numerically approximate the real roots of an expression using Steffensen's method
Steffensen(f, x=a, opts)
Steffensen(f, a, opts)
algebraic; expression in the variable x representing a continuous function
name; the independent variable of f
numeric; the initial approximate root
(optional) equation(s) of the form keyword=value, where keyword is one of fixedpointiterator, functionoptions, lineoptions, maxiterations, output, pointoptions, showfunction, showlines, showpoints, stoppingcriterion, tickmarks, caption, tolerance, verticallineoptions, view; the options for approximating the roots of f
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 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.
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 : pn−pn−1pn < tolerance
absolute : pn−pn−1 < tolerance
function_value : f⁡pn < 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 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 Δ2 technique of accelerating sequential convergence.
Given an expression f and an initial approximate a, the Steffensen 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 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⁡x=0 to a problem of finding a fixed point for the function g⁡x, where g⁡x=x−f⁡x and f⁡x is specified by f and x.
The user can specify a custom iterator function g⁡x by omitting the first argument f and supplying the fixedpointiterator = g option. The right-hand side expression g specifies a function g⁡x, and this procedure will aim to find a root to f⁡x = x−g⁡x=0 by way of solving the fixed-point problem g⁡x=x.
When output = plot or output = animation is specified, both the function f⁡x and the fixed-point iterator function g⁡x will be plotted and correspondingly labelled.
The tolerance option, when stoppingcriterion = function_value, applies to the function f⁡x in the root-finding form of the problem.
f ≔ x23−1:
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