ABSTRACT
The Lanczos algorithm of tridiagonalization is the key step for converting the original (presumably large) matrix into its corresponding sparse J-matrix. This chapter derives an equation that defines the Lanczos polynomial propagation of an initial wave packet. It introduces a pair of solutions, one regular and the other irregular, called the polynomials of the first and the second kind, respectively, within second-order differential equations satisfied by the classical orthogonal polynomials. The optimal method for finding all the roots of a given polynomial is the use of the corresponding Hessenberg matrix to solve its well-conditioned eigenvalue problem.
