PHYCS 498CQM: Homework 6

1. Show by induction that each vector psi_n generated by the Lanczos algorithm is orthogonal to all the previous vectors, and that the Hamiltonian has the tridiagonal form. Regarding the problem that orthogonality is guaranteed only for infinite numerical precision, discuss how errors in each step can accumulate in the deviations from orthogonality.
2. Derive the continued fraction representation of the spectral function using the Lanczos tridiagonal form. Hint: The key is the form of the inverse of a tridiagonal matrix T. You can generate the continued fraction representation by generating the T^-1_1,1 in terms of 1/T_1,1, T_1,2, and the inverse of T' where T' is the matrix formed by eliminating the first row and column. This can be continued indefinitely.