# How would you compute the eigenvalues of a finite difference operator?

Having read: What are eigenvalues of higher order finite differences matrices? I am still unclear how you would do this for an arbitrary matrix: $$A=\begin{pmatrix} a&b&e&0&0&0&0\\ d&a&b&e&0&0&0\\ c&d&a&b&e&0&0\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ 0&0&c&d&a&b&e\\ 0&0&0&c&d&a&b\\ 0&0&0&0&c&d&a \end{pmatrix}\in\mathbb{R}^{n\times n}$$ Especially if there are boundary conditions imposed on your system. I have seen an example pop up on wikipedia over the summer which also uses Chebyshev polynomials https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative but there's no reference or explanation where the Chebyshev polynomials come from.

For example, lets take the original problem to be $$u_{t}=Du_{xx}-c u_{x}$$ where $D,c$ are constants, and Boundary conditions: $$u(t,0)=0,\quad u_{x}(t,L)=0$$ and some well defined initial condition, $u(0,x)=...$ and there's some n-diagonal matrix, as above, representing some finite difference scheme on a uniform grid on $[0,1]$. So that you get some ODE system $$u_{t} = Au + b$$ $b$ being the boundary data, (I realize that the first and last rows should account for the BC's). So it's respective eigenvalue problem should be something like: $$(A-\lambda I)v=0$$

Can someone explain where the Chebyshev polynomials come from and how to apply the resulting recursion to them? Is there an alternative method to compute the $k$-th Eigenvalue?

• The Chebyshev polynomials are just coming from the recurrence for the Chebyshev polynomials. The point of making this identification instead of just working with the recursion directly is that the Chebyshev polynomials have an alternate form which makes finding their roots far easier than for a general high degree polynomial. – Ian Aug 21 '17 at 16:04
• Could you give an example? Or show how the chebyshev polynomial should be constructed from a recursion. – DanBernou Aug 22 '17 at 10:50
• The Chebyshev polynomials are defined by a recursion, which happens to be the same recursion that falls out of the problem mentioned in that article. Again that observation by itself is useless, since it is just assigning a name to something. The observation is only useful because the Chebyshev polynomials are related to trig functions. – Ian Aug 22 '17 at 15:31
• So simply by observation, we notice the recursion we get from the Eigenvalue problem fitted with BC conditions can be replaced with Chebyshev polynomials, which is nice because we want our eigenvalues to be defined by a trig function? I still don't see how to adjust The Chebyshev polynomials for arbitrary real coefficients on the diagonals. Also neither article makes mention of re-scaling the domain from [0,1] to [-1,1] which is required for Chebyshev right? Could you complete the example above or even just the 3 stencil (d,a,b) equivalent? – DanBernou Aug 23 '17 at 7:28