numerically approximate the real roots of an expression using the fixed point iteration method - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Education : Student Package : Numerical Analysis : Visualization : Student/NumericalAnalysis/FixedPointIteration

Student[NumericalAnalysis][FixedPointIteration] - numerically approximate the real roots of an expression using the fixed point iteration method

Calling Sequence

FixedPointIteration(f, x=a, opts)

FixedPointIteration(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 FixedPointIteration command numerically approximates the roots of an algebraic function, f by converting the problem to a fixed-point problem.

• 

Given an expression f and an initial approximate a, the FixedPointIteration 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.

• 

The first argument f may be substituted with an option of the form fixedpointiterator = fpexpr. See Notes.

• 

The FixedPointIteration command is a shortcut for calling the Roots command with the method=fixedpointiteration 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:=xcosx:

FixedPointIterationf,x=1.0,tolerance=102

0.7414250866

(1)

FixedPointIterationf,x=1.0,tolerance=102,output=sequence,maxiterations=20

1.0,0.5403023059,0.8575532158,0.6542897905,0.7934803587,0.7013687737,0.7639596829,0.7221024250,0.7504177618,0.7314040424,0.7442373549,0.7356047404,0.7414250866

(2)

FixedPointIterationf,x=1.0,tolerance=102,output=plot,stoppingcriterion=function_value,maxiterations=20

FixedPointIterationf,x=1.0,tolerance=103,output=animation,stoppingcriterion=absolute,maxiterations=20

To find a root of fx=x22x3 using the fixed-point iterator function gx=2x+3, use the fixedpointiterator = g option.

g:=2x+312:

FixedPointIterationfixedpointiterator=g,x=4.0,tolerance=102

3.011440019

(3)

FixedPointIterationfixedpointiterator=g,x=4.0,tolerance=102,output=sequence

4.0,3.316624790,3.103747667,3.034385495,3.011440019

(4)

FixedPointIterationfixedpointiterator=g,x=4.0,output=plot,stoppingcriterion=function_value,maxiterations=10

FixedPointIterationfixedpointiterator=g,x=4.0,output=animation,stoppingcriterion=absolute

See Also

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


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam