We show how they can be described by n x n matrices with polynomial entries and use this description to derive a recursion relation for the iterations of the CA.
We will deduce informally the basic structure of the spacetime diagrams from a simple recursion relation for the [2.
Then, in section 4, I derive a recursion relation for computing [OMEGA] at increasing values of an integer "time"M.
i], that can be obtained via repeated actions of the matrix [OMEGA] in the recursion relation (10.
Ghosh develops the theory of skew-orthogonal polynomials and obtains recursion relations
which (unlike orthogonal polynomials) depend on weight functions.
In recursion relations
, given by Eq 17, the initial values of vectors at k = 0 are assumed equal to zero.
In , Cullum and Willoughby show that if the coefficients of the recursion relations are chosen so that the tridiagonal matrix is symmetric (but complex) accurate eigenvalues can be computed for some test problems.
To a large degree this problem can be overcome by choosing the coefficients in the recursion relations so that the tridiagonal matrix is complex symmetric , but to minimize CPU and memory costs we wish to use real arithmetic.