Finding error and partial sum in a Fourier Series

I am looking to claculate the eror of this fourier series:

The intended goal of this project is to perform the partial sum from 1 to some value of n which yields a root mean square error(RMSE) smaller thatn 10^-6. I have been working alongisde some people and this is what we've developed as our code.

I am not sure how to include the partial sum for each of these values and I am attempting to find a way to extrapolate the value of n where error is smaller than 10^-6.

Initially I had a for loop for with different conditions of n and x but it would not produce a reasonable error.

Any advice on how to simplify this process or extrapolate error would be greatly appreciated!

close all; clc;
L=10;
nx=7000 %% number of data points

nx = 7000

x = (linspace(0,L,nx)).' ; 
n_max=1000 %% maximum boundary condition of summation

n_max = 1000

n=1:n_max

n = 1×1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

f =(((-1).^n)-1).*(cos(n.*x))./((n.^2)*(3.14159))+((1-(2.*(-1).^n)).*(sin(n.*x))./n) ;
f_final=-pi/4+sum(f,2)

f_final = 7000×1

-1.5705 -0.2936 0.2702 0.1342 -0.1200 -0.0950 0.0736 0.0978 -0.0177 -0.0626

plot(x, f_final)

error=sqrt((f(n_max)-f(n_max-1)).^2)./n_max

error = 1.5159e-06

Answer 1

[Ishu]

Hi Ryan,

I understand that you are trying to calculate the “Root Mean Square Error” smaller than 10^-6 from 1 to some value of n. So, in this case you can use "while" loop, whenever the error is less than 10^-6 then you can break the loop otherwise you can increment the value of n by 1.

To calculate the partial sum you can use "cumsum" function.

I have used the same values of ‘nx’ and ‘L’ as provided by you and then calculated the partial sums using these values for every value of ‘n’ starting from 1 until the error comes less than 10^-6.

L=10;
nx=7000;
x = (linspace(0,L,nx)).'; % column vector with 7000 values
n=1;
error_max = 10e-6; % error limit

while true
    
    f =(((-1).^n)-1).*(cos(n.*x))./((n.^2)*(3.14159))+((1-(2.*(-1).^n)).*(sin(n.*x))./n);
    f_final = -pi/4+cumsum(f,2);  %partial sums

    % Calculate the error
    mean_value = mean(f_final);
    error = sqrt(sum(((f_final - mean_value).^2))./length(f_final));  % Root mean square error function

    % Check if the error is smaller than the error limit
    if error < error_max
        break;  % Exit the loop if the error is below the limit
    end

    n = n + 1;  % Increment n for the next iteration
end

error

For further information on cumsum you can refer this documentation:

https://www.mathworks.com/help/matlab/ref/cumsum.html

Hope this helps!