Sparse linear system of equations

I'm given the following: [I_N - (N+1)^2]*B_N = V_N where N ≥ 3, I_N is and N x N identify matrix, B_N is the N x N matrix with (i,j)-th entry

B_N (i, j) =
0 if |i − j| > 1
−1 if |i − j| = 1
2 if |i − j| = 0
1<= i,j <= N

V_N is given and U_N is unknown. How would I right a function that sets up the B_N piece-wise and solved the linear system?

[James Tursa]

Please edit your question to make the problem more clear. E.g., you mention U_N is unknown, but U_N doesn't seem to appear in your problem statement.

[Jim Oste]

I apologize. The given equation should be: [I_N - ((N+1)^2)*B_N]*U_N = V_N

Answer 1

[James Tursa]

Are you just asking how to construct B_N? E.g.,

B_N = spdiags(repmat([-1 2 -1],N,1),[-1 0 1],N,N);

A_N = speye(N) - ((N+1)^2)*B_N;