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1.
Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Padé fractions in the main diagonal resp. in the diagonal below the main diagonal of the Padé table for the exponential function play an essential role. 相似文献
2.
《Applied Mathematics Letters》2001,14(3):333-340
Here we consider a numerical procedure to interpolate on matrix Lie groups. By using the exponential map and its (1, 1) diagonal Padé approximant, piecewice interpolants may be derived. The approach based on the Padé map has the advantage that the computation of exponentials and logarithms of matrices are reduced. We show that the updating technique proposed by Enright in [1] may be applied when a dense output is required. The application to the numerical solution of a system ODEs on matrix group and to a classical interpolation problem are reported. 相似文献
3.
Arieh Iserles 《Journal of Computational and Applied Mathematics》1978,4(2):143-146
It is considered whether a linear combination of three A-acceptable Padé approximations to the exponential function remains A-acceptable when it is exponentially fitted to two distinct negative points. The results of Iserles [5] regarding linear combinations of two A-acceptable Padé approximations are generalized. 相似文献
4.
H. Stahl 《Constructive Approximation》1997,13(2):287-292
The Baker-Gammel-Wills Conjecture states that if a functionf is meromorphic in a unit diskD, then there should, at least, exist an infinite subsequenceN ⊆N such that the subsequence of diagonal Padé approximants tof developed at the origin with degrees contained inN converges tof locally uniformly inD/{poles off}. Despite the fact that this conjecture may well be false in the general Padé approximation in several respects. In the present
paper, six new conjectures about the convergence of diagonal Padé approximants are formulated that lead in the same direction
as the Baker-Gammel-Wills Conjecture. However, they are more specific and they are based on partial results and theoretical
considerations that make it rather probable that these new conjectures hold true. 相似文献
5.
The quadratic approximation is a three dimensional analogue of the two dimensional Padé approximation. A determinantal expression for the polynomial coefficients of the quadratic approximation is given. A recursive algorithm for the construction of these coefficients is derived. The algorithm constructs a table of quadratic approximations analogous to the Padé table of rational approximations. 相似文献
6.
This paper clears up to the following three conjectures:
- The conjecture of Ehle [1] on theA-acceptability of Padé approximations toe z , which is true;
- The conjecture of Nørsett [5] on the zeros of the “E-polynomial”, which is false;
- The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist.
7.
A. Bultheel 《Journal of Computational and Applied Mathematics》1980,6(4):259-266
It is known [26] that the Viskovatoff algorithm can be generalized to cover the computation of continued fractions whose successive convergents form the Padé approximants of a descending staircase or diagonal, even in the case of a non-normal Padé table. It is the intention of the author to generalize this idea to other paths of the Padé table and in this way link together some algorithms scattered in literature. 相似文献
8.
Summary.
For univariate functions the Kronecker theorem, stating the equivalence
between the existence of an infinite block in the table of Padé approximants
and the approximated function being rational, is well-known.
In [Lubi88] Lubinsky proved that if is not rational, then its Padé table
is normal almost everywhere: for an at most countable set of points the
Taylor series expansion of is such that it generates a non-normal
Padé table. This implies that the Padé operator is an almost always
continuous operator because it is continuous when computing a normal
Padé approximant [Wuyt81].
In this paper we generalize the above results to the case of multivariate
Padé approximation. We distinguish between two different approaches for
the definition of multivariate Padé approximants: the general order one
introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed
in [Cuyt84].
Received December 19, 1994 相似文献
9.
A. Zappavigna P. Colaneri S. Kirkland R. Shorten 《Linear algebra and its applications》2012,436(9):3425-3442
In this paper the discretisation of switched and non-switched linear positive systems using Padé approximations is considered. Padé approximations to the matrix exponential are sometimes used by control engineers for discretising continuous time systems and for control system design. We observe that this method of approximation is not suited for the discretisation of positive dynamic systems, for two key reasons. First, certain types of Lyapunov stability are not, in general, preserved. Secondly, and more seriously, positivity need not be preserved, even when stability is. Finally we present an alternative approximation to the matrix exponential which preserves positivity, and linear and quadratic stability. 相似文献
10.
