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Generalization Of Taylor's Theorem And Newton's Method Via A New Family Of Determinantal Interpolation Formulas
 J. of Comp. and Appl. Math
, 1997
"... The general form of Taylor's theorem gives the formula, f = Pn +Rn , where Pn is the Newton 's interpolating polynomial, computed with respect to a confluent vector of nodes, and Rn is the remainder. When f 0 6= 0, for each m = 2; : : : ; n + 1, we describe a "determinantal interpo ..."
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Cited by 12 (12 self)
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The general form of Taylor's theorem gives the formula, f = Pn +Rn , where Pn is the Newton 's interpolating polynomial, computed with respect to a confluent vector of nodes, and Rn is the remainder. When f 0 6= 0, for each m = 2; : : : ; n + 1, we describe a "determinantal interpolation formula", f = P m;n +R m;n , where P m;n is a rational function in x and f itself. These formulas play a dual role in the approximation of f or its inverse. For m = 2, the formula is Taylor's and for m = 3 it gives Halley's iteration function, as well as a Pad'e approximant. By applying the formulas to Pn , for each m 2, Pm;m\Gamma1 ; : : : ; Pm;m+n\Gamma2 , is a set of n rational approximations that includes Pn , and may provide a better approximation to f , than Pn . Thus each Taylor polynomial unfolds into an infinite spectrum of rational approximations. The formulas also give an infinite spectrum of rational inverse approximations, as well as a family of iteration functions for real or complex ...
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...
AN INFINITE FAMILY OF BOUNDS ON ZEROS OF ANALYTIC FUNCTIONS AND RELATIONSHIP TO SMALE’S BOUND
"... Abstract. Smale’s analysis of Newton’s iteration function induce a lower bound on the gap between two distinct zeros of a given complexvalued analytic function f(z). In this paper we make use of a fundamental family of iteration functions Bm(z), m ≥ 2, to derive an infinite family of lower bounds o ..."
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Abstract. Smale’s analysis of Newton’s iteration function induce a lower bound on the gap between two distinct zeros of a given complexvalued analytic function f(z). In this paper we make use of a fundamental family of iteration functions Bm(z), m ≥ 2, to derive an infinite family of lower bounds on the above gap. However, even for m =2,whereB2(z) coincides with Newton’s, our lower bound is more than twice as good as Smale’s bound or its improved version given by Blum, Cucker, Shub, and Smale. When f(z) isacomplex polynomial of degree n, for small m the corresponding bound is computable in O(n log n) arithmetic operations. For quadratic polynomials, as m increases the lower bounds converge to the actual gap. We show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of f(z). In particular, using the latter result, we show that, given a complex polynomial f(z) =anzn + ···+ a0, ana0 ̸ = 0,foreach m ≥ 2 we can compute upper and lower bounds Um and Lm such that the roots of f(z) lie in the annulus {z: Lm ≤z  ≤Um}. In particular, L2 =
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, ...
On Homogeneous Linear Recurrence Relations and Approximation of Zeros of Complex Polynomials
 Department of Computer Science, Rutgers University
, 2000
"... . Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a  p(a)Dm2 (a)/Dm1 (a)}, where Dm (a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each ..."
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. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a  p(a)Dm2 (a)/Dm1 (a)}, where Dm (a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm (a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m # 2 and an appropriat...
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
New Formulas For Approximation Of Pi And Other Transcendental Numbers
, 1999
"... We derive many new formulas for approximation of pi. The formulas make use of a sequence of iteration functions called the Basic Family; a nontrivial determinantal generalization of Taylor's theorem; other ingredients; as well as several new results presented in the present paper. In one scheme ..."
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We derive many new formulas for approximation of pi. The formulas make use of a sequence of iteration functions called the Basic Family; a nontrivial determinantal generalization of Taylor's theorem; other ingredients; as well as several new results presented in the present paper. In one scheme, one evaluates members of the Basic Family, for an appropriately selected function, all at the same input. This scheme generates almost a fixed and preselected number of digits in each successive evaluation. The computation amounts to the evaluation of a recursive formula and is equivalent to the computation of special Toeplitz matrix determinants. The approximations of pi obtained via this scheme are within simple algebraic extensions of the rational field. In a second scheme, the fixedpoint iteration is applied to any fixed member of the Basic Family, while selecting an appropriate function. In this scheme we prove highorder of convergence from the initial point. We report on some prelimina...