numerically approximate the real roots of an expression using the modified Newton's method - Maple Help

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Student[NumericalAnalysis][ModifiedNewton] - numerically approximate the real roots of an expression using the modified Newton's method

 Calling Sequence ModifiedNewton(f, x=a, opts) ModifiedNewton(f, a, opts)

Parameters

 f - algebraic; expression in the variable x representing a continuous function x - name; the independent variable of f a - numeric; the initial approximate root opts - (optional) equation(s) of the form keyword=value, where keyword is one of fixedpointiterator, functionoptions, lineoptions, maxiterations, output, pointoptions, showfunction, showlines, showpoints, showverticalline, stoppingcriterion, tickmarks, caption, tolerance, verticallineoptions, view; the options for approximating the roots of f

Description

 • The ModifiedNewton command numerically approximates the roots of an algebraic function, f, using a variation on the classical Newton-Raphson method. If the classical Newton-Raphson method produces only a linearly convergent sequence toward a root of multiplicity m > 1, then this "accelerated" Newton-Raphson method can produce a quadratically convergent sequence toward that root.
 • Given an expression f and an initial approximate a, the ModifiedNewton command computes a sequence ${p}_{k}$, $k$=$0..n$, of approximations to a root of f, where $n$ is the number of iterations taken to reach a stopping criterion. For sufficiently well-behaved functions and sufficiently good initial approximations, the convergence of ${p}_{k}$ toward the exact root is quadratic.
 • The ModifiedNewton command is a shortcut for calling the Roots command with the method=modifiednewton option.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
 > $f:={ⅇ}^{x}+{2}^{-x}+2\mathrm{cos}\left(x\right)-6:$
 > $\mathrm{ModifiedNewton}\left(f,x=2.0,\mathrm{tolerance}={10}^{-2}\right)$
 ${1.829383413}$ (1)
 > $\mathrm{ModifiedNewton}\left(f,x=2.0,\mathrm{tolerance}={10}^{-2},\mathrm{output}=\mathrm{sequence}\right)$
 ${2.0}{,}{1.805613641}{,}{1.828908757}{,}{1.829383413}$ (2)
 > $\mathrm{ModifiedNewton}\left(f,x=2,\mathrm{output}=\mathrm{plot},\mathrm{stoppingcriterion}=\mathrm{function_value}\right)$
 > $\mathrm{ModifiedNewton}\left(f,x=1.3,\mathrm{output}=\mathrm{animation},\mathrm{stoppingcriterion}=\mathrm{absolute}\right)$