Laurent Padé-Chebyshev rational approximants,A
m
(z,z
−1)/B
n
(z, z
−1), whose Laurent series expansions match that of a given functionf(z,z
−1) up to as high a degree inz, z
−1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by
Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm
and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z
−1)B
n
(z, z
−1)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth
kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for
developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8]
but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of
equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for
explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable
improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper
[7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared
with those of the Maehly type. 相似文献
11.
Algorithms are developed to compute simultaneously the poles of functions represented by continued fractions whose approximants lie on the main diagonal of two-point Padé tables. The algorithms are based on equations recently developed by McCabe and Murphy for producing continued fraction expansions of a pair of power series. Sufficient conditions are given to ensure that the computations can be carried out and that the resulting approximations converge geometrically to the desired poles. As a by-product, an algorithm is obtained for computing zeros of polynomials. The theory and methods are illustrated by means of numerical examples. 相似文献
12.
Grigori L. Litvinov 《Central European Journal of Mathematics》2003,1(1):36-60
The error autocorrection effect means that in a calculation all the intermediate errors compensate each other, so the final
result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of
interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall
discuss a very strong form of the effect which appears in rational approximations to functions. The error autocorrection effect
occurs in all efficient methods of rational approximation (e.g., best approxmations, Padé approximations, multipoint Padé
approximations, linear and nonlinear Padé-Chebyshev approximations, etc.), where very significant errors in the approximant
coefficients do not affect the accuracy of this approximant. The reason is that the errors in the coefficients of the rational
approximant are not distributed in an arbitrary way, but form a collection of coefficients for a new rational approximant
to the same approximated function. The understanding of this mechanism allows to decrease the approximation error by varying
the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The effect
of error autocorrection indicates that variations of an approximated function under some deformations of rather a general
type may have little effect on the corresponding rational approximant viewed as a function (whereas the coefficients of the
approximant can have very significant changes). Accordingly, while deforming a function for which good rational approximation
is possible, the corresponding approximant’s error can rapidly increase, so the property of having good rational approximation
is not stable under small deformations of the approximated functions. This property is “individual”, in the sense that it
holds for specific functions. 相似文献
13.
The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Padé approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Padé approximants, to the application of multipoint Padé approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Padé-Hermite approximation for systems of Markov type functions. 相似文献
14.
《Numerical Methods for Partial Differential Equations》2018,34(2):760-798
In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust. 相似文献
15.
B. A. Wade A. Q. M. Khaliq M. Siddique M. Yousuf 《Numerical Methods for Partial Differential Equations》2005,21(3):553-573
We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
16.
Fractional derivatives: Probability interpretation and frequency response of rational approximations
J.A. Tenreiro Machado 《Communications in Nonlinear Science & Numerical Simulation》2009,14(9-10):3492-3497
The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts. 相似文献
17.
《Journal of Computational and Applied Mathematics》2006,187(2):142-170
New compact approximation schemes for the Laplace operator of fourth- and sixth-order are proposed. The schemes are based on a Padé approximation of the Taylor expansion for the discretized Laplace operator. The new schemes are compared with other finite difference approximations in several benchmark problems. It is found that the new schemes exhibit a very good performance and are highly accurate. Especially on large grids they outperform noncompact schemes. 相似文献
18.
《Journal of Computational and Applied Mathematics》1988,21(3):343-355
In this paper the concept of partial Padé approximation, introduced by Claude Brezinski, is generalised to the case of simultaneous rational approximation with common denominator. The use of information about known poles and/or zeros, can lead to approximants with a better numerical behaviour than in the case of ordinary simultaneous Padé approximation. 相似文献
19.
Y. V. Novak 《Mathematical Notes》2008,84(5-6):821-825
20.
《Journal of Computational and Applied Mathematics》2002,139(1):75-94
A one-parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered. The corresponding dynamics of the Schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi-discrete modified KdV equation, namely, the Schur flow. A discrete analogue of the Miura transformation is found. An integrable discretization of the Schur flow enables us to compute a Padé approximation of the Carathéodory functions, or equivalently, to compute a Perron–Carathéodory continued fraction in a polynomial time. 相似文献