The Root Finding Song

(to the tune of "Twinkle Twinkle Little Star")
If the root of f one seeks,
There are good and bad techniques.
Each one has points good and bad,
To cheer you up and make you mad.
Listen on, and you shall see
What those points turn out to be.

Bisection is very slow
But converges, this we know.
Divide the interval in two,
Then select half to pursue.
The root is always within sight,
So this method is all right.

Newton came up with a plan
For fixed point iteration.
It involves finding f prime,
But error is squared each time.
Sadly, there's no guarantee
That x will end where it should be.

What if we approximate
f prime at each iterate?
The secant method does this thing
Through finite differencing.
But like Newton, it may fail
and follow a diverging trail.

For roots of f of more than x,
The situation is complex.
No analog for bisection,
Jacobian used in Newton.
Broyden's method is like secant,
Convergence guarantees are scant.

"The Root Finding Song" Copyright (c) 2002-2007 Rebecca Hartman-Baker.

Last updated January 22, 2007

hartmanbakrj@ornl.gov