Student[NumericalAnalysis]
Secant
numerically approximate the real roots of an expression using the secant method
Calling Sequence
Parameters
Options
Description
Notes
Examples
Secant(f, x=[a, b], opts)
Secant(f, [a, b], opts)
f
-
algebraic; expression in the variable x representing a continuous function
x
name; the independent variable of f
a
numeric; one of two initial approximates to the root
b
numeric; the other of the two initial approximates to the root
opts
(optional) equation(s) of the form keyword=value, where keyword is one of functionoptions, lineoptions, maxiterations, output, pointoptions, showfunction, showlines, showpoints, showverticallines, stoppingcriterion, tickmarks, caption, tolerance, verticallineoptions, view; the options for approximating the roots of f
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. 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.
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 : fpn < 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 plot view of the plot when output = plot. See plot/options for more information.
The Secant command numerically approximates the roots of an algebraic function, f, using a technique similar to Newton's method but without the need to evaluate the derivative of f.
Given an expression f and an initial approximate a, the Secant 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.
The Secant command is a shortcut for calling the Roots command with the method=secant option.
The secant method has the limitation that it may diverge when the initial approximates a and b are not sufficiently close to the root.
withStudentNumericalAnalysis:
f≔x3−7x2+14x−6:
Secantf,x=2.7,3.2,tolerance=10−2
3.005775850
Secantf,x=2.7,3.2,tolerance=10−2,output=sequence
2.7,3.2,3.100884956,2.858406793,3.026267866,3.005775850
Secantf,x=2.7,3.2,tolerance=10−2,stoppingcriterion=absolute
2.999670283
To play the following animation in this help page, right-click (Control-click, on Macintosh) the plot to display the context menu. Select Animation > Play.
Secantf,x=3.2,4.0,output=animation,stoppingcriterion=function_value
Secantf,x=2.9,3.1,tolerance=10−3,output=plot
See Also
Student[NumericalAnalysis][Roots]
Student[NumericalAnalysis][VisualizationOverview]
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