The Arithmetic - Geometric Mean and Fast Computation of Elementary Functions
نویسندگان
چکیده
We produce a self contained account of the relationship between the Gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for r is one of the by-products.
منابع مشابه
A remark on the means of the number of divisors
We obtain the asymptotic expansion of the sequence with general term $frac{A_n}{G_n}$, where $A_n$ and $G_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$, with $d(n)$ denoting the number of positive divisors of $n$. Also, we obtain some explicit bounds concerning $G_n$ and $frac{A_n}{G_n}$.
متن کاملFast Multiprecision Evaluation of Series of Rational Numbers
We describe two techniques for fast multiple-precision evaluation of linearly convergent series, including power series and Ramanujan series. The computation time for N bits is O((logN)M(N)), whereM(N) is the time needed to multiply twoN -bit numbers. Applications include fast algorithms for elementary functions, π, hypergeometric functions at rational points, ζ(3), Euler’s, Catalan’s and Apéry...
متن کاملComputing the Matrix Geometric Mean of Two HPD Matrices: A Stable Iterative Method
A new iteration scheme for computing the sign of a matrix which has no pure imaginary eigenvalues is presented. Then, by applying a well-known identity in matrix functions theory, an algorithm for computing the geometric mean of two Hermitian positive definite matrices is constructed. Moreover, another efficient algorithm for this purpose is derived free from the computation of principal matrix...
متن کاملA new and efficient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers
There are certain problems in the subtraction operator, division operator and obtaining the membership functions of operators and above all, dependence effect in the fuzzy arithmetic operations using the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of α cuts). In this regard, this paper provide a new method regarding the effective pra...
متن کاملA Note on the First Geometric-Arithmetic Index of Hexagonal Systems and Phenylenes
The first geometric-arithmetic index was introduced in the chemical theory as the summation of 2 du dv /(du dv ) overall edges of the graph, where du stand for the degree of the vertex u. In this paper we give the expressions for computing the first geometric-arithmetic index of hexagonal systems and phenylenes and present new method for describing hexagonal system by corresponding a simple g...
متن کامل