how to fill a matrix without using loop in matlab?

I want to find the coefficient of this fourier series without using loop.I mean filling an,bn and after that is it possible to plot hplot step by step without using loop

close all; clear; clc;
N = 6;
f = @(x) rectangularPulse(-1,1,x);
x = -2:0.001:2;
%2*p is the period
p = pi;
% the main function
plot(x,f(x),'LineWidth',2);
grid;
hold on;
grid minor;
xlim([-2 2]);
ylim([-0.1 1.1]);
x = linspace(-2,2,100).';
y = linspace(0,1,0.2);
a0 = (1/(2*p))*integral(f,-p,p);
an = zeros(1,N);
bn = zeros(1,N);

% calculate an and bn till N
for n=1:N
    fan = @(x) rectangularPulse(-1,1,x).*cos((n*pi/p)*x);
    an(1,n) = (1/p)*integral(fan,-p,p);
    fbn = @(x) rectangularPulse(-1,1,x).*sin((n*pi/p)*x);
    bn(1,n) = (1/p)*integral(fbn,-p,p);
end
% create the gif
for n = 1:N
    An = an(:,(1:n));
    Bn = bn(:,(1:n));
    fs = a0 + sum(An.*cos((1:n).*x) + Bn.*sin((1:n).*x),2);
    hPlot = plot(x,fs,'color','red','LineWidth',2);
    drawnow;
    if(n~=N)
        delete(hPlot);
    end
end

Answer 1

[Torsten]

If you want to plot the partial sums of the Fourier series, you will have to keep the last loop, I guess.

close all; clear; clc;
N = 6;
f = @(x) rectangularPulse(-1,1,x);
x = -2:0.001:2;
%2*p is the period
p = pi;
% the main function
plot(x,f(x),'LineWidth',2);
grid;
hold on;
grid minor;
xlim([-2 2]);
ylim([-0.1 1.1]);
x = linspace(-2,2,100).';
y = linspace(0,1,0.2);
a0 = (1/(2*p))*integral(f,-p,p);
an = 1/p * integral(@(x) f(x).*cos((1:N)*pi/p*x),-p,p,'ArrayValued',1);
bn = 1/p * integral(@(x) f(x).*sin((1:N)*pi/p*x),-p,p,'ArrayValued',1);
fs = a0 + sum(an.*cos((1:N).*x) + bn.*sin((1:N).*x),2);
plot(x,fs,'color','red','LineWidth',2);

[Niloufar]

This isn't what I want.I want to create a gif to show gibbs phenomenon with this function as I sent the code.but the speed of the program is slow when I change N to larger numbers, I wanted to avoid using loops to speed it up

[Torsten]

If you want to plot the fourier series for different values of n, you will have to use a loop.

The loop to generate the coefficients an and bn is superfluous, as you can see above.

[Image Analyst]

How many iterations do you have? Billions? If you use for n = 1 : 6, the "overhead" time to do six iterations is negligible. Your computer will do 6 iterations in nanoseconds. The bottleneck is what's happening inside the loop, not the looping itself.