Results 11  20
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52
On The ArithmeticGeometric Mean Inequality And Its Relationship To Linear Programming, Matrix Scaling, And Gordan's Theorem
, 1998
"... It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the ..."
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It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrary linear form, the new minimization is a nontrivial problem. We prove a generalization of this inequality, also relating it to linear programming, to the diagonal matrix scaling problem, as well as to Gordan's theorem. Linear programming is equivalent to the search for a nontrivial zero of a linear or positive semidefinite quadratic form over the nonnegative points of a given subspace. The goal of this paper is to present these intricate, surprising, and significant relationships, called scaling dualities, and via an elementary proof. Also, to introduce two conceptually simple polynomialtime alg...
Halley's Method as the First Member of an Infinite Family of Cubic Order Rootfinding Methods
, 1998
"... For each natural number m # 3, we give a rootfinding method Hm , with cubic order of convergence for simple roots. However, for quadratic polynomials the order of convergence of Hm is m. Each Hm depends on the input, the corresponding function value, as well as the first two derivatives. We shall ..."
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Cited by 6 (6 self)
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For each natural number m # 3, we give a rootfinding method Hm , with cubic order of convergence for simple roots. However, for quadratic polynomials the order of convergence of Hm is m. Each Hm depends on the input, the corresponding function value, as well as the first two derivatives. We shall refer to this family as Halley Family, since H 3 is the wellknown method of Halley. For all m # 4, the asymptotic error constant of Hm is the same constant. Each Hm is described in terms of determinants that are computable recursively. The Halley Family and their derivativefree variants o#er alternatives to the traditional rootfinding methods, such as secant, Newton, and Muller methods, as well as Halley's method itself. Keywords: Rootfinding, Halley's Method, Order of Convergence. AMS Subject Classification. 65H05. 1 Introduction. In this paper we consider the rootfinding problem for smooth functions of a single variable. For each natural number m # 3, we give a rootfinding metho...
A Basic Family Of Iteration Functions For Polynomial Root Finding And Its Characterizations
 J. of Comp. and Appl. Math
, 1997
"... Let p(x) be a polynomial of degree n 2 with coefficients in a subfield K of the complex numbers. For each natural number m 2, let Lm (x) be the m2m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 0 1, its jth subdiagonal entries are p (j) (x)=j!. For i = 1; ..."
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Cited by 17 (12 self)
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Let p(x) be a polynomial of degree n 2 with coefficients in a subfield K of the complex numbers. For each natural number m 2, let Lm (x) be the m2m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 0 1, its jth subdiagonal entries are p (j) (x)=j!. For i = 1; 2, let L (i) m (x) be the matrix obtained from Lm (x) by deleting its first i rows and its last i columns. L (1) 1 (x) j 1. Then, the function Bm (x) = x 0 p(x) det(L (1) m01 (x))=det(L (1) m (x)) is a member of S(m; m + n 0 2), where for any M m, S(m; M) is the set of all rational iteration functions such that for all roots ` of p(x) , g(x) = ` + P M i=m fl i (x)(` 0 x) i , with fl i (x)'s also rational and welldefined at `. Given g 2 S(m; M), and a simple root ` of p(x), g (i) (`) = 0, i = 1; : : : ; m0 1, and fl m (`) = (01) m g (m) (`)=m!. For Bm (x) we obtain fl m (`) = (01) m det(L (2) m+1 (`))=det(L (1) m (`)). For m = 2 and 3, Bm (x) coincides with Newton'...
Approximation of Polynomial Root Using a Single Input and the Corresponding Derivative Values
, 1998
"... A new formula for the approximation of root of polynomials with complex coe#cients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the c ..."
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Cited by 9 (8 self)
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A new formula for the approximation of root of polynomials with complex coe#cients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the corresponding derivative values. Each element of the sequence is defined in terms of the quotient of two determinants, computable via a recursive formula. Convergence is proved by deriving an explicit error estimate. For special polynomials explicit neighborhoods and error estimates are derived that depend only on the initial error. In particular, the latter applies to the approximation of root of numbers. The proof of convergence utilizes a family of iteration functions, called the Basic Family; a nontrivial determinantal generalization of Taylor's theorem; a lower bound on determinants; Gerschgorin's theorem and Hadamard's inequality; as well as several new key results. The convergence resu...
A Lower Bound On Determinants From Linear Programming
, 1997
"... We derive a lower bound on determinants, utilizing a theorem of linear programming. Let l and u be positive lower and upper bounds on the moduli of the eigenvalues of a real or complex invertible matrix A. If the modulus of the trace of A is at least nl, n the dimension of A, we derive a better low ..."
