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A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion

Jum E., Kobayashi K.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 2

201--220

This paper establishes a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift, where the time change is given by a general inverse subordinator. The scheme involves two types of errors: one generated by application of the Euler-Maruyama scheme and the other ascribed to simulation of the inverse subordinator. With the two errors carefully examined, the orders of strong and weak convergence are established. In particular, an improved error estimate for the Euler-Maruyama scheme is derived, which is required to guarantee the strong convergence. Numerical examples are attached to support the convergence results.

A Note on the Singh Six-order Variant of Newton’s Method

Marciniak A., Wolf M.

Computational Methods in Science and Technology

2015

Vol. 21, No. 4

261--261

In 2009 in this journal it was published the paper of M. K. Singh [1], in which the author presented a six-order variant of Newton’s method. Unfortunately, in this paper there were a number of printer errors and a serious error in the proof of theorem on the order of the method proposed. Therefore, we have opted for presenting the correct proof of this theorem.

A Six-order Variant of Newton’s Method for Solving Nonlinear Equations

Singh M. K.

Computational Methods in Science and Technology

2009

Vol. 15, nr 2

185-193

A new variant of Newton's method based on contra harmonic mean has been developed and its convergence properties have been discussed. The order of convergence of the proposed method is six. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first order derivatives per iteration. This implies that efficiency index of our method is 1.5651. The proposed method is comparable with the methods of Parhi, and Gupta [15] and that of Kou and Li [8]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev-Halley type methods. The efficiency of the method is tested on a number of numerical examples. It is observed that our method takes lesser number of iterations than Newton’s method and the other third order variants of Newton´s method. In comparison with the sixth order methods, it behaves either similarly or better for the examples considered.