * pi/180;

fundecoy = @(thetaPL)EnergyFunc1(thetaPL,Sigma_Ku,Sigma_Angle,Hteste,thetaH);
[thetaval,energyval]=fminsearch(fundecoy,0);

[Pedro Araújo]

Thanks for the help!

I have just a question why is it assumed that is a matrix operation?

                                                                                                                                                                               %v why not added here
EnergyFunc1 = @(theta,SigmaKu,SigmaTheta0,H,thetaH) -mu0*H*MS_AMR*t_AMR*cos(thetaH-theta) + integral(@(phi) 1/(SigmaTheta0*sqrt(2*pi))*exp(-1/2*(phi/SigmaTheta0).^2).*KuAVG_AMR*t_AMR.*(sin(theta-phi)).^2,0,2*pi);
                                                                                                                                                                                     %^ added this dot 

Why for example you didnt put the dot in the first term? and only in the last one

[Ameer Hamza]

The reason is a bit intricate but I can give you general detail. In MATLAB (*) operator is defined for matrix multiplication and (.*) is defined for element-wise multiplication. For a scalar number it does not make any difference. For example, I evalaute following example function without dots

fun = @(x,a,b) a*b*x*x;
fun(2,1,1)
ans =
     4

and with dot

fun = @(x,a,b) a*b*x.*x;
fun(2,1,1)
ans =
     4

No problem!!! But suppose I want to input x which is a vector. The first one will fail

fun = @(x,a,b) a*b*x*x;
fun([1 2 3],1,1)
Error using  * 
Incorrect dimensions for matrix multiplication. Check that the number of columns
in the first matrix matches the number of rows in the second matrix. To perform
elementwise multiplication, use '.*'.
Error in @(x,a,b)a*b*x*x 

Because 1x3 matrix cannot multiply with 1x3 matrix according to the rule of matrix multiplication. But element-wise multiplication works

fun = @(x,a,b) a*b*x.*x;
fun([1 2 3],1,1)
ans =
     1     4     9

Now consider the function in your question. It gives a scalar function. If you input a scalar value, it will give a scalar output. However, to make the integral() and fminsearch function computationally efficient, MATLAB calls your function with theta and phi as vectors, not scalars. If your function does not support element-wise operations then it will fail for a vector input.

I didn't use .* in the first term because all the term multipliying the cos() function -mu0*H*MS_AMR*t_AMR are themself scalar, so it does not matter wather you use (.*) or (*). You can use either.

To summarize, you should always use element-wise operators when you are trying to integrate a scalar function using integral.

[Pedro Araújo]

Thanks a lot!

I will progress much faster!!!

[Ameer Hamza]

Glad to be of help.