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Student[NumericalAnalysis]

  

ModifiedNewton

  

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

 

Calling Sequence

Parameters

Options

Description

Examples

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

Options

• 

functionoptions = list

  

A list of options for the plot of the expression f. By default, f is plotted as a solid red line.

• 

lineoptions = list

  

A list of options for the lines on the plot. By default the lines are solid blue.

• 

maxiterations = posint

  

The maximum number of iterations to to perform. The default value of maxiterations depends on which type of output is chosen:

– 

output = value: default maxiterations = 100

– 

output = sequence: default maxiterations = 10

– 

output = information: default maxiterations = 10

– 

output = plot: default maxiterations = 5

– 

output = animation: default maxiterations = 10

• 

output = value, sequence, plot, animation, or information

  

The return value of the function. The default is value.

– 

output = value returns the final numerical approximation of the root.

– 

output = sequence returns an expression sequence pk, k=0..n that converges to the exact root for a sufficiently well-behaved function and initial approximation.

– 

output = plot returns a plot of f with each iterative approximation shown and the relevant information about the numerical approximation displayed in the caption of the plot.

– 

output = animation returns an animation showing the iterations of the root approximation process.

– 

output = information returns detailed information about the iterative approximations of the root of f.

• 

plotoptions = list

  

The final plot options when output = plot or output = animation.

• 

pointoptions = list

  

A list of options for the points on the plot. By default, the points are plotted as green circles.

• 

showfunction = truefalse

  

Whether to display f on the plot or not.  By default, this option is set to true.

• 

showpoints = truefalse

  

Whether to display the points at each approximate iteration on the plot when output = plot. By default, this option is set to true.

• 

showverticallines = truefalse

  

Whether to display the vertical lines at each iterative approximation on the plot when output = plot.

• 

stoppingcriterion = relative, absolute, or function_value

  

The criterion that the approximations must meet before discontinuing the iterations. The following describes each criterion:

– 

relative : pnpn1pn < tolerance

– 

absolute : pnpn1 < tolerance

– 

function_value : fpn < tolerance

  

By default, stoppingcriterion = relative.

• 

tickmarks = list

  

The tickmarks when output = plot or output = animation. By default, tickmarks are placed at the initial and final approximations with the labels p0 (or a and b for two initial approximates) and pn, where n is the total number of iterations used to reach the final approximation. See plot/tickmarks for more detail on specifying tickmarks.

• 

caption = string

  

A caption for the plot. The default caption contains general information concerning the approximation. For more information about specifying a caption, see plot/typesetting.

• 

tolerance = positive

  

The error tolerance of the approximation. The default value is 110000.

• 

verticallineoptions = list

  

A list of options for the vertical lines on the plot. By default, the lines are dashed and blue.

• 

view = [realcons..realcons, realcons..realcons]

  

The plot view of the plot when output = plot.  See plot/options for more information.

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&lsqb;NumericalAnalysis&rsqb;&colon;

f&ExponentialE;x&plus;2x&plus;2cosx6&colon;

ModifiedNewtonf&comma;x&equals;2.0&comma;tolerance&equals;102

1.829383413

(1)

ModifiedNewtonf&comma;x&equals;2.0&comma;tolerance&equals;102&comma;output&equals;sequence

2.0&comma;1.805613641&comma;1.828908757&comma;1.829383413

(2)

ModifiedNewtonf&comma;x&equals;2&comma;output&equals;plot&comma;stoppingcriterion&equals;function_value

ModifiedNewtonf&comma;x&equals;1.3&comma;output&equals;animation&comma;stoppingcriterion&equals;absolute

See Also

Student[NumericalAnalysis]

Student[NumericalAnalysis][Roots]

Student[NumericalAnalysis][VisualizationOverview]

 


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