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Approximation by Rational Functions
Journal of the London Mathematical Society, 1977This paper contains eight theorems on the rational approximation of \(e^{-x}\) . We cite one of them by way of an example: ''Let \(p(x)\) and \(q(x)\) be any polynomials of degress at most \(n-1\) where \(n\geq 2\). Then we have \[ \left\|e^{-x}-\frac{p(x)}{q(x)}\right\|_{l_{\infty}(N)}\geq\frac{(e-1)^ne^{-4n}2^{-7n}}{n(3+2\sqrt2)^{n-1}}.'', \] (\(N ...
Erdős, Paul +2 more
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Evaluation of a rational function
Numerical Algorithms, 1992The aim of this paper is to present four algorithms for the evaluation of a rational function. (Horner's method, direct evaluation through a continued fraction representation or evaluation by a subsequent process that uses three-term recurrences.) The algorithms are presented and demonstrated. They are implemented using a parallel architecture (a four-
Alen George Law +3 more
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Evaluation of Rational Functions
1972In the first part of this paper the complexity (with respect to multiplication and division) of a general continued fraction and of arbitrary quadratic forms is determined with the help of Pan’s method. In the second part some results on avoiding division and on the multiplication of matrices from an algebraic group are reported.
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Representations of rational functions
24th Annual Symposium on Foundations of Computer Science (sfcs 1983), 1983Fast parallel algorithms for various problems in algebraic computation are presented. Two of the algorithms convert the coefficient representation of a rational function into a base representation, and vice versa. Combining them yields an algorithm which converts the representation of a rational function in one base of polynomials into that in another ...
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Minimization of Rational Word Functions
SIAM Journal on Computing, 1991Summary: Rational functions from a free monoid into another are characterized be the finiteness of the index of some congruence naturally associated with the function. A sequential bimachine is constructed computing the function, which is completely canonical, and in some sense minimal. This generalizes the Nerode criterion and the minimal automaton of
Reutenauer, Christophe +1 more
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Rational Functions and Rational Maps
1992Let X ⊂ A n be an irreducible affine variety. Since its coordinate ring A(X) is an integral domain, we can form its quotient field; this is called the rational function field of X and is usually denoted K(X); its elements are called rational functions on X.
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Zeta functions of rational functions are rational
1994Let \(f\) be a rational function of degree \(d\) in the Riemann sphere where \(d \geq 2\). Let \(N_ n\) denote the number of distinct fixed points of the \(n\)th iterate \(f^ n\) of \(f\). A fixed point \(z\) of \(f^ n\) whose multiplier \((f^ n)'(z)\) is equal to 1 is called a parabolic fixed point of some \(f^ n\), with \(f^ p(z)=z\), is called a ...
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