Topic
Partial fraction decomposition
About: Partial fraction decomposition is a research topic. Over the lifetime, 623 publications have been published within this topic receiving 9993 citations. The topic is also known as: partial fraction & partial fraction expansion.
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31 Dec 1935
1,323 citations
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01 Jan 1992TL;DR: This book discusses Polynomials GCD Computation, the construction of bases for Polynomial Ideals, and the Risch Integration Algorithm, which automates the process of solving Systems of Equation.
Abstract: Preface. 1. Introduction to Computer Algebra. 2. Algebra of Polynomials, Rational Functions, and Power Series. 3. Normal Forms and Algebraic Representations. 4. Arithmetic of Polynomial, Rational Functions, and Power Series. 5. Homomorphisms and Chinese Remainder Algorithms. 6. Newton's Iteration and the Hensel Construction. 7. Polynomials GCD Computation. 8. Polynomial Factorization. 9. Solving Systems of Equation. 10. Grobner Bases for Polynomial Ideals. 11. Integration of Rational Functions. 12. The Risch Integration Algorithm. Notation. Index.
680 citations
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TL;DR: This paper takes a new look at numerical techniques for solving parabolic equations by the method of lines by approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace.
Abstract: This paper takes a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus the resulting approximation consists of applying an evolution operator of very small dimension to a known vector, which is, in turn, computed accurately by exploiting high-order rational Chebyshev and Pade approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications and, as a result, the algorithm can easily be parallelized and vectorized. Further parallelism is introduced by expanding the rational approximations into partial fractions. Some relevant approximation and ...
342 citations
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301 citations
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10 Apr 1980TL;DR: In this paper, a Hankel-norm optimal solution for multivariable system reduction problems is presented, with the optimality verified by a complete error analysis, based on the scalar ease approach.
Abstract: This paper represents a first attempt to derive a closed-form (Hankel-norm) optimal solution for multivariable system reduction problems. The basic idea is to extend the scalar ease approach in [5] to deal with the multivariable systems. The major contribution lies in the development of a minimal degree approximation (MDA) theorem and a computation algorithm. The main theorem describes a closed-form formulation for the optimal approximants, with the optimality verified by a complete error analysis. In deriving the main theorem, some useful singular value/vector properties associated with block-Hankel matrices are explored and a key extension theorem is also developed. Imbedded in the polynomial-theoretic derivation of the extension theorem is an efficient approximation algorithm. This algorithm consists of three steps: i) compute the minimal basis solution of a polynomial matrix equation; ii) solve an algebraic Riccati equation; and iii) find the partial fraction expansion of a rational matrix.
288 citations