Finite Difference Matrix Help

So I have a finite difference problem with beam bending. I am trying define a matrix that follows the 4th order ODE for a Central Difference formula.

n= 10 %number of nodes in the beam

%% Step 2: Define the A Matrix
A = (((2*eye(N) +... 
         diag(ones(N-1,1),1)))+...
         diag(ones(N-1,1),-1));

Now this generates a matrix of size N with '2' along the main diagonal. However, a 4th order ODE is different. So I guess my question is, is there a way to add in values '2' off the main diagonal? I've been trying to mess around with the code above, but it keeps saying the "matrix size dimensions must agree.

[Sindar]

check out spdiags. The first example creates the second order second derivative

[Srivardhan Gadila]

@Justin Yeung are you looking for a matrix which has zeros as diagonal elements, 2's as non-diagonal elements and add this matrix to some other matrix? If not, can you please be more specific or can you give an example?

Answer 1

[Fabio Freschi]

Edit: I changed my answer including a reference and the second order derivative

The coefficients for central differences of different order of accuracy with uniform spacing can be found on wikipedia here.

I assume you need a second order derivative. If it is the case, you can build the matrix using spdiags:

N = 10;

% coefficients (Derivative 2, Accuracy 4 of the wikipedia table)
C = [ones(N,1)/12 4*ones(N,1)/3 -5*ones(N,1)/2 4*ones(N,1)/3 ones(N,1)/12];

% positions along the diagonal
idiag = -2:2;

% matrix
A = spdiags(C,idiag,N,N);

Remember to divide the matrix by the step size dx^2.

The matrix created in this way is sparse (as it is usually done with these problems).