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 Δ2 technique of accelerating sequential convergence.

• 

Given an expression f and an initial approximate a, the Steffensen command computes a sequence pk, 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 pk 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 fx=0 to a problem of finding a fixed point for the function gx, where gx=xfx and fx is specified by f and x.

• 

The user can specify a custom iterator function gx by omitting the first argument f and supplying the fixedpointiterator = g option. The right-hand side expression g specifies a function gx, and this procedure will aim to find a root to fx = xgx=0 by way of solving the fixed-point problem gx=x.

  

When output = plot or output = animation is specified, both the function fx and the fixed-point iterator function gx will be plotted and correspondingly labelled.

  

The tolerance option, when stoppingcriterion = function_value, applies to the function fx in the root-finding form of the problem.

Examples

withStudent[NumericalAnalysis]:

f:=x231:

Steffensenf,x=2.0,tolerance=102

1.732049797

(1)

Steffensenf,x=2.0,tolerance=102,output=sequence

2.0,1.727272728,1.732049797

(2)

Steffensenf,x=2,output=plot,stoppingcriterion=function_value

Steffensenf,x=1.3,output=animation,stoppingcriterion=absolute

See Also

Student[NumericalAnalysis], Student[NumericalAnalysis][Roots], Student[NumericalAnalysis][VisualizationOverview]


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