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9 Fourier Bounds for the Newton-Raphson Iteration

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62695-1
  • Mathematics


Publisher Summary This chapter in discussion of Fourier bounds for the Newton–Raphson iteration considers a function f(x) in the interval J0—where x inclusively lies between x0 and y0 and J0 being an interval on the X-axis. In this discussion, it is assumed that the “Fourier conditions” are satisfied in J0. The sequence xν (ν = 0,1,…) is defined by the Newton–Raphson formula. The sequence yν introduced by Fourier is defined in the chapter. From the relation obtained in the course of discussion it is evident that the rule for diminishing of the distance between xν and yν is quadratic; also that the xν tend monotonically to a number ζ ≤ ζ and the yν to a number η ≥ ζ. The limitation of ζ by yν would be sufficiently accurate if λν + 1 have been available. The chapter, however, shows that λν tends to infinity and even λν+1+ 2 tends to 1. As [(1+λν/λν)] –t > 0 for 0 ≤t ≤ 1, the generalized mean value theorem of the integral calculus is applied in the current study.

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