A fast algorithm for evaluation of normalized Hermite functions

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szegő recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we explore those of the Racah and q-Racah types. For the M-indexed (q-)Racah polynomials, we derive 3 + 2M term recurrence relations with variable dependent coefficients and 1 + 2L term (L ≥ M + 1) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi-indexed (q-)Racah polynomials are obtained. In Appendix B and Appendix C, we present a proof and some data of the recurrence relations with constant coefficients for the multi-indexed Wilson and Askey-Wilson polynomials.

Matrix polynomials satisfying first order differential equations and three term recurrence relations

This chapter contains sections titled: Introduction Round-Off Error Propagation Induced by FP Computation The CESTAC Method, a Stochastic Approach for Analyzing Round-Off Error Propagation Stochastic Arithmetic Validation and Implementation of the CESTAC Method Discrete Stochastic Arithmetic (DSA) Taking into Account Data Errors The CADNA Library [36] The Use of the CADNA Library Can the CADNA Library Fail? Conclusion References

An algorithm for computing polynomial zeros, based on Aberth's method, is presented. The starting approximations are chosen by means of a suitable application of Rouche's theorem. More precisely, an integerq ≥ 1 and a set of annuliAi,i=1,...,q, in the complex plane, are determined together with the numberki of zeros of the polynomial contained in each annulusAi. As starting approximations we chooseki complex numbers lying on a suitable circle contained in the annulusAi, fori=1,...,q. The computation of Newton's correction is performed in such a way that overflow situations are removed. A suitable stop condition, based on a rigorous backward rounding error analysis, guarantees that the computed approximations are the exact zeros of a “nearby” polynomial. This implies the backward stability of our algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of any degree, not necessarily monic, whose zeros and coefficients are representable as floating point numbers. In all the tests performed with more than 1000 polynomials having degrees from 10 up to 25,600 and randomly generated coefficients, the Fortran 77 implementation of our algorithm computed approximations to all the zeros within the relative precision allowed by the classical conditioning theorems with 11.1 average iterations. In the worst case the number of iterations needed has been at most 17. Comparisons with available public domain software and with the algorithm PA16AD of Harwell are performed and show the effectiveness of our approach. A multiprecision implementation in MATHEMATICA™ is presented together with the results of the numerical tests performed.

Orthogonal matrix polynomials and higher-order recurrence relations

A three term recurrence relation is found forwhen a + d = b + c. This includes the recurrence relations of Apéry associated with ζ(3), ζ(2) and log 2 as special or limiting cases.

A Generalization of Favard′s Theorem for Polynomials Satisfying a Recurrence Relation

A favard theorem for rational functions

The main difference between certain spectral problems for linear Schrödinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across ℤ and in the latter only across ℤ+. We present a technique that, by a mixture of Dirichlet and Taylor expansions, translates the almost Mathieu equation and its generalizations to three term recurrence relations. This opens up the possibility of exploiting the full power of the theory of orthogonal polynomials in the analysis of Schrödinger spectra.Aforementioned three-term recurrence relations share the property that their coefficients are almost periodic. In the special case when they form a periodic sequence, the support can be explicitly identified by a technique due to Geronimus. The more difficult problem, when the recurrence is almost periodic but fails to be periodic, is still open and we report partial results.The main promise of the technique of this article is that it can be extended to deal with multivariate extensions of the almost Mathieu equations. However, important theoretical questions need be answered before it can be implemented to the analysis of the spectral problem for Schrödinger operators.KeywordsOrthogonal PolynomialSpectral ProblemChebyshev PolynomialBorel MeasureEssential SpectrumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There are two definitions for the discrete q-Hermite polynomials, one of them is defined for \(0 1\). This paper is devoted to extend these definitions to the discrete q-Hermite matrix polynomials by means of the generating matrix functions. Explicit expressions and Rodrigues-type formulas for the discrete q-Hermite matrix polynomials are obtained. Some recurrence relations for these matrix polynomials, in particular the three terms recurrence relations are given. Furthermore, some identities are proved.

In this paper, we discuss the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. Some numerical examples have been solved to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. The error bound and convergence of the method have also been established.

An accurate Fortran code for computing hydrogenic continuum wave functions at a wide range of parameters

Relaxation dynamics of a particle in the presence of an external potential: exact solution in terms of matrix continued fractions

A stronger notion of nonnegative linearization of orthogonal polynomials is introduced. It requires that also the associated polynomials of any order have nonnegative linearization property. This turns out to be equivalent to a maximal principle of a discrete boundary value problem associated with orthogonal polynomials through the three term recurrence relation. The property is stable for certain perturbations of the recurrence relation. Criteria for the strong nonnegative linearization are derived. The range of parameters for the Jacobi polynomials satisfying this new property is determined.