Troisième Journées d' Approximation, Lille, May, 15-16, 2008

Conference:

Troisième Journées d' Approximation location:Lille date:15-16 May 2008

Abstract:

Suppose A is a large Hermitian NxN matrix and v an N-vector. Then de space K_n(A,v)={v_0,...,v_{n-1}} with n << N, v_0=v, v_k=A(I-α_k A)^{-1} v_{k-1} and α_k real is called a rational Krylov subspace. Because of its construction we can also write v_k as v_k=r(A)v with r_k a rational function of the form p_k(z)/[(1-α_1 z)...(1-α_kz)], whete p_k is a polynomial of degree k at most. After orthogonalizing v_k with respect to the various vectors, we obtain a vector q_k = φ_k(A)v, where again φ_k is a rational function of the same form as r_k.

If α_k=0 for all positive k, the rational functions r_k and φ_k reduce to respectively the polynomials p_k and φ_k so that v_k=p_k(A)v and q_k = φ_k(A)v. The orthogonality of the vectors q_k is then equivalent to the orthogonality of the polynomials φ_k with respect to the inner product <φ_k,φ_l> = M(φ_k (φ_l)*), where the linear functional M is defined on the space of polynomials by its moments M(z^k)=v*A^kv. Since the classical moment matrix has a Hankel structure, this theory will be related to the orthogonality of polynomials on the real line. Thus, in the classical Lanczos method for Hermitian matrices, the three-term recurrence relation for orthogonal polynomials (OP) leads to a short recurrence between the successive vectors q_k, meaning that q_k can be computed from q_{k-1} and q_{k-2} without the need for a full Gram-Schmidt orthogonalization.

Orthogonal rational functions (ORF) on the real line are a generalization of OP on the real line in such a way that the OP return if all the poles 1/&alpha_k are at infinity. Consequently, if the α_k are arbitrary real, it will be obvious that the orthogonality of the q_k will lead to the orthogonality of the rational functions φ_k, so that a simple recurrence of the ORF will lead to an efficient implementation of the rational Lanczos algorithm (RLA). We use this relationship between ORF and RLA to find numerical approximants to matrix functions that appear in the solution of various differential problems.