The Roots command numerically approximates the roots of an algebraic function, f, using the specified method and returns the specified outputs.

Given an expression f and an initial approximate a or a pair of initial approximates [a, b], the Roots command computes a sequence , =, of approximations to a root of f, where  is the number of iterations taken to reach a stopping criterion.

See method in the Options section to see which methods require one initial approximate (and hence the last two calling sequences) and which methods require a pair of initial approximates (and hence the first two calling sequences).

If method = fixedpointiteration or method = steffensen is specified, the first argument f may be substituted with an option of the form fixedpointiterator = fpexpr. See method = fixedpointiteration in the Options section for details.

fixedpointiterator = algebraic (optional)

Can only be specified if method = steffensen or method = fixedpointiteration.

The expression on the right-hand side will be used to generate the fixed-point iteration sequence. If this option is specified, the first argument, f, must be omitted. See the method option under method = fixedpointiteration for more details.

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 for the Newton-Raphson, Modified Newton-Raphson, Secant, Steffensen, and False Position methods and dotted blue for the Bisection and Fixed-Point Iteration methods.

maxiterations = posint

The maximum number of iterations 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

method = newton, modifiednewton, bisection, secant, fixedpointiteration, steffensen, or falseposition

The method used to approximate the root(s) of f numerically.

newton: Newton-Raphson Method

This method requires one initial approximate.

modifiednewton: Modified Newton-Raphson Method

This method requires one initial approximate.

bisection: Bisection Method

This method requires a pair of initial approximates.

secant: Secant Method

This method requires a pair of initial approximates.

fixedpointiteration: Fixed-Point Iteration Method

This method requires one initial approximate.

This method first converts the problem of finding a root to the equation  to a problem of finding a fixed point for the function , where  and  is specified by f and x.

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

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

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

The information in the preceding paragraphs applies to Steffensen's method as well.

steffensen: Steffensen's Method

This method requires one initial approximate.

See the fixed-point iteration method above (fixedpointiteration) for important information that also applies to this method.

falseposition: Method of False Position

This method requires a pair of initial approximates.

By default, the Newton-Raphson Method is used.

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 , =, where, depending on the method, either form successive intervals that bracket the exact root or a sequence of successively more accurate approximate roots.

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.

showlines = truefalse

Whether to display lines that accentuate each approximate iteration when output = plot. This option is effective with every method except for the Modified Newton-Raphson method. By default, this option is set to true.  To control the vertical lines, see the showverticallines and verticallineoptions options.

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.  This option is only effective when method is one of: newton, modifiednewton, secant, or falseposition. By default this option is set to true.

stoppingcriterion = relative, absolute, or function_value

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

relative :  < tolerance

absolute :  < tolerance

function_value :  < 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  (or a and b for two initial approximates) and , where  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 .

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.

Both Newton's method and the secant method have the limitation that they may produce a divergent sequence of approximates if the initial approximates a and b are not sufficiently close to the root.