Publications of Adhemar Bultheel | |
Suppose A is a large Hermitian NxN matrix and v an N-vector. Then de space Kn(A,v)={v0,...,vn-1} with n << N, v0=v, vk=A(I-αk A)-1 vk-1 and αk real is called a rational Krylov subspace. Because of its construction we can also write vk as vk=r(A)v with rk a rational function of the form pk(z)/[(1-α1 z)...(1-αkz)], whete pk is a polynomial of degree k at most. After orthogonalizing vk with respect to the various vectors, we obtain a vector qk = φk(A)v, where again φk is a rational function of the same form as rk.
If αk=0 for all positive k, the rational functions rk and φk reduce to respectively the polynomials pk and φk so that vk=pk(A)v and qk = φk(A)v. The orthogonality of the vectors qk is then equivalent to the orthogonality of the polynomials φk with respect to the inner product <φk,φl> = M(φk (φl)*), where the linear functional M is defined on the space of polynomials by its moments M(zk)=v*Akv. 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 qk, meaning that qk can be computed from qk-1 and qk-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/αk are at infinity. Consequently, if the αk are arbitrary real, it will be obvious that the orthogonality of the qk will lead to the orthogonality of the rational functions φk, 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.
Status:
Published
BiBTeX entry:
@unpublished{AbsDB08b, author = "K. Deckers* and A. Bultheel", title = "Rational {K}rylov sequences and orthogonal rational functions", note = "Presentation at the 13th Internatl. Conf. on Computational and Applied Mathematics (ICCAM), Ghent, Belgium", year = "2008", month = "July, 7-11", url = "http://nalag.cs.kuleuven.be/papers/ade/ICCAM08/index.html", LIMO = "1945454", }File(s): abstract.pdf (101K) | slides.pdf (1.8M)