Backward and Central Difference

Given that x =10 and delta_x = 0.4,

Is there a better way of writing this code?

x = 10; 
delta_x = 0.4;

backward_difference = ((2*f(x)-5*f(x-dx)+4*f(x-2*dx)-f(x-3*dx))/dx^2);
central_difference = (-f(x+2*dx)+16*f(x+dx)-30*f(x)+16*f(x-dx)-f(x-2*dx))/(12*(dx^2));

[Joseph Cheng]

Have you already defined "f" as an anonymous function or symbolic function? Otherwise if "f" is an array you would be indexing "f" in a non-integer value

[Anna Lin]

Yes, I have already defined f as an anonymous function.

f=@(x) x.^3+sin(x)

Answer 1

[J. Alex Lee]

I guess the answer depends what you want to do with those finite difference approximations. If you want to use it in an algorithm to solve ODEs, your strategy won't work because you don't a priori have a functional form.

This would be a typical matrix math way (assuming your coefficients are correct, i won't check)

cb = [-1,4,-5,2];
cc = [-1,16,-30,16,-1]/12;

fun = @(x) x.^3+sin(x);
funp = @(x) 3*x.^2 + cos(x);
funpp = @(x) 6*x - sin(x);

dx = 0.5;
x0 = 10;

% create stencils on x to define discrete f
xb = x0 - (3:-1:0)'*dx;
xc = x0 + (-2:2)'*dx;

% generate discrete f
fb = fun(xb);
fc = fun(xc);

% execute finite differences
fbpp = cb*fb/dx^2

fbpp = 60.5508

fcpp = cc*fc/dx^2

fcpp = 60.5437

backward_difference = ((2*fun(x0)-5*fun(x0-dx)+4*fun(x0-2*dx)-fun(x0-3*dx))/dx^2)

backward_difference = 60.5508

central_difference = (-fun(x0+2*dx)+16*fun(x0+dx)-30*fun(x0)+16*fun(x0-dx)-fun(x0-2*dx))/(12*(dx^2))

central_difference = 60.5437

fpp = funpp(x0)

fpp = 60.5440

[Anna Lin]

I'm a bit confused about this 2 lines.

% cb = [-1,4,-5,2];
% xb = x0 - (3:-1:0)'*dx;

% If I wrote it like this? 
cb = [2,-5,4,-1];
xb = x0 - (0:3)'*dx;

fbpp = cb*fb/dx^2 % This will not be 60.5508

[J. Alex Lee]

it is not natural to order it that way (from right node to left note). But it should still work:

fun = @(x) x.^3+sin(x);
dx = 0.5;
x0 = 10;
cb = [2,-5,4,-1];
xb = x0 - (0:3)'*dx

xb = 4×1

10.0000 9.5000 9.0000 8.5000

fb = fun(xb);
fbpp = cb*fb/dx^2 % This will not be 60.5508

fbpp = 60.5508

[Anna Lin]

Thank you.