Rational Krylov sequences and Orthogonal Rational Functions
نویسندگان
چکیده
In this paper we study the relationship between spectral decomposition, orthogonal rational functions and the rational Lanczos algorithm, based on a simple identity for rational Krylov sequences.
منابع مشابه
Orthogonal rational functions on the unit circle: from the scalar to the matrix case
The purpose of these lecture notes is to give a short introduction to the theory of orthogonal rational functions (ORF) on the unit circle. We start with the classical problem of linear prediction of a stochastic process to give a motivation for the study of Szegő’s problem and to show that in this context it will turn out that not as much the ORF but rather the reproducing kernels will play a ...
متن کاملOn the computation of orthogonal rational functions
Several techniques are known to compute a new orthogonal polynomial φk+1 of degree k + 1 from Lk := span{φ0, ..., φk} in case of (discrete) orthogonality on the real line. In the Arnoldi approach one chooses Φk ∈ Lk and makes xΦk orthogonal against φ0, ..., φk. By taking as Φk a linear combination of φk and the kernel (or GMRES) polynomial ψk(x) = ∑k j=0 φj(0)φj(x), one needs to orthogonalize o...
متن کاملThe implicit application of a rational lter in the RKS methodGorik
The implicitly restarted Arnoldi method implicitly applies a polynomial lter to the Arnoldi vectors by use of orthogonal transformations. In this paper, an implicit ltering by rational functions is proposed for the rational Krylov method. This ltering is performed in an eecient way. Two applications are considered. The rst one is the ltering of unwanted eigenvalues using exact shifts. This appr...
متن کاملHermite Orthogonal Rational Functions
We recount previous development of d-fold doubling of orthogonal polynomial sequences and give new results on rational function coefficients, recurrence formulas, continued fractions, Rodrigues’ type formulas, and differential equations, for the general case and, in particular, for the d-fold Hermite orthogonal rational functions.
متن کاملSuperlinear convergence of the rational Arnoldi method for the approximation of matrix functions
A superlinear convergence bound for rational Arnoldi approximations to functions of matrices is derived. This bound generalizes the well-known superlinear convergence bound for the CG method to more general functions with finite singularities and to rational Krylov spaces. A constrained equilibrium problem from potential theory is used to characterize a max-min quotient of a nodal rational func...
متن کامل