• 제목/요약/키워드: asymptotic error constant

검색결과 26건 처리시간 0.017초

ASYMPTOTIC ERROR ANALYSIS OF k-FOLD PSEUDO-NEWTON'S METHOD LOCATING A SIMPLE ZERO

  • Kim, Young Ik
    • 충청수학회지
    • /
    • 제21권4호
    • /
    • pp.483-492
    • /
    • 2008
  • The k-fold pseudo-Newton's method is proposed and its convergence behavior is investigated near a simple zero. The order of convergence is proven to be at least k + 2. The asymptotic error constant is explicitly given in terms of k and the corresponding simple zero. High-precison numerical results are successfully implemented via Mathematica and illustrated for various examples.

  • PDF

Asymptotic Characteristics of MSE-Optimal Scalar Quantizers for Generalized Gamma Sources

  • 이재건;나상신
    • 한국통신학회논문지
    • /
    • 제37권5A호
    • /
    • pp.279-289
    • /
    • 2012
  • Characteristics, such as the support limit and distortions, of minimum mean-squared error (MSE) N-level uniform and nonuniform scalar quantizers are studied for the family of the generalized gamma density functions as N increases. For the study, MSE-optimal scalar quantizers are designed at integer rates from 1 to 16 bits/sample, and their characteristics are compared with corresponding asymptotic formulas. The results show that the support limit formulas are generally accurate. They also show that the distortion of nonuniform quantizers is observed to converge to the Panter-Dite asymptotic constant, whereas the distortion of uniform quantizers exhibits slow or even stagnant convergence to its corresponding Hui-Neuhoff asymptotic constant at the studied rate range, though it may stay at a close proximity to the asymptotic constant for the Rayleigh and Laplacian pdfs. Additional terms in the asymptote result in quite considerable accuracy improvement, making the formulas useful especially when rate is 8 or greater.

THE ORDER AND SPEED OF CONVERGENCE FOR THE k-FOLD PSEUDO-OLVER'S METHOD LOCATING A SIMPLE REAL ZERO

  • Kim, Young Ik
    • 충청수학회지
    • /
    • 제19권1호
    • /
    • pp.49-56
    • /
    • 2006
  • A convergence behavior is under investigation near a simple real zero for the k-fold pseudo-Olver's method defined by extending the classical Olver's method. The order of convergence is shown to be at least k+3. The asymptotic error constant is explicitly given in terms of k and the corresponding simple zero. Various numerical examples with a proposed zero-finding algorithm are successfully confirmed with the use of symbolic and computational ability of Mathematica.

  • PDF

A FOURTH-ORDER FAMILY OF TRIPARAMETRIC EXTENSIONS OF JARRATT'S METHOD

  • Kim, Young Ik
    • 충청수학회지
    • /
    • 제25권3호
    • /
    • pp.579-587
    • /
    • 2012
  • A fourth-order family of triparametric extensions of Jarratt's method are proposed in this paper to find a simple root of nonlinear algebraic equations. Convergence analysis including numerical experiments for various test functions apparently verifies the fourth-order convergence and asymptotic error constants.

ERROR ANALYSIS OF k-FOLD PSEUDO-HALLEY'S METHOD FINDING A SIMPLE ZERO

  • Kim, Young Ik
    • 충청수학회지
    • /
    • 제20권1호
    • /
    • pp.11-21
    • /
    • 2007
  • Given a nonlinear function f : $\mathbb{R}{\rightarrow}\mathbb{R}$ that has a simple real zero ${\alpha}$, a new numerical method to be called k-fold pseudo- Halley's method is proposed and it's error analysis is under investigation to confirm the convergence behavior near ${\alpha}$. Under the assumption that f is sufficiently smooth in a small neighborhood of ${\alpha}$, the order of convergence is found to be at least k+3. In addition, the corresponding asymptotic error constant is explicitly expressed in terms of k, ${\alpha}$ and f as well as the derivatives of f. A zero-finding algorithm is written and has been successfully implemented for numerous examples with Mathematica.

  • PDF

ON THE HIGH-ORDER CONVERGENCE OF THE k-FOLD PSEUDO-CAUCHY'S METHOD FOR A SIMPLE ROOT

  • Kim, Young Ik
    • 충청수학회지
    • /
    • 제21권1호
    • /
    • pp.107-116
    • /
    • 2008
  • In this study the k-fold pseudo-Cauchy's method of order k+3 is proposed from the classical Cauchy's method defined by an iteration $x_{n+1}=x_n-{\frac{f^{\prime}(x_n)}{f^{{\prime}{\prime}}(x_n)}}{\cdot}(1-{\sqrt{1-2f(x_n)f^{{\prime}{\prime}}(x_n)/f^{\prime}(x_n)^2}})$. The convergence behavior of the asymptotic error constant is investigated near the corresponding simple zero. A root-finding algorithm with the k-fold pseudo-Cauchy's method is described and computational examples have successfully confirmed the current analysis.

  • PDF

A STUDY ON CONVERGENCE OF EXTENDED LEAP-FROGGING NEWTON'S METHOD LOCATING MULTIPLE ZEROS

  • Geum, Young Hee
    • 충청수학회지
    • /
    • 제22권2호
    • /
    • pp.217-227
    • /
    • 2009
  • Assuming that a given nonlinear function f : $\mathbf{R}{\rightarrow}\mathbf{R}$ has a zero $\alpha$with integer multiplicity $m{\geq}1$ and is sufficiently smooth in a small neighborhood of $\alpha$, we define extended leap-frogging Newton's method. We investigate the order of convergence and the asymptotic error constant of the proposed method as a function of multiplicity m. Numerical experiments for various test functions show a satisfactory agreement with the theory presented in this paper and are throughly verified via Mathematica programming with its high-precision computability.

  • PDF

A THIRD-ORDER VARIANT OF NEWTON-SECANT METHOD FINDING A MULTIPLE ZERO

  • Kim, Young Ik;Lee, Sang Deok
    • 충청수학회지
    • /
    • 제23권4호
    • /
    • pp.845-852
    • /
    • 2010
  • A nonlinear algebraic equation f(x) = 0 is considered to find a root with integer multiplicity $m{\geq}1$. A variant of Newton-secant method for a multiple root is proposed below: for n = 0, 1, $2{\cdots}$ $$x_{n+1}=x_n-\frac{f(x_n)^2}{f^{\prime}(x_n)\{f(x_n)-{\lambda}f(x_n-\frac{f(x_n)}{f^{\prime}(x_n)})\}$$, $$\lambda=\{_{1,\;if\;m=1.}^{(\frac{m}{m-1})^{m-1},\;if\;m{\geq}2$$ It is shown that the method has third-order convergence and its asymptotic error constant is expressed in terms of m. Numerical examples successfully verified the proposed scheme with high-precision Mathematica programming.

A NOTE ON THE PAPER ENTITLED SIXTEENTH-ORDER METHOD FOR NONLINEAR EQUATIONS

  • Kim, Young Ik
    • 충청수학회지
    • /
    • 제25권2호
    • /
    • pp.359-365
    • /
    • 2012
  • The purpose of this paper is to provide some corrections regarding algebraic flaws encountered in the paper entitled "Sixteenth-order method for nonlinear equations" which was published in January of 2010 by Li et al.[9]. Further detailed comments on their error equation are stated together with convergence analysis as well as high-precision numerical experiments.