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 $f\left(x\right)=0$ to a problem of finding a fixed point for the function $g\left(x\right)$, where $g\left(x\right)=x-f\left(x\right)$ and $f\left(x\right)$ is specified by f and x.

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

When output = plot or output = animation is specified, both the function $f\left(x\right)$ and the fixed-point iterator function $g\left(x\right)$ will be plotted and correspondingly labeled.

The tolerance option, when stoppingcriterion = function_value, applies to the function $f\left(x\right)$ 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 $p_k$, $k$=$0..n$, 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 : $\frac{\left|p_n-p_{n-1}\right|}{\left|p_n\right|}$ < tolerance

absolute : $\left|p_n-p_{n-1}\right|$ < tolerance

function_value : $\left|f\left(p_n\right)\right|$ < 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 $p_0$ (or a and b for two initial approximates) and $p_n$, 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 $\frac{1}{10000}$.

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.

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 $p_k$, $k$=$0..n$, of approximations to a root of f, where $n$ 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.

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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\&colon;$

$f≔x^3-7x^2+14x-6$

$f≔x^3-7x^2+14x-6$

$\mathrm{Roots}\left(f\&comma;x=3.2\&comma;\mathrm{tolerance}={10}^{-2}\right)$

$3.416190743$

$\mathrm{Roots}\left(f\&comma;x=\left[2.7\&comma;3.2\right]\&comma;\mathrm{method}=\mathrm{bisection}\&comma;\mathrm{tolerance}={10}^{-2}\&comma;\mathrm{output}=\mathrm{information}\right)$

$\left[\begin{array}{cccccc}n& a_n& b_n& p_n& f\left(p_n\right)& \mathrm{relative\; error}\\ 1& 2.7& 3.2& 2.950000000& 0.054875000& 0.08474576271\\ 2& 2.950000000& 3.2& 3.075000000& \mathrm{-0.063328125}& 0.04065040650\\ 3& 2.950000000& 3.075000000& 3.012500000& \mathrm{-0.012185547}& 0.02074688797\\ 4& 2.950000000& 3.012500000& 2.981250000& 0.019446526& 0.01048218029\\ 5& 2.981250000& 3.012500000& 2.996875000& 0.003144514& 0.005213764338\end{array}\right]$

$f≔x^2-3x-2\&colon;$

$\mathrm{Roots}\left(f\&comma;x=1\&comma;\mathrm{output}=\mathrm{sequence}\right)$

$1.,\mathrm{-3.000000000},\mathrm{-1.222222222},\mathrm{-0.6417233560},\mathrm{-0.5630533137},\mathrm{-0.5615533585},\mathrm{-0.5615528128}$

$\mathrm{Roots}\left(f\&comma;x=0.3\&comma;\mathrm{method}=\mathrm{modifiednewton}\&comma;\mathrm{output}=\mathrm{plot}\&comma;\mathrm{stoppingcriterion}=\mathrm{absolute}\right)$

$\mathrm{Roots}\left(f\&comma;x=0.3\&comma;\mathrm{method}=\mathrm{modifiednewton}\&comma;\mathrm{output}=\mathrm{plot}\&comma;\mathrm{stoppingcriterion}=\mathrm{absolute}\&comma;\mathrm{verticallineoptions}=\left[\mathrm{color}=\mathrm{gold}\&comma;\mathrm{linestyle}=\mathrm{dash}\right]\right)$

$\mathrm{Roots}\left(f\&comma;x=\left[1.3\&comma;5\right]\&comma;\mathrm{method}=\mathrm{secant}\&comma;\mathrm{output}=\mathrm{animation}\&comma;\mathrm{stoppingcriterion}=\mathrm{function\_value}\&comma;\mathrm{tickmarks}=\left[5\&comma;5\right]\right)$