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  technique of accelerating sequential convergence.

Given an expression f and an initial approximate a, the Steffensen command computes a sequence , =, of approximations to a root of f, where  is the number of iterations taken to reach a stopping criterion. For sufficiently well-behaved functions and sufficiently good initial approximations, the convergence of  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.

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

This procedure 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 labelled.

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