Results 221 to 230 of about 18,915 (259)

A noncontinuous generalization of the arithmetic–geometric mean

Applied Mathematics and Computation, 2001
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Aniko Ekart, S Z Nemeth
exaly   +4 more sources

The arithmetic-geometric mean of Gauss (1984)

2016
Paper 3: David A. Cox, “The arithmetic-geometric mean of Guass,” L’Enseignement Mathematique, vol. 30 (1984), p. 275–330. Reprinted by permission.
Cox David A
exaly   +2 more sources

Comparison of Arithmetic, Geometric, and Harmonic Means

Mathematical Notes, 2021
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openaire   +1 more source

On arithmetic-geometric-mean polynomials

International Journal of Mathematical Education in Science and Technology, 2016
ABSTRACTWe study here an aspect of an infinite set P of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of P for which every element may be expressed as a finite sum of squares of real polynomials.
Martin Griffiths, Des MacHale
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Notes on the arithmetic–geometric mean inequality

Aequationes mathematicae
Starting with the scalar inequality \[ |(ab)^{1/2}x+(ab)^{-1/2}y|\le \frac{|ax + b^{-1}y|+|bx + a^{-1}y|}{2}, \] where \(a, b\) are positive real numbers and \(x, y\) are complex numbers, the authors present matrix versions of it and some generalizations.
Al-Natoor, Ahmad, Hirzallah, Omar, Kittaneh, Fuad   +2 more
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The Arithmetic, Geometric and Harmonic Means

1988
This chapter is devoted to the properties and inequalities of the classical arithmetic, geometric and harmonic means. In particular the basic inequality between these means, the Geometric Mean-Arithmetic Mean Inequality, is discussed at length with many proofs being given.
P. S. Bullen, D. S. Mitrinović, P. M. Vasić   +2 more
openaire   +1 more source

The Arithmetic-Geometric Mean of Gauss

1997
This paper is an expository account of the arithmetic-geometric mean M(a,b) of two numbers a,b. For \(a,b>0\) define \(a_ 0=a\), \(b_ 0=b\) and \(a_{n+1}=(a_ n+b_ n)/2,\quad b_{n+1}=(a_ nb_ n)^{1/2},\quad n=0,1,2,\ldots.\) It follows by elementary methods that the two sequences \(a_ n\), \(b_ n\) have a common limit M(a,b).
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An approximation to the arithmetic-geometric mean

The Mathematical Gazette, 2014
Summary: Given positive numbers \(a>b\), consider the `agm iteration' given by \(a_0=a\), \(b_0=b\) and \[ a_{n+1}=\frac{1}{2}(a_n+b_n), \quad b_{n+1}=(a_nb_n)^{1/2}. \] At each stage, the two new numbers are the arithmetic and geometric means of the previous two.
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A Harmonic Measure Interpretation of the Arithmetic–Geometric Mean

The American Mathematical Monthly, 2007
(2007). A Harmonic Measure Interpretation of the Arithmetic–Geometric Mean. The American Mathematical Monthly: Vol. 114, No. 7, pp. 610-622.
Byron L. Walden, Lesley A. Ward
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More Matrix Forms of the Arithmetic-Geometric Mean Inequality

SIAM Journal on Matrix Analysis and Applications, 1993
The authors prove the following arithmetic-geometric mean inequality: \(2||| A^* XB||| \leq ||| AA^* X+XBB^*|||\) for arbitrary \(n\times n\) matrices \(A\), \(B\), \(X\). They also show that the real function \(f(p):=||| A^{1+p} XB^{1-p}+A^{1-p} XB^{1+p}|||\), \(A,B\geq 0\) is convex on \([-1,1]\) and takes its minimum at \(p=0\), where \(|||\cdot|||\)
Rajendra Bhatia, Chandler Davis
openaire   +3 more sources