Orthogonal Rational Functions Research Articles

Growth properties of Nevanlinna matrices for rational moment problems

We consider rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna-type parameterization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove results related to the growth at the singularities of the functions in a Nevanlinna matrix, and in particular provide bounds on the growth analogous to the corresponding result in the classical polynomial case, when the number of singularities is finite.

Approximations to the magic formula

Pacejka’s tire model is widely used and well-known by the automotive engineering community. The magic formula describes the brake force, side force and self-aligning torque in terms of the longitudinal slip and slip angle, plus several corrections. This paper uses approximation theory to obtain different types of approximations to the magic formula: rational functions (RA) resulting from the Remez algorithm, expansions in a series of Chebyshev polynomials (ACh), a series of Chebyshev rational polynomials (ARChPs), a series of rational orthogonal functions (ORF) and a series of ARChPs that result from grade-1 ORFs. The last expansion shows the fastest convergence and most effective computation. Jacobi rational polynomials can also be obtained to complement this expansion and facilitate fine-tuning in specific areas of the error curve. This work is complemented by obtaining the original rational approximations to the inverse tangent function, which take advantage of the curve symmetry to reduce the computation load and provide models that include the influence of the vertical load. The convergence properties of the development in series and the error values resulting from numeric examples for the three types of stress are shown. The proposed final ARChP expressions show very low error (1%) compared to the original magic formula. They can be computed 20 times faster; they can be evaluated, derived and integrated analytically easily; and their coefficients can be obtained from tests using common least-squares algorithms.

Parametric identification of fractional-order systems using a fractional Legendre basis

This paper deals with the identification of fractional-order systems through orthogonal rational functions. Fractional systems are characterized by their non-exponential aperiodic multimodes; therefore, fractional orthogonal rational functions provide better approximating models with fewer parameters. In spite of the fact that the Laguerre-based model is simple, it is, to some extent, deficient at high frequencies. Motivated by this objective, the use of a Legendre basis which has progressive pole locations and can be expected to perform better at high frequencies is studied.

The linear pencil approach to rational interpolation

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils z B − A of two tridiagonal matrices A , B , following Spiridonov and Zhedanov. In the present paper, as well as revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Padé approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to ∞ , we compute the spectrum and the numerical range of the underlying linear pencil explicitly.

Schur–Nevanlinna–Potapov sequences of rational matrix functions

We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur–Nevanlinna–Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur–Nevanlinna–Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna–Pick type for matricial Schur functions.

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented.

Orthogonal rational functions with complex poles: The Favard theorem

Let { φ n } be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions, the rational functions φ n form an orthonormal system with respect to a Hermitian positive-definite inner product.

Orthogonal Rational Functions with real coefficients and semiseparable matrices

When one wants to use Orthogonal Rational Functions (ORFs) in system identification or control theory, it is important to be able to avoid complex calculations. In this paper we study ORFs whose numerator and denominator polynomial have real coefficients. These ORFs with real coefficients (RORFs) appear when the poles and the interpolation points appear in complex conjugate pairs, which is a natural condition. Further we deduce that there is a strong connection between RORFs and semiseparable matrices.

A rational spectral method for the KdV equation on the half line

A Petrov–Galerkin method using orthogonal rational functions is proposed for the Korteweg–de Vries (KdV) equation on the half line with initial-boundary values. The nonlinear term and the right-hand side term are treated by Chebyshev rational interpolation explicitly, and the linear terms are computed with the Galerkin method implicitly. Such an approach is applicable using fast algorithms. Numerical results are presented for problems with both exponentially and algebraically decaying solutions, respectively, highlighting the performance of the proposed method.

Computing rational Gauss–Chebyshev quadrature formulas with complex poles: The algorithm

We provide an algorithm to compute arbitrarily many nodes and weights for rational Gauss–Chebyshev quadrature formulas integrating exactly in spaces of rational functions with complex poles outside [−1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O ( n ) . This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions on [−1, 1] with arbitrary complex poles outside this interval.

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Mobius transformations of Hessenberg matrices and also as the eigenvalues of the operator Mobius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.

Orthogonal rational functions and rational modifications of a measure on the unit circle

In this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure.

Rational Functions with a General Distribution of Poles on the Real Line Orthogonal with Respect to Varying Exponential Weights: I

Orthogonal rational functions are characterized in terms of a family of matrix Riemann–Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of ‘model’ matrix Riemann–Hilbert problems which are amenable to asymptotic analysis via the Deift–Zhou non-linear steepest-descent method.

Evaluating of Dawson’s Integral by solving its differential equation using orthogonal rational Chebyshev functions

Dawson’s Integral is u ( y ) ≡ exp ( - y 2 ) ∫ 0 y exp ( z 2 ) d z . We show that by solving the differential equation d u / d y + 2 yu = 1 using the orthogonal rational Chebyshev functions of the second kind, SB 2 n ( y ; L ) , which generates a pentadiagonal Petrov–Galerkin matrix, one can obtain an accuracy of roughly ( 3 / 8 ) N digits where N is the number of terms in the spectral series. The SB series is not as efficient as previously known approximations for low to moderate accuracy. However, because the N -term approximation can be found in only O ( N ) operations, the new algorithm is the best arbitrary-precision strategy for computing Dawson’s Integral.

Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

An interpolation problem of Nevanlinna–Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix-valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non-degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix-valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix-valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

Rational Gauss-Chebyshev quadrature formulas for complex poles outside $[-1,1

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [-1,1] to arbitrary complex poles outside [-1,1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [-1,1].

Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind.