Activities

Numerical methods of solving equations - Newton's method - complex numbers - Newton Basins

Newton's method also works for complex functions.

Of course, the concept of drawing a tangent is no longer appropriate for a function of a complex variable.

Cubic equation with three real solutions

Consider the cubic equation with solutions 0, -1 and 1:

The formula for Newton's method is given by:
The initial guess can be any complex number. Shown below are four starting numbers that are close together on the Argand diagram (complex plane) but whose sequences do not all converge to the same solution.
 
z0
1+1.5i
1.1+1.5i
1.2+1.5i
1.3+1.5i
solution
0
-1
0
1

Find starting numbers whose sequences converge to the different solutions (example).

Newton Basins

Those starting numbers in the complex plane that lead to the same solution are said to lie in a Newton Basin. If each basin is painted the same colour, then interesting fractal patterns emerge. If there are any starting numbers that do not lead to a solution, the points can be left uncoloured.

For more details, see Newton Basins by David Joyce.
 
Returning to the equation :

A Newton basin diagram is shown opposite (larger version).

The diagram represents points:
   x from -2 to 2, y from -2 to 2

Which colour represents each solution 0, -1 and 1?

Explain the results for the starting numbers:
   1+1.5i, 1.1+1.5i, 1.2+1.5i, 1.3+1.5i 
 

What is the equation for this diagram?

The diagram represents points:
   x from -2 to 2, y from -2 to 2

Compare the diagram with that for .

Check your answer with the Newton Basin generator.
   OR
Download Dynamical Systems software.
(NB. "Save" feature does not work.)

Newton Basin - a fractal

Look at the Newton Basin diagram for  at the point (0.5,0). Do you see a green blob? Use the Newton Basin generator to keep zooming in. What do you notice?

Pick any point on the boundary between two basins. Keep zooming in. No matter how closely you zoom in, you should see all three basins in the picture.

Cubic equation with real coefficients - one real solution

Find a cubic function the graph of which intersects the x-axis at only one point.

Each of the coefficients a,b,c,d should be a real number (a not 0).

Use Newton's method to solve the equation. Produce a Newton Basin diagram.

The complex solutions form a conjugate pair? What about the complex solutions of quartic equations with real coefficients? Does this result apply for all polynomial equations with real coefficients?

Cubic equation with complex coefficients

Choose a cubic equation for which not all the coefficients are real.

At least one of the coefficients a,b,c,d should be a complex number (a not 0).

Use Newton's method to find the three solutions of the equation. Produce a Newton Basin diagram.

What is the connection between the value of b/a and the sum of the solutions? Explain.

What is the connection between the value of d/a and the product of the solutions? Explain.

Other complex equations

Invent your own!

Are there any complex numbers such that ?

Newton Basins - orbits
 
Consider the Newton basin diagram for :

Take your starting number in the largest green blob and investigate the sequence z0, z1, z2, z3 ... (the orbit).

The Dynamical Systems software is ideal for this purpose.
(NB. "Save" feature does not work.)

There is a main set of green globs that appear to be increasing in size. Choose a starting number in the largest blob between two main blobs and investigate the orbit.

To investigate orbits on the Newton Basin for z3 - 1 = 0, use this java applet. Click inside the diagram for an orbit of five values. Click on the last of the five values to continue the orbit.