How to write the linear equations in the form of matrix

I have a linear equations: -i*c*sqrt((nx+1)*ny)*u_{nx+1,ny-1}+i*c*sqrt(nx*(ny+1))*u_{nx-1,ny+1}=E*u_{nx,ny}, where nx,ny=0,1,2,3,...,N are integers, c is a constant. As well known, these linear equations can be written in the form of matrix that is A*u=E*u, with u=(u_{0,0},u_{0,1},u_{0,2},...,u_{1,0},u_{1,1},u_{1,2},...u_{nx,ny},...u_{N,N})^T, and A is a matrix with dimension (N+1)*(N+1). I want to know how to construct the matrix A fastly for a very large N in the Matlab. Thanks very much.

Answer 1

[Torsten]

If you want it fast, you will have to define the matrix once by hand. Number the unknowns

U1 = u_{0,0}, U2 = u_{0,1},...,U(N+1)^2 = u_{N,N}

Proceed the same way with the equations EQN1,EQN2,...,EQN(N+1)^2 for U1,U2,...,U(N+1)^2.

Then your matrix will have an entry C in position (i,j) if the coefficient in front of Uj in EQNi equals C.

A slower, but more comfortable way is to write your equations from above as a function:

function res = fun(u)

and define

res(nx,ny) =  E*u(nx,ny) - (-i*c*sqrt((nx+1)*ny)*u(nx+1,ny-1)+i*c*sqrt(nx*(ny+1))*u(nx-1,ny+1))

Then use "jacobianest" from the file exchange (or some faster routine) to calculate the Jacobian matrix A of the

function defined in "fun".