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

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

 Calling Sequence Steffensen(f, x=a, opts) Steffensen(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, stoppingcriterion, tickmarks, caption, tolerance, verticallineoptions, view; the options for approximating the roots of f

Description

 • The Steffensen command numerically approximates the roots of an algebraic function, f, using fixed-point iteration coupled with a slightly modified version of Aitken's ${\mathrm{\Delta }}^{2}$ technique of accelerating sequential convergence.
 • Given an expression f and an initial approximate a, the Steffensen 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 first argument f may be substituted with an option of the form fixedpointiterator = fpexpr. See Notes.
 • The Steffensen command is a shortcut for calling the Roots command with the method=steffensen option.

Notes

 • This procedure first converts the problem of finding a root to the equation $f\left(x\right)=0$ to a problem of finding a fixed point for the function $g\left(x\right)$, where $g\left(x\right)=x-f\left(x\right)$ and $f\left(x\right)$ is specified by f and x.
 • The user can specify a custom iterator function $g\left(x\right)$ by omitting the first argument f and supplying the fixedpointiterator = g option. The right-hand side expression g specifies a function $g\left(x\right)$, and this procedure will aim to find a root to $f\left(x\right)$ = $x-g\left(x\right)=0$ by way of solving the fixed-point problem $g\left(x\right)=x$.
 When output = plot or output = animation is specified, both the function $f\left(x\right)$ and the fixed-point iterator function $g\left(x\right)$ will be plotted and correspondingly labelled.
 The tolerance option, when stoppingcriterion = function_value, applies to the function $f\left(x\right)$ in the root-finding form of the problem.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
 > $f:=\frac{{x}^{2}}{3}-1:$
 > $\mathrm{Steffensen}\left(f,x=2.0,\mathrm{tolerance}={10}^{-2}\right)$
 ${1.732049797}$ (1)
 > $\mathrm{Steffensen}\left(f,x=2.0,\mathrm{tolerance}={10}^{-2},\mathrm{output}=\mathrm{sequence}\right)$
 ${2.0}{,}{1.727272728}{,}{1.732049797}$ (2)
 > $\mathrm{Steffensen}\left(f,x=2,\mathrm{output}=\mathrm{plot},\mathrm{stoppingcriterion}=\mathrm{function_value}\right)$
 > $\mathrm{Steffensen}\left(f,x=1.3,\mathrm{output}=\mathrm{animation},\mathrm{stoppingcriterion}=\mathrm{absolute}\right)$