# How to do forward, backward and central difference

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Kennedy Oparka on 19 Sep 2019
Edited: Bruno Luong on 19 Sep 2019
I am working on an assignment to to create plot showing forward, backward and centeral differenciation using f=sin(pi*x) [-1:1] for different values of n.
This is what i've written for n=10 with plot
yf=zeros(1,10);
yb=zeros(1,10);
y=zeros(1,10);
for j=1:11
% n=10
xb=-1+(j-1)*.2;
xbb=-1+(j-2)*.2;
xf=-1+j*.2;
y(j)=sin(pi*xb); %u(xi)
yb(j)=sin(pi*xbb); %u(xi-dx)
yf(j)=sin(pi*xf); %u(xi+dx)
end
bd=y-yb; %backward dif
fd=yf-y; %forward dif
cd=(yf-yb)/2; %central diff
n=-1:.2:1;
f1=figure
fplot(@(t) pi*cos(pi*t),[-1,1]);
title('n=10')
hold on
plot(n,bd)
plot(n,fd)
plot(n,cd)
legend('derivative','backward','forward','center')
As I increase n to 100 as seen below, the curve becomes flatter (I would expect it to follow the curve more closely). Am I missing something conseptually or does the code not reflect the equations for forward, backward, and central difference.
for j=1:101
%n=100
xb=-1+(j-1)*.02;
xbb=-1+(j-2)*.02;
xf=-1+j*.02;
z(j)=sin(pi*xb); %u(xi)
zb(j)=sin(pi*xbb); %u(xi-dx)
zf(j)=sin(pi*xf); %u(xi+dx)
end
bd_1=z-zb;
fd_1=zf-z;
cd_1=(zf-zb)/2;
N=-1:.02:1; %hundo
f2=figure
fplot(@(t) pi*cos(pi*t),[-1,1]);
title('n=100')
hold on
plot(N,bd_1)
plot(N,fd_1)
plot(N,cd_1)
legend('derivative','backward','forward','center')

Bruno Luong on 19 Sep 2019
Edited: Bruno Luong on 19 Sep 2019
You forget to divide the differences by the time step (dt)
dt = 0.02
for j=1:101
xb=-1+(j-1)*dt;
xbb=-1+(j-2)*dt;
xf=-1+j*dt;
z(j)=sin(pi*xb); %u(xi)
zb(j)=sin(pi*xbb); %u(xi-dx)
zf(j)=sin(pi*xf); %u(xi+dx)
end
bd_1=(z-zb)/dt; % bug was HERE
fd_1=(zf-z)/dt; % bug was HERE
cd_1=((zf-zb)/2)/dt; % bug was HERE
N=-1:dt:1; %hundo
f2=figure
fplot(@(t) pi*cos(pi*t),[-1,1]);
title('n=100')
hold on
plot(N,bd_1)
plot(N,fd_1)
plot(N,cd_1)
legend('derivative','backward','forward','center')

Image Analyst on 19 Sep 2019
Once you have y, or z, why not just compute differences numerically using conv()?
n = 11; % Whatever
kernel = zeros(1, 2*n+1);
kernel(n+1) = 1; % Center element of window
kernel(end) = -1; % Kernel will subtract first element in window from center element. Remember, convolution flips kernel!
yBackwards = conv(y, kernel, 'same');

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