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 when numerically approximating a root of f. 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, 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 the first $n-1$ elements are the subintervals that contain an approximation and the $\mathrm{nth}$ element is the final approximate root.

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 of the numerical approximation on the plot of f.

output=information returns detailed information about the iterative approximations of the root of f.

pointoptions = list

A list of options for the points on the plot. By default, the points are plotted as green circles.

showfunction = true or false

Whether to display f on the plot or not.  By default, this option is set to true.

showlines = true or false

Whether to display lines that accentuate each approximate iteration when output=plot. By default, this option is set to true. To control the vertical lines, see the showverticallines and verticallineoptions options.

showpoints = true or false

Whether to display the points at each approximate iteration on the plot when output=plot. By default, this option is set to true.

showverticallines = true or false

Whether to display the vertical lines at each iterative approximation on the plot when output=plot. By default, this option is set to true.

stoppingcriterion = relative, absolute, function_value

The criterion that the approximation 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 a and b for the two initial approximates and the label $p_n$ for the final approximation, where $n$ is the total number of iterations used to reach the final approximation.  If the stopping criterion is not met, no tickmark will be placed on the last 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 FalsePosition command numerically approximates the roots of an algebraic function, f, using a technique similar to the Secant method, but bracketing is incorporated.

Given an expression f and an initial approximate a, the FalsePosition 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.

The FalsePosition command is a shortcut for calling the Roots command with the method=falseposition option.

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

$f≔x^3-7x^2+14x-6\&colon;$

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

$3.014898139$

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

$\left[2.7\&comma;3.2\right],\left[2.7\&comma;3.100884956\right],\left[2.7\&comma;3.041032849\right],3.014898139$

$\mathrm{FalsePosition}\left(f\&comma;x=\left[2.7\&comma;3.2\right]\&comma;\mathrm{tolerance}={10}^{-2}\&comma;\mathrm{stoppingcriterion}=\mathrm{absolute}\right)$

$3.005163289$

$\mathrm{FalsePosition}\left(f\&comma;x=\left[3.2\&comma;4.0\right]\&comma;\mathrm{output}=\mathrm{animation}\&comma;\mathrm{tolerance}={10}^{-2}\&comma;\mathrm{stoppingcriterion}=\mathrm{function\_value}\right)$

$\mathrm{FalsePosition}\left(f\&comma;x=\left[2.9\&comma;3.1\right]\&comma;\mathrm{output}=\mathrm{plot}\right)$