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High-Order Iterative Methods Repeating Roots a Constructive Recapitulation

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Applied Mathematics 2022年第2期13卷 131-146页

作者： isaac fried Department of Mathematics Boston University Boston MA USA

This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.

关键词： Roots of Nonlinear Equations Multiple Roots Multiplicity Index of a Root Estimation of the Multiplicity Index of a Root High-Order Iterative Methods Root Bracketing Alternatingly Converging Methods Contrarily Converging Methods

Newton’s Method and an Exact Opposite That Average into Halley’s Method

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Applied Mathematics 2017年第10期8卷 1427-1436页

作者： isaac fried Department of Mathematics Boston University Boston MA USA

This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an oscillating super-linear method approaching the targeted root alternatingly from above and from below. Further extension of Newton’s method creates an oppositely converging quadratic counterpart to it. This new method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method. This note examines also the creation of high order iterative methods by a repeated specification of undetermined coefficients.

关键词： Iterative Methods Alternating Methods Opposite Methods Root Bounds Undetermined Coefficients

Newton, Halley, Pell and the Optimal Iterative High-Order Rational Approximation of √<span style='margin-left:-2px;margin-right:2px;border-top:1px solid black'>N</span>

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Applied Mathematics 2018年第7期9卷 861-873页

作者： isaac fried Department of Mathematics Boston University Boston MA USA

In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating rational approximations p/q that are optimal in the sense... 详细信息

关键词： Iterative Methods Super-Linear and Super-Quadratic Methods Square Roots Pell’s Equation Optimal Rational Iterants Root Bounds

Consistency and Stability Issues in the Numerical Integration of the First and Second Order Initial Value Problem

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Applied Mathematics 2019年第8期10卷 676-690页

作者： isaac fried Department of Mathematics Boston University Boston MA USA

In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). Included are remarks on multiple solutions, multi-step methods, effect of initial value perturbations, as well as slowing and advancing the computed motion in second order problems.

关键词： Initial Value Problems Numerical Integration Consistency Stability Multiple Solutions Sensitivity to Initial Conditions Slowing and Advancing the Computed Motion

Weighted Least-Squares for a Nearly Perfect Min-Max Fit

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Applied Mathematics 2017年第5期8卷 645-654页

作者： isaac fried Ye Feng Department of Mathematics Boston University Boston MA USA

In this note, we experimentally demonstrate, on a variety of analytic and nonanalytic functions, the novel observation that if the least squares polynomial approximation is repeated as weight in a second, now weighted, least squares approximation, then this new, second, approximation is nearly perfect in the uniform sense, barely needing any further, say, Remez correction.

关键词： Least Squares-Approximation of Functions Weighted Approximations Nearly Perfect Uniform Fits

Gershgorin and Rayleigh Bounds on the Eigenvalues of the Finite-Element Global Matrices via Optimal Similarity Transformations

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Applied Mathematics 2020年第9期11卷 922-941页

作者： isaac fried Roberto Riganti Chen Yu Department of Mathematics Boston University Boston MA USA

The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenvalue bounding theorem to this matrix inherently fails to foresee its positive definiteness, predictably, and routinely failing to produce a nontrivial lower bound on the least eigenvalue of this, theoretically assured to be positive definite, matrix. Considered here are practical methods for producing an optimal similarity transformation for the finite-elements global stiffness matrix, following which non trivial, realistic, lower bounds on the least eigenvalue can be located, then further improved. The technique is restricted here to the common case of a global stiffness matrix having only non-positive off-diagonal entries. For such a matrix application of the Gershgorin bounding method may be carried out by a mere matrix vector multiplication.

关键词： Finite Elements Global Stiffness Matrix Gershgorin and Rayleigh Computed Upper and Lower Bounds on the Extremal Eigenvalues Similarity Transformations

Linear and Quadratic Leap-and-Land Trajectory Tracking Algorithms

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Consistency and Stability Issues in the Numerical Integration of the First and Second Order Initial Value Problem

Applied Mathematics（应用数学） 2021年第7期

作者： isaac fried

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Gershgorin and Rayleigh Bounds on the Eigenvalues of the Finite-Element Global Matrices via Optimal Similarity Transformations

Applied Mathematics（应用数学） 2019年第8期

作者： isaac fried

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