Degree Of Denominator Research Articles

Rational interpolation and recursive solution of Löwner–Vandermonde systems of equations

Rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure: the first columns are of Vandermonde type whereas the last columns form a Löwner matrix. Here three-term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix. On this base an O( n 2) algorithm for the solution of n× n nonhomogeneous Löwner–Vandermonde systems is obtained.

Optimal Rational Functions for the Generalized Zolotarev Problem in the Complex Plane

It has been long recognized that the determination of optimal parameters for the classical alternating direction implicit (ADI) method leads to the Zolotarev problem \[ \min_{r \in R_{nn}} \frac{\max \{|r(z)|, z \in E \}}{\min \{|r(z)|, z \in F\}} \] for disjoint compact sets E, F \subseteq \mathbb{C}$, where Rnn isthe collection of rational functions of order n. In the case where E and F are real intervals, it was more recently pointed out that if they have different lengths, it is of interest to generalize the foregoing problem to the set Rmn with unequal numerator degree m and denominator degree n. The object of the paper is to investigate the generalized Zolotarev problem in the complex plane. A method is proposed to construct the optimal rational function, which is then applied to the particular example of two line segments---E on the real axis, F parallel to the imaginary axis. Numerical experiments show that the improvement on the classical solution may be amazingly great. Explicit expressions are also provided for special values of data.

Optimum parameters for the generalized ADI method

In the context of the generalized ADI method, we are concerned with the problem of finding in the set of rational functions r with numerator degree m and denominator degree n an element \(r^*\) that minimizes \(\) where E,F are disjoint real intervals. By extending a recent analysis by Levin and Saff, we present an explicit formula for choosing the pair (m,n) for given m +n. Furthermore, we provide a characterization of \(r^*\) and a Remes type algorithm for its determination. Extensive numerical computations furnish some comparison of \(r^*\) with asymptotically optimal solutions based on Fejer-Walsh and Leja-Bagby points.

Ray Sequences of Best Rational Approximants For |x|α

AbstractThe convergence behavior of best uniform rational approximations with numerator degree m and denominator degree n to the function |x|α, α > 0, on [-1, 1] is investigated. It is assumed that the indices (m, n) progress along a ray sequence in the lower triangle of the Walsh table, i.e. the sequence of indices {(m, n)} satisfiesIn addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function on [-1, 1] will be studied. The results will be compared with those for paradiagonal sequences (m = n + 2[α/2]) and for sequences of best polynomial approximants.

Matrices for the direct determination of the barycentric weights of rational interpolation

Let x 0,…, x N be N + 1 interpolation points (nodes) and 0,…, N be N + 1 interpolation data. Then every rational function r with numerator and denominator degress ⩽ N interpolating these values can be written in its barycentric form completely determined by a vector u of N + 1 barycentric weights u k . Finding u is therefore an alternative to the determination of the coefficients in the canonical form of r; it is advantageous inasmuch as u contains information about unattainable points and poles. In classical rational interpolation the numerator and the denominator of r are made unique (up to a constant factor) by restricting their respective degrees. We determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u' s . The method is of complexity O( n 3) in terms of the denominator degree n; it seems on the other hand to be among the most stable ones.

A program for solving the L2 reduced-order model problem with fixed denominator degree

A set of necessary conditions that must be satisfied by the L2 optimal rational transfer matrix approximating a given higher-order transfer matrix, is briefly described. On its basis, an efficient iterative numerical algorithm has been obtained and implemented using standard MATLAB functions. The purpose of this contribution is to make the related computer program available and to illustrate some significant applications.

Degree of Approximation by Rational Functions with Prescribed Numerator Degree

AbstractWe prove a Jackson type theorem for rational functions with prescribed numerator degree: For continuous functions f: [—1,1] —> ℝ with ℓ sign changes in (—1,1), there exists a real rational function Rℓ,n(x) with numerator degree ℓ and denominator degree at most n, that changes sign exactly where f does, and such thatHere C is independent of f, n and ℓ, and ωφ is the Ditzian-Totik modulus of continuity. For special functions such as f(x) = sign(x)|x|α we consider improvements of the Jackson rate.

Comments on "Extreme point results for robust stabilization of interval plants with first-order compensators" [with reply

The commenters identify an extremely minor error in the above-titled paper by B.R. Barmish et al. (ibid., vol.37, no.6, p.707-14, June 1992), i.e. a counterexample which does not satisfy the statement regarding the simplification of the results to the interval plant with denominator degree less than four. In their reply, Barmish et al. agree and point out that if 'less than three' is substituted for 'less than four', their statement is completely accurate.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Szegö polynomials associated with Wiener-Levinson filters

Szegö polynomials are studied in connection with Wiener–Levinson filters formed from discrete signals x N ={ x N ( k)} N−1 k=0 . Our main interest is in the frequency analysis problem of finding the unknown frequencies ω j , when the signal is a trigonometric polynomial x N(k)= ∑ j=−I I α j e iω jk . Associated with this signal is the sequence of monic Szegö polynomials { ρ n ( ψ N ; z)} ∞ n=0 orthogonal on the unit circle with respect to a distribution function ψ N ( θ). Explicit expressions for the weight function ψ′ N ( θ) and associated Szegö function D N ( z) are given in terms of the Z-transform X N ( z) of the signal x N . Several theorems are given to support the following conjecture which was suggested by numerical experiments: As N and n increase, the 2 I + 1 zeros of ρ n ( ψ N ; z) of largest modulus approach the points e iω j . We conclude by showing that the reciprocal polynomials ρ ∗ n(ψ N; z)≔z n ρ n(ψ N; 1 z ) are Padé numerators for Padé approximants (of fixed denominator degree) to a meromorphic function related to D N ( z).

