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 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 ModifiedNewton command is a shortcut for calling the Roots command with the method=modifiednewton option.

Examples

withStudent[NumericalAnalysis]:

f:=ⅇx+2x+2cosx6:

ModifiedNewtonf,x=2.0,tolerance=102

1.829383413

(1)

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

2.0,1.805613641,1.828908757,1.829383413

(2)

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

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

See Also

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


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