fixedpointiterator = algebraic (optional)

An expression that will be used to generate the fixed-point iteration sequence. If this option is specified, the first argument, f, must be omitted. See the Notes section 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.

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 $p_k$, $k$=$0..n$ 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.

showlines = truefalse

Whether to display lines that accentuate each approximate iteration when output = plot. 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.

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

The Steffensen command numerically approximates the roots of an algebraic function, f, using fixed-point iteration coupled with a slightly modified version of Aitken's ${\mathrm{\Delta }}^2$ technique of accelerating sequential convergence.

Given an expression f and an initial approximate a, the Steffensen 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. For sufficiently well-behaved functions and sufficiently good initial approximations, the convergence of $p_k$ toward the exact root is quadratic.

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

The Steffensen command is a shortcut for calling the Roots command with the method=steffensen option.

This procedure 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 labelled.

The tolerance option, when stoppingcriterion = function_value, applies to the function $f\left(x\right)$ in the root-finding form of the problem.

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

$f≔\frac{x^2}{3}-1\&colon;$

$\mathrm{Steffensen}\left(f\&comma;x=2.0\&comma;\mathrm{tolerance}={10}^{-2}\right)$

$1.732049797$

$\mathrm{Steffensen}\left(f\&comma;x=2.0\&comma;\mathrm{tolerance}={10}^{-2}\&comma;\mathrm{output}=\mathrm{sequence}\right)$

$2.0,1.727272728,1.732049797$

$\mathrm{Steffensen}\left(f\&comma;x=2\&comma;\mathrm{output}=\mathrm{plot}\&comma;\mathrm{stoppingcriterion}=\mathrm{function\_value}\right)$

$\mathrm{Steffensen}\left(f\&comma;x=1.3\&comma;\mathrm{output}=\mathrm{animation}\&comma;\mathrm{stoppingcriterion}=\mathrm{absolute}\right)$