The density of alternation points in rational approximation

We investigate the behavior of the equioscillation (alternation) points for the error in best uniform rational approximation on $[-1,1]$. In the context of the Walsh table (in which the best rational approximant with numerator degree $\leq m$, denominator degree $\leq n$, is displayed in the $n$th row and the $m$th column), we show that these points are dense in $[-1,1]$, if one goes down the table along a ray above the main diagonal $\left ( {n = \left [ {cm} \right ],c < 1} \right )$. A counterexample is provided showing that this may not be true for a subdiagonal of the table. In addition, a Kadec-type result on the distribution of the equioscillation points is obtained for asymptotically horizontal paths in the Walsh table.

A General Nested Split-Plot Analysis of Covariance

Abstract We present a general split-plot analysis of covariance (ANOCOVA) model for unbalanced data. The imbalance arises from assigning a different number of experimental units to the levels of the whole-plot factor A. The values of the covariable measurements on each experimental unit are allowed to vary across the levels of the split-plot factor B. Exact test statistics and confidence intervals are derived. Exact F test statistics for the A and B main effects and the A × B interaction each have one less numerator and denominator degree of freedom than for the corresponding analysis of variance (ANOVA). Reduced versions of the general model are given in which the lost numerator degrees of freedom are regained. Competing approximate tests are discussed, including generalized least squares. For the nested split-plot ANOVA model for balanced data, the numerator sums of squares for the A and B main effects and the A × B interaction, together with the A and B error sums of squares, add up to the total sum of...

Row convergence theorems for generalised inverse vector-valued Padé approximants

The approximants mentioned in the title are related to vector-valued continued fractions and the vector ϵ-algorithm devised by Wynn in 1963. Here we establish a unitary invariance property of these approximants and describe how the classical (1-dimensional) Padé approximants can be obtained as a special case. The main results of the paper consist of De Montessus—De Ballore type convergence theorems for row sequences (having fixed denominator degree) of vector-valued approximants to meromorphic vector functions.

Rational Approximations to the Exponential Function with Two Complex Conjugate Interpolation Points

We examine rational approximations to the exponential function which satisfy $R(z) = \exp (z) + O(z^{2m - 1} )$ and $R(z_0 ) = \exp (z_0 ),R(\bar z_0 ) = \exp (\bar z_0 )$, where m denotes the maximal degree of numerator and denominator. It is proved that a curve bisects the complex plane so that if $z_0 $ lies to its left the approximation is A-acceptable, while if it is to the right of this curve then $R(z)$ is non-A-acceptable.

Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods

All rational approximations to exp(z) of order ?2m?β (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving β real parameters. Using the theory of order stars [9], necessary and sufficient conditions forA-stability (respectivelyI-stability) are given. On the basis of this characterization relations between the concepts ofA-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducibleA-stableR(z) of oderp?0 there exist algebraically stable Runge-Kutta methods of the same order withR(z) as stability function.

Simplification of linear time-invariant systems by moment approximants †

Abstract Given a system S characterized by a transfer function, a practical method is developed for finding a system S∗ with proper rational transfer function of small denominator degree, such that the step response of S∗ is, in an appropriate sense, the baat approximant to the step response of S. The method is extended to multivariable systems. Various relations between moment and Padé approximants are derived.

Use of a Luenberger observer to control the numerator and denominator of a transfer function

It is shown that a Luenberger observer may be used to convert a transfer function having a numerator of degree m and a denominator of degree n (with m≤n−1) into one having a numerator of degree n+m−1 and a denominator of degree 2n−1, with the denominator polynomial and n−1 of the numerator coefficients assignable arbitrarily.

Program for numerical inversion of the Laplace transform

The letter describes a numerical Laplace-transform inversion based on the Padé rational approximation of exp (z) with any degree of numerator and denominator. User-oriented information is given. The program is intended mainly for time responses of networks with lumped-distributed and active elements where partial-fraction expansions are not feasible.

Driving-point-function synthesis using a unity-gain amplifier

A unity-gain amplifier together with passive RC elements is shown to realize driving-point admittance functions having a numerator degree equal to, or one greater than, the denominator degree; poles on the negative real axis, excluding the origin; and zeros anywhere in the s plane excepting the positive real axis. Synthesis procedure is discussed and illustrated with two examples of current interest concerning realization of a solid-state inductance and one example of synthesis of a nonpositive real function.