Matlab simple loop for different function variables (Finite Difference)

Question

cagri aydin 2017-3-8

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/328722-matlab-simple-loop-for-different-function-variables-finite-difference

评论： Rahul Kalampattel 2017-3-8

So, i wrote a simple matlab script to evaluate forward, backward and central difference approximations of first and second derivatives for a spesific function (y = x^3-5x) at two different x values (x=0.5 and x = 1.5) and for 7 different step sizes (h) and compare the relative errors of the approximations to the analytical derivatives.

However, i need to enter x and h values manually every time. Question is, how do i create a loop for 7 different h values and 2 different x values and get all the results as a matrix?

Script:

clc
clear all
close all
h = 0.00001;  %step size
x1 = 0.5;     %x value
y = @(x) x.^3 - 5*x;   %main function
dy = @(x) 3*x.^2 - 5;  %first derivative
ddy = @(x) 6*x;        %second derivative
d1 = dy(x1);
d2 = ddy(x1);
%Forward Differencing
f1 = (y(x1+h) - y(x1))/h;
f2 = (y(x1+2*h) - 2*y(x1+h) + y(x1))/(h.^2);
%Central Differencing
c1 = (y(x1+h)-y(x1-h))/(2*h);
c2 = (y(x1+h)-2*y(x1)+y(x1-h))/(h.^2);
% Backward Differencing
b1 = (y(x1) - y(x1-h))/h;
b2 = (y(x1)-2*y(x1-h)+y(x1-2*h))/(h.^2);
% Relative Errors
ForwardError1 = (f1 - dy(x1))/dy(x1);
ForwardError2 = (f2 - ddy(x1))/ddy(x1);
CentralError1 = (c1 - dy(x1))/dy(x1);
CentralError2 = (c2 - ddy(x1))/ddy(x1);
BackwardError1 = (b1 - dy(x1))/dy(x1);
BackwardError2 = (b2 - ddy(x1))/ddy(x1);

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Answer 1

Rahul Kalampattel 2017-3-8

3
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/328722-matlab-simple-loop-for-different-function-variables-finite-difference#answer_257822

在 MATLAB Online 中打开

Something like this? Or did you want all the error terms in one 3D matrix?

clc
clearvars
close all
hVec = [0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01];   %step size
x1Vec = [0.5 1.5];                                         %x value
y = @(x) x.^3 - 5*x;   %main function
dy = @(x) 3*x.^2 - 5;  %first derivative
ddy = @(x) 6*x;        %second derivative
for i = 1:2
    for j=1:7
          x1 = x1Vec(i);
          h = hVec(j);
          d1 = dy(x1);
          d2 = ddy(x1);
          %Forward Differencing
          f1 = (y(x1+h) - y(x1))/h;
          f2 = (y(x1+2*h) - 2*y(x1+h) + y(x1))/(h.^2);
          %Central Differencing
          c1 = (y(x1+h)-y(x1-h))/(2*h);
          c2 = (y(x1+h)-2*y(x1)+y(x1-h))/(h.^2);
          % Backward Differencing
          b1 = (y(x1) - y(x1-h))/h;
          b2 = (y(x1)-2*y(x1-h)+y(x1-2*h))/(h.^2);
          % Relative Errors
          ForwardError1(i,j) = (f1 - dy(x1))/dy(x1);
          ForwardError2(i,j) = (f2 - ddy(x1))/ddy(x1);
          CentralError1(i,j) = (c1 - dy(x1))/dy(x1);
          CentralError2(i,j) = (c2 - ddy(x1))/ddy(x1);
          BackwardError1(i,j) = (b1 - dy(x1))/dy(x1);
          BackwardError2(i,j) = (b2 - ddy(x1))/ddy(x1);
      end
  end

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cagri aydin 2017-3-8

Thank you, this is quite enough of an answer for me. Although, if possible, it would be very helpful to learn how to get all the error terms in one 3D matrix.

Rahul Kalampattel 2017-3-8

在 MATLAB Online 中打开

Just change the last bit to

% Relative Errors
RelativeError(i,j,1) = (f1 - dy(x1))/dy(x1);      % Forward, first
RelativeError(i,j,2) = (f2 - ddy(x1))/ddy(x1);    % Forward, second 
RelativeError(i,j,3) = (c1 - dy(x1))/dy(x1);      % Central, first
RelativeError(i,j,4) = (c2 - ddy(x1))/ddy(x1);    % Central, second
RelativeError(i,j,5) = (b1 - dy(x1))/dy(x1);      % Backward, first
RelativeError(i,j,6) = (b2 - ddy(x1))/ddy(x1);    % Backward, second

RelativeError will be a 2x7x6 matrix. Each of the 6 layers corresponds to a different error, and within a layer the rows correspond to x and the columns correspond to h.

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