Pade Approximants,Symbolic Evaluations,and Computation of Solitons in Two-Field Antiferromagnet Model

An efficient analytic–numerical method for finding soliton solutions in the gauge-invariant Heisenberg antiferromagnet model is suggested. The method is based on power and asymptotic series and on the analytic continuation technique: re-expansions and Pade approximants. Symbolic evaluations are used both for constructing the series and for efficient finding initial approximations to the solutions.     

关于矩阵指数的PADE逼近新算法

基于广义逆矩阵Pade逼近的特点是在保持逼近阶的前提下,在构造过程中不需要 用到矩阵的乘法运算.利用该结果建立矩阵指数etA的一种新的非线性逼近算法.该方法与原 Pade近似法相比具有明显的优点,即它对奇异矩阵和高阶矩阵是适用的,并且所得到的算法 适合编程上机进行计算.给出的一个计算实例说明了算法的有效性.逼近公式的存在性和唯 一性得到了证明.     

Convergence of a class of Padé approximations for delay systems

This paper concerns the convergence of a class of rational approximations for delay systems of the form exp (— sT) M(s), where M(s) is a strictly proper rational transfer matrix. The rationale for reducing the order of G(s) is to replace exp (— sT) in G(s) by the class of all-pass/low-pass Pade approximations. The L₂ and L_m convergence in the frequency domain are established under mild conditions on M(s) (and hence impulse response in the L₂ case). For scalar M(s), the convergence is achieved at an optimal rate. Error bounds for the approximants are obtained which provide a priori estimates for the errors.     

A practical two-dimensional Thiele-type matrix Pade/spl acute/ approximation

In view of several potential applications in multivariable two-dimensional (2-D) systems theory, a practical 2-D matrix Pade/spl acute/ approximation is introduced by using a generalized inverse of the matrices. The approximants are expressed in the form of the 2-D Thiele-type continued fractions and are computed by an efficient recursive algorithm. As it's an application, the state-space realization problem of the 2-D filters is discussed.     

Balanced realization of Pade approximants of e-sT

An algorithm to calculate the balanced realization of all-pass Pade approximants of e^-sT is presented. The algorithm makes use of a regular C-type continued fraction expansion of e^-sT and a properly scaled recurrence formula is used to generate the coefficients of the Pade denominator polynomial. A truncation property of the balanced realization is discussed     

Construction of the characteristic polynomial of a matrix

A connection between Padé approximants and characteristic polynomial of a matrix with complex, in general, elements is established. A system of homogeneous, first-order ordinary differential equations is associated with the given matrix and after Laplace transform the individual transformed components are shown to be special Padé approximants.     

Padé error estimates for the logarithm of a matrix

The error of Padé approximations to the logarithm of a matrix and related hypergeometric functions is analysed. By obtaining an exact error expansion with positive coefficients, it is shown that the error in the matrix approximation at X is always less than the scalar approximation error at x, when ∥X∥ < x. A more detailed analysis, involving the interlacing properties of the zeros of the Padé denominator polynomials, shows that for a given order of approximation, the diagonal Padé approximants are the most accurate. Similarly, knowing that the denominator zeros must lie in the interval (1,∞) leads to a simple upper bound on the condition number of the matrix denominator polynomial, which is a crucial indicator of how accurately the matrix Padé approximants can be evaluated numerically. In this respect the Padé approximants to the logarithm are very well conditioned for ∥X∥ < 0·25. This latter condition can be ensured by using the ‘inverse scaling and squaring’ procedure for evaluating the logarithm.     

On the Convergence of Global Rational Approximants for Stochastic Discrete Event Systems

Difficulties often arise in analyzing stochastic discrete event systems due to the so-called curse of dimensionality. A typical example is the computation of some integer-parameterized functions, where the integer parameter represents the system size or dimension. Rational approximation approach has been introduced to tackle this type of computational complexity. The underline idea is to develop rational approximants with increasing orders which converge to the values of the systems. Various examples demonstrated the effectiveness of the approach. In this paper we investigate the convergence and convergence rates of the rational approximants. First, a convergence rate of order O(1/ ) is obtained for the so-called Type-1 rational approximant sequence. Secondly, we establish conditions under which the sequence of [n/n] Type-2 rational approximants has a convergence rate of order .     

Elimination and resultants. 1. Elimination and bivariate resultants

We discuss the relevance of elimination theory and resultants in computing, especially in computer graphics and CAGD. We list resultant properties to enhance overall understanding of resultants. For bivariate resultants, we present two explicit expressions: the Sylvester and the Bezout determinants. The Sylvester matrix is easier to construct, but the symmetrical Bezout matrix is structurally richer and thus sometimes more revealing. It let Kajiya (1982) observe directly that a line and a bicubic patch could intersect in at most 18 points, not 36 points, as a naive analysis would presume. For Bezier curves, there is an interesting algebraic and geometric relationship between the implicit equation in Bezout determinant form and the properties of end point interpolation and de Casteljau subdivision. When the two polynomials are of different degrees, the Bezout resultant suffers from extraneous factors. Fortunately, we can easily discard these factors. For problems related to surfaces, we need multivariate resultants: in particular, multivariate resultants for three homogeneous polynomials in three variables     

Recursive formulation of the matrix Padé approximation in packed storage

The Extended Euclidean algorithm for matrix Padé approximants is applied to compute matrix Padé approximants when the coefficient matrices of the input matrix polynomial are triangular. The procedure given by Bjarne S. Anderson et al. for packing a triangular matrix in recursive packed storage is applied to pack a sequence of lower triangular matrices of a matrix polynomial in recursive packed storage. This recursive packed storage for a matrix polynomial is applied to compute matrix Padé approximants of the matrix polynomial using the Matrix Padé Extended Euclidean algorithm in packed form. The CPU time and memory comparison, in computing the matrix Padé approximants of a matrix polynomial, between the packed case and the non-packed case are described in detail.