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Cited by 4 (4 self)
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We derive a lower bound on determinants, utilizing a theorem of linear programming. Let l and u be positive lower and upper bounds on the moduli of the eigenvalues of a real or complex invertible matrix A. If the modulus of the trace of A is at least nl, n the dimension of A, we derive a better lower bound on the determinant of A than the trivial lower bound, l n . In particular, given a diagonally dominant real matrix with positive diagonal entries, our lower bound is applicable and improves the lower bound derivable from Gerschgorin's circle theorem. We describe the application of the lower bound as an auxiliary result in representing square roots, the number ß, and e x , as the limiting quotient of Toeplitz determinants. Keywords: Eigenvalues, Linear Programming. AMS Subject Classification. 65F40, 65H17, 90C05. 1. Introduction. While there are upper bounds on the modulus of matrix determinants, e.g. from Hadamard's determinant inequality, nontrivial lower bounds are difficul...
A Computational Comparison of the First Nine Members of a Determinantal Family of Rootfinding Methods
"... For each natural number m greater than one, and each natural number k less than or equal to m, there exists a rootfinding iteration function, B (k) m defined as the ratio of two determinants that depend on the first m  k derivatives of the given function. This infinite family is derived in [4] ..."
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Cited by 5 (5 self)
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For each natural number m greater than one, and each natural number k less than or equal to m, there exists a rootfinding iteration function, B (k) m defined as the ratio of two determinants that depend on the first m  k derivatives of the given function. This infinite family is derived in [4] and its order of convergence is analyzed in [5]. In this paper we give a computational study of the first nine rootfinding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials B (k1) m is more e#cient than B (k) m , but as the degree increases, B (k) m becomes more e#cient than B (k1) m . The most e#cient of the nine methods is B (4) 4 , having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least e#cient method. Keywords: Polynomial Zeros, Order of Convergence. AMS Subject Classification. 65H05, ...
Approximating the diameter of a set of points in the Euclidean
 C. Faloutsos and
, 1989
"... Given a set P with n points in R li, its diameter d, is the maximum of the Euclidean distances between its points. We describe an algorithm that in m < n iterations obtains r, < rs <.. < r,,, < d,, < min ( fir,, dr,,,). For k fixed, the cost of each iteration is O(n). In particula ..."
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Cited by 7 (0 self)
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Given a set P with n points in R li, its diameter d, is the maximum of the Euclidean distances between its points. We describe an algorithm that in m < n iterations obtains r, < rs <.. < r,,, < d,, < min ( fir,, dr,,,). For k fixed, the cost of each iteration is O(n). In particular, the first approximation r, is within fi of dp, independent of the dimension k.
EDITED BY
, 2005
"... Available electronically at www.ams.org/mcom / Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics ..."
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Available electronically at www.ams.org/mcom / Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology. Reviews of books in areas related to computational mathematics are also included. Submission information. See Information for Authors at the end of this issue. Publisher Item Identifier. The Publisher Item Identifier (PII) appears at the top of the first page of each article published in this journal. This alphanumeric string of characters uniquely identifies each article and can be used for future cataloging, searching, and electronic retrieval. Postings to the AMS website. Articles are posted to the AMS website individually after proof is returned from authors and before appearing in an issue. Subscription information. Mathematics of Computation is published quarterly. Beginning in January 1996 Mathematics of Computation is accessible from www.ams.org/
Polynomiography: A New Intersection between Mathematics and Art 1
"... Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions. ” An individual image is called a “polynomiograph.” The word p ..."
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Cited by 3 (2 self)
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Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions. ” An individual image is called a “polynomiograph.” The word polynomiography is a combination of the word “polynomial ” and the suffix “graphy. ” It is meant to convey the idea that it represents a certain graph of polynomials, but not in the usual sense of graphing, say a parabola for a quadratic polynomial. Polynomiographs are obtained using algorithms requiring the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial, it is possible to obtain beautiful images on a laptop computer in less time than a TV commercial. Polynomials form a fundamental class of mathematical objects with diverse applications; they arise in devising algorithms for such mundane task as multiplying two numbers, much faster than the ordinary way we have all learned to do this task (FFT). According to the Fundamental Theorem of Algebra, a polynomial of degree n, with real or complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of the zeros of polynomials is a problem that was known to Sumerians (third millennium B.C.). This problem
On The Order Of Convergence Of A Determinantal Family Of RootFinding Methods
 BIT
, 1997
"... For each natural number m greater than one, and each natural number k less than or equal to m, there exists a rootfinding iteration function, defined as the ratio of two determinants that depend on the first m \Gamma k derivatives of the given function, and for k = 1 are Toeplitz determinants. In t ..."
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Cited by 5 (4 self)
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For each natural number m greater than one, and each natural number k less than or equal to m, there exists a rootfinding iteration function, defined as the ratio of two determinants that depend on the first m \Gamma k derivatives of the given function, and for k = 1 are Toeplitz determinants. In this paper we analyze the order of convergence of this fundamental family. For fixed m, as k increases, the order decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order increases in m. The asymptotic error constant is defined in terms of Toeplitz determinants. Newton's method, Halley's method, and their multipoint versions are members of the family. Keywords: Iteration Functions, Roots, Taylor's Theorem, Newton's Method, Halley's Method, Interpolation, Generalized Fibonacci Numbers. AMS Subject Classification. 65H05, 65D05, 30C15, 11B39. 1. Introduction. In [24] it is shown that for each natural number m gre...
Results 11  20
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52