Function Repository Resource:

NewtonMethodPlot

Source Notebook

Plot the function together with a graphical display of the Newton iterations approximating its root

Contributed by: Dennis M Schneider

ResourceFunction["NewtonMethodPlot"][f,{x,x_min,x_max},pt]

returns a plot of f from x=x_min to x=x_max, together with illustrations representing the iterations of Newton’s root-finding method, starting at x=pt.

Details and Options

Starting with an initial guess x₀ for the zero of the function, the next approximation x₁ is the horizontal intercept of the tangent line at x₀. The process is repeated until a sufficiently accurate value is obtained.

The following options are accepted:

"DrawGraph"

whether to include the graph

"Iterations"

5

the number of iterations to perform

"LineStyle"

graphics directive to specify the style for the line

"PointStyle"

PointSize[0.016]

graphics directive to specify the style for the point

"PrintDisplay"

whether to include the table of iteration values

WorkingPrecision

MachinePrecision

determines the number of digits used in internal calculations

Examples

Basic Examples (4)

Illustrate Newton’s method on a simple function:

In[1]:=

$ResourceFunction["NewtonMethodPlot"][x^3 - 2 + 3 x, {x, -3, 5}, 4]$

Out[1]=

Increase the number of iterations:

In[2]:=

$ResourceFunction["NewtonMethodPlot"][x^3 - 2 + 3 x, {x, -3, 5}, 4, "Iterations" -> 10]$

Out[2]=

Newton’s method can cycle:

In[3]:=

$ResourceFunction[ "NewtonMethodPlot"][-x^3 + 2 x^2 + 5 x + 6, {x, -2, 2}, 1, PlotRange -> All]$

Out[3]=

Newton’s method may "converge" to a cycle:

In[4]:=

$ResourceFunction["NewtonMethodPlot"][-x^3 + 2 x - 2, {x, -2, 2}, .1, "Iterations" -> 15, PlotRange -> All]$

Out[4]=

Possible Issues (2)

Use Surd when necessary:

In[5]:=

$ResourceFunction["NewtonMethodPlot"][x^(1/3) + 1, {x, -2, 0}, -3/2]$

In[6]:=

Out[6]=

Three examples that show that Newton’s method may converge to an unexpected root when it begins or lands near a zero of the derivative:

In[7]:=

$ResourceFunction["NewtonMethodPlot"][Sin[x^2], {x, 0, 3}, 1.28, "Iterations" -> 10]$

Out[7]=

In[8]:=

$ResourceFunction["NewtonMethodPlot"][Sin[x^2], {x, 0, 3}, 1.29, "Iterations" -> 10]$

Out[8]=

In[9]:=

$ResourceFunction["NewtonMethodPlot"][Sin[x^2], {x, 0, 3}, 1.3, "Iterations" -> 10]$

Out[9]=

Publisher

Dennis M Schneider

Related Links

Newton's Method–Wolfram MathWorld

Version History

1.0.0 – 20 August 2019

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License