Converting Equation of Two Variables Into Matrix Elements - ERROR: Invalid MATLAB indexing or symbolic function definition

Hello everyone,

I am trying to output a square matrix that is N x M in size, where N = M. In my case, N = 3. So this is a "3x3" matrix

I have a variable that's called "Kernel". This variable is calculated as an equation in terms of "z" and "z_prime".

My values in z_prime will vary from L / 2N to (2N-1)*(L/2N) , in steps of "L/N".

My "z" variable will be used as an "integration variable" as seen below.

I am trying to calculate an output called the "Z Matrix" which will integrate the "Kernal" variable across specific ranges.

I attached a .pdf file to clarify my end goal.

See the .pdf file attached:

Z_Matrix_Output.pdf

My Integration Limits for "z" will be going from:

z_start = 0 : L/N : (N-1) * (L/N)

z_end = L/N : L/N : L

As of right now, I'm getting the error:

Invalid MATLAB indexing or symbolic function definition

This is after I tried using "function handles" in my code.

In my attempt, I have tried to use for loops and created function handles to make it easier to generate this matrix.However, I didn't end up with a "N X M" matrix, I just ended up with a "1 X N" matrix.

I know that there is a way to use "nested" for loops to generate a square matrix of an "N X M" size. However, I wasn't quite sure on how to implement that in MATLAB for my case.

I have tried to look through different questions/answers on the Mathworks Forum regarding converting equations of two variables into "matrix elements". However, I wasn't able to find anything that was relevant to my case.

I attached my MATLAB Code for reference.

See MATLAB .m file attached:

Z_Calculation.m

Answer 1

[Shashi Kiran]

Hi Ammar,

The code looks interesting.

I understand that you are encountering errors while using MATLAB's 'integral' function for integration.

According to the documentation, the 'integral' function requires the input argument 'integrand' to be a function handle.

Based on the code and theory you have provided,

The Kernel and function integral can be defined as below.

% Kernel function
Kernel = F_1 * F_2 * (F_3 * F_4 + F_5);

% Convert Kernel to a function handle
Kernel_handle = matlabFunction(Kernel, 'Vars', [z, z_prime]);

% Define the integrand as a function handle
integrand = @(z) Kernel_handle(z, loop_z_prime);
        
% Integrate using the defined function handle
Z(i,j) = integral(integrand, loop_lower_z, loop_upper_z);

Below is complete implementation of your code using the above arguments

%% Starting Timer and Clearing Plots/Command Line
tic
close all; clc;

%% Fixed Variables and Parameters
% Value of j
j = 1i;

% Speed of light = c = 3*10^8 m/s
c = 3*10^8;

% Frequency = 300 MHz = 300*10^6 Hz = 3*10^8 Hz
f = 3*10^8;

% Value of lambda = c / f 
lambda = c/f;

% Wavenumber value = k
k = (2*pi)/lambda;

% Intrinsic Impedance Value = eta
eta = 120*pi;

% Radius of the wire
a = 0.005*lambda;

% Length of the dipole antenna
L = 0.5*lambda;

% Number of segments 
N = 3;

% Number of columns
M = N;

% Declaring "z" and "z_prime" as symbolic variables
syms z z_prime;

%% Calculating the Kernel Factor
% R equation
R = sqrt(a^2 + (z - z_prime)^2);

% Factor_1 = (-j*eta) / k
F_1 = (-j*eta) / k;

% Factor_2 = (exp(-j*k*R)) / (4*pi*(R)^5)
F_2 = exp(-j*k*R) / (4*pi*(R)^5);

% Factor_3 = 1 + jkR
F_3 = 1 + (j*k*R);

% Factor_4 = 2*R^2 - 3*a^2
F_4 = (2*R^2) - (3*a^2);

% Factor_5 = (k*a*R)^2
F_5 = (k*a*R)^2;

% Kernel function
Kernel = F_1 * F_2 * (F_3 * F_4 + F_5);

% Convert Kernel to a function handle
Kernel_handle = matlabFunction(Kernel, 'Vars', [z, z_prime]);

% Upper and lower z integration boundaries
delta_z = L / N;
upper_z = 0 : delta_z : L - delta_z;
lower_z = delta_z : delta_z : L;

% Start and stop values for z_prime
z_prime_start = L / (2*N);
delta_z_prime = L / N;
z_prime_end = L - z_prime_start;
z_prime = z_prime_start : delta_z_prime : z_prime_end;

% Pre-allocating the Z Matrix into an N x N Matrix
Z = zeros(N, N);

for i = 1:N
    for j = 1:M
        loop_z_prime = z_prime(i);
        loop_upper_z = upper_z(j);
        loop_lower_z = lower_z(j);
        
        % Define the integrand as a function handle
        integrand = @(z) Kernel_handle(z, loop_z_prime);
        
        % Integrate using the defined function handle
        Z(i,j) = integral(integrand, loop_lower_z, loop_upper_z);
    end
end

toc

Elapsed time is 0.501875 seconds.

disp(Z)

   1.0e+02 *

   1.3037 - 2.4675i   1.1676 + 7.6634i   0.8191 + 0.6541i
   1.1676 + 7.6634i   1.3037 - 2.4675i   1.1676 + 7.6634i
   0.8191 + 0.6541i   1.1676 + 7.6634i   1.3037 - 2.4675i

Refer to the following references for further help.

1. https://www.mathworks.com/help/matlab/ref/integral.html?s_tid=doc_ta#btbbkta-3
2. https://www.mathworks.com/help/matlab/matlab_prog/creating-a-function-handle.html

Hope this answers your query.

[Ammar]

Hi @Shashi Kiran, Thank you this works for me.