Continued Fraction Inversion Research Articles

A Computer-Aided Design Toolkit For Continuous/Discrete Systems Based On Continued Fractions

This paper presents new algorithms (1) to generate complete mathematical symbolic expressions of continued fraction expansion coefficients in the s-domain, (2) to determine their numeric values and (3) to perform continued fraction inversion in the s-domain. A software package in PASCAL and LISP to implement these algorithms is developed. In addition, a simple z-domain inversion algorithm used in the computer implementation of bilinear s-z transformation is also included in the software package. These algorithms play an important role in the analysis and synthesis of complex electrical networks and control systems. Especially, the s-domain expansion and inversion algorithms have potential applications in model simplification and system order reductions. the paper also shows that the package, as such, serves as a comprehensive computer-aided analysis and design (CAD) toolkit for both continuous and discrete systems. the software is interactive and runs on computers equipped with a PASCAL or LISP compiler. It is noted that the iterative implementation of these methods using the new continued fraction algorithms saves considerable memory space and processing time. Numerical examples and computer data are given to demonstrate the development of the new algorithms and the usefulness of the software toolkit in the CAD design of continuous and discrete systems.

The computer implementation of bilinear s-z transformation using new continued fraction algorithms

The application of continued fractions to the bilinear transformation of polynomials is introduced. The operations to perform the transformation are continued fraction expansion in the s domain and continued fraction inversion in the z domain. Novel algorithms and computer programs to implement the expansion and the inversion are presented. In particular, the expansion algorithm provides closed forms for the continued fraction quotients in terms of the corresponding rational function coefficients. It is shown that the computer implementation of this algorithm iteratively results in a simple procedure for evaluating the quotients by hand and saves considerable computation time. It is especially noted that the algorithmic character of continued fractions leads to a simple systematic computation scheme. The proposed method of bilinear transformation capitalizes on the knowledge gained in continuous-time stability theory and introduces the topic of z-domain continued fractions and their applications. Numerical examples and computer data are provided. >

Multidimensional continued fraction inversion

A computationally simple algorithm for the inversion of multidimensional (mD) continued fraction expansions is presented. The approach is based on the interpretation of an mD continued fraction expansion as a driving-point admittance. To facilitate the inversion procedure a cyclic function Ti(z1, z2,…, zm) is introduced. Several examples are given for inverting 3-D and 4-D systems to illustrate the efficiency of the algorithm.

A simplified viewpoint of two-dimensional continued fraction inversion based on matrix formulation

A computationally improved version of a recent method for the inversion of a two-dimensional continued fraction is pointed out. The number of multiplications and additions required is reduced, which will allow a more compact computer program to be obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two-dimensional continued fraction inversion

A computationally simple method is presented for the inversion of a two-dimensional (2-D) continued fraction. The procedure is based on the interpretation of a 2-D continued fraction expansion as a driving point admittance and uses results from the theory of two-port networks. The simplicity and the efficiency of the method are illustrated by two examples.

An algorithm for a 2-D continued fraction inversion

This paper indicates that the one-shot algorithm proposed recently by John and Parthasarathy for inverting a one-dimensional (1-D) continued fraction can be extended to the two-dimensional (2-D) case by merely using polynomial coefficient array (PCA) to represent a 2-D polynomial.

Continued fraction expansion and inversion by ‘rearranged’ Routh tables

The Routh algorithm is known to be the simplest method for continued fraction expansion and inversion but it faces a serious limitation when the first column entry of any row of the Routh table becomes zero (but not a zero row). In this paper a remedy for such situations is proposed by rearranging the coefficients of the row in which zero entry has occurred in the first column of the Routh table. All possible cases, where the first column entry of the Routh table may become zero, are discussed for Cauer's first and second forms and are illustrated by numerical examples. The coefficients of the rearranged Routh table are also calculated by the usual Routh algorithm, hence it is equally suitable for digital computation.

A note on continued fraction inversion

An extension of the Routh algorithm [1] to the inversion of continued fraction terminated into a rational transfer function is presented.

Continued fraction inversion and expansion

The algorithm proposed by Parthasarthy and Singh for inverting continued fraction expansion in the Cauer first form is modified. Its use is extended for inverting a number of continued fraction expansions commonly encountered in engineering applications. An algorithm for converting a rational function into continued fraction is also given.

An improved algorithm for continued fraction inversion

An algorithm for the continued fraction inversion is presented, which is an improvement over a previous algorithm [1].

A simplified approach to continued fraction inversion based on matrix formulation

An improved algorithm for the continued fraction inversion based on matrix formulation [1] is presented.

On continued fraction inversion by Routh's algorithm

This note points out that the generalized algorithm proposed recently by Chao et al. for Cauer second form can be applied for inverting a continued fraction in the Cauer first form by merely writing the transfer function as a ratio of two polynomials arranged in the descending powers of " <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> ."

A matrix method for a general continued fraction inversion

A recently developed method, based on A, B, C, and D parameters, deals with a generalized continued fraction interpreting its form as a driving-point admittance. This letter presents another A, B, C, and D matrix method for inverting a general continued fraction assuming its form as a driving-point impedance.

On the inversion of matrix Routh array

Abstract A new matrix which is the product of a chain of factored matrices is constructed in this paper to find the matrix Routh array and the matrix continued fraction inversion for multivariate systems if the partial matrix coefficients are known. A linear transformation matrix ia established from tho matrix Routh array developed to transform the second Cauer matrix form in state-space coordinates to tho phase variable matrix form.

On continued fraction inversion by Routh's algorithm

This correspondence presents an algorithm for inverting a continued fraction given in the Cauer first form, in the light of the method proposed recently by Rao and Lamba for the Cauer second form.

An algorithm for the inversion of continued fractions

A simple procedure for the inversion of a general continued fraction is presented.

Matrix continued fraction expansion and inversion by the generalized matrix Routh algorithm

Abstract A generalized matrix Mouth algorithm is established to expand n matrix transfer function into the matrix continued fraction of three matrix Cauer forms. By the use of the generalized matrix Routh algorithm and state-space techniques. a method is established for performing the matrix continued fraction inversion. The procedure is amenable to digital computation.

A note on continued fraction inversion by Routh's algorithm

This correspondence presents an algorithm for finding continued fraction inversion by Routh's algorithm. A recursive relation is given to implement the algorithm in a very simple way. This approach requires less computation when compared with the existing techniques. The method is demonstrated by means of a numerical example.

A matrix method for continued fraction inversion

Previously used methods for converting a continued fraction function into a rational transfer function have been restricted to a special kind of continued fraction. This letter presents a method for inverting a general fraction; each partial quotient including the remainder could be a polynomial of s or s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> or both.

A chain of factored matrices for Routh array inversion and continued fraction inversion†

A chain of factored matrices is derived for formulating the Routh array if the first column of the array is known. The factored matrices may be used to perform the conversion of a continued fraction into a rational function. A set of tables based on the second Cauer form of continued fraction expansion is also included. These tables are the approximated rational functions for the commonly used irrational function √ and the transcendental function e s .