<title>Algorithms for solving rational interpolation problems related to fast and superfast solvers for Toeplitz systems</title>

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.

Thiele-type continued fractions interpolation may be the classical rational interpolation and plays critical role in image interpolation and numerical analysis. Different from the traditional method, a new Thiele type continued fractions rational interpolation method with parameters was presented to address the interpolation problem efficiently. Firstly, in order to gain neat expressions in terms of inverse differences, we chose the multiplicity of the points strategically. Secondly, we constructed a univariate Thiele type continued fractions rational interpolation with parameters, which can satisfy the interpolation condition. We also discussed the interpolation algorithm, interpolation theorem. Numerical examples were given to show that the presented method achieves state-of-the-art performance.

In this paper, a theory of polynomial and rational matrix interpolation is introduced and applied to problems in systems and control. The polynomial matrix interpolation theory is first outlined and then applied to solving matrix equations; it is also used in pole assignment and other control problems. Rational matrix interpolation is also discussed and it is used to solve rational matrix equations including the model matching problem.

A framework for analyzing nonlinear eigenproblems and parametrized linear systems

I recently came across a very appealing result about the product of the lengths of the chords of a regular polygon with n sides inscribed in a unit circle. If one vertex is selected and the product of the chord lengths from that vertex to each of the others is evaluated, the result is just n . There is a simple proof. Consider such a polygon and denote its vertices by the points A 0 , A 1 , … , A n . Suppose that O is the centre of the circle and the origin of the complex plane. The roots of the equation are just where ω is a primitive nth root of unity. Each of these roots represents one of the vertices. Suppose for convenience that the real number 1 is represented by A 0 .

We propose a new method in the spectral analysis of noisy time-series data for damped oscillators. From the Jacobi three terms recursive relation for the denominators of the Padé approximations built on the well-known Z-transform of an infinite time series, we build a Hilbert space operator, a J-operator, where each bound state (inside the unit circle in the complex plane) is simply associated with one damped oscillator while the essential spectrum of the J-operator, which lies on the unit circle itself, is shown to represent the noise. Signal and noise are thus clearly separated in the complex plane. For a finite time series of length 2N, the J-operator is replaced by a finite order J-matrix JN, having N eigenvalues which are time reversal covariant. Different classes of input noise, such as blank (white and uniform), Gaussian and pink, are discussed in detail, the J-matrix formalism allowing us to efficiently calculate hundreds of poles of the Z-transform. Evidence of a universal behavior in the final statistical distribution of the associated poles and zeros of the Z-transform is shown. In particular, the poles and zeros tend, when the length of the time series goes to infinity, to a uniform angular distribution on the unit circle. Therefore at finite order, the roots of unity in the complex plane appear to be noise attractors. We show that the Z-transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property. A few basic examples are given to suggest the power of the proposed method.

Generalizing the Walsh theorem, E. B. Saff, A. Sharma, and R. S. Varga showed that there is a close relation between the rational interpolants in roots of unity and Pade approximants of certain meromorphic functions. The purpose of this paper is to extend this result, replacing the Pade approximant with other rational functions so as to obtain a larger region of equiconvergence.

Rational interpolation via orthogonal polynomials

We present numerical algorithms for solving the rational interpolation problem (r.i.p.) when both the numerators and the denominators are expressed with respect to the Newton basis. These algorithms generalize some classical procedures which were developed for dealing with polynomials given in power form. Our approach is based on the reduction of the r.i.p. to computing a block triangular decomposition of certain structured matrices. These matrices generalize the Hankel structure in the sense that they belong to the kernel of a suitable displacement operator which reduces to that defining the Hankel structure when the interpolation nodes collapse to zero. The computational cost and the stability properties of the resulting algorithms depend on the criterion we choose for the step size regulation. Jumping from a nonsingular leading principal submatrix to the next one leads to an algorithm for r.i.p. which reaches the optimal sequential complexity. Conversely, jumping from a well-conditioned leading principal submatrix to a next one allows us to control the accuracy of the computed approximations.

Each g ∊ ℤ[x] defines a homeomorphism of a compact space We investigate the isomorphism classes of the C*-crossed product algebra Bg associated with the dynamical system An isomorphism invariant Eg of the algebra Bg is shown to determine the algebra Bg up to * or * anti-isomorphism if degree g ≤ 1 and 1 is not a root of g or if degree g = 2 and g is irreducible. It is also observed that the entropy of the dynamical system is equal to the growth rate of the periodic points if g has no roots of unity as zeros. This slightly extends the previously known equality of these two quantities under the assumption that g has no zeros on the unit circle.

Power integral bases in prime-power cyclotomic fields

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Matrix polynomials orthogonal on the unit circle and accuracy of autoregressive models

This paper contains further steps towards a Szegö theory for orthogonal rational matrix‐valued functions on the unit circle 𝕋. It continues the investigations started in [18]–[20]. Hereby we are guided by former work of Bultheel, González–Vera, Hendriksen, and Njåstad on scalar orthogonal rational functions on the one side and by research of Delsarte, Genin, and Kamp on matrix polynomials on the other side. In this paper we derive recursion formulas for Christoffel–Darboux pairs of rational matrix functions which lead us to jqq ‐recursively connected pairs of rational matrix functions. Moreover, we prove a Favard‐type theorem for rational matrix functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

When machining a series of short linear segments, the computer numerical control system interpolates the machining path with the pre-specified spline curve to obtain continuous feed motion. However, to obtain adequate smooth tool-paths to smooth the feed velocity and acceleration, the computer-aided manufacturing system needs to use a higher order spline curve, which will be limited by some technical bottlenecks. In this article, a new real-time interpolation algorithm is proposed from the perspective of kinematics to achieve uninterrupted feed motion throughout the global tool-path. At the same time, the acceleration profile achieves G2 continuous to avoid unnecessary feed frustration and inertial impact, reaching the balance between time-optimal and motion performance. First, the jounce-limited acceleration curve is blended at the corner of the machining path, and the optimal cornering transition velocity is obtained by adding the velocity, acceleration and contour error constraints to the corner transition motion. Then, according to the linear segments with different lengths between the corners, combined with the feed motion around the corner, a look-ahead interpolation algorithm is proposed to calculate the maximum feed rate with the constraint of the linear segment length and kinematic boundary conditions. At last, for the linear segments whose corner contours overlap with each other after interpolation, the smooth transition between the two corners can be realized by mixing the feed motion of the next corner. Compared with non-uniform rational B-spline interpolation algorithm, the proposed algorithm reduces the total machining time by 14% and the computer numerical control system improves the computational efficiency by 11%. It proves that the proposed algorithm has better application value in the manufacturing of complex parts.