TME 310 - Computational Physical Modeling

Open methods (root-finding)

Lorne Arnold, PhD, PE

University of Washington Tacoma

Root-finding methods

We have a few options:

1. Graphical methods
2. Bracketed methods
3. Open methods
   1. Newton-Raphson
   2. Secant method

Open methods

Algorithms for searching from a single starting point

• Newton-Raphson:
  • Choose a starting point.
  • Look for the root where a line tangent (i.e., local derivative) to the function intersects horizontal axis.
  • Repeat until tolerance is satisfied.
• Secant method
  • Same thing, but with a secant line from an approximate derivative.

Newton-Raphson

Pros and cons (Newton-Raphson)

Pros

• Fast convergence (faster than bisection)
• Simple
• Unbounded

Cons

• Need to know the derivative!
• Can’t start where derivative is zero
• It can get caught in a local maximum/minimum

Local maxima/minima problem

Secant method

Pros and cons (secant)

Pros

• Same as Newton-Raphson, plus…
• Don’t need to know the derivative
• Less strict about starting points

Cons

• It can get caught in a local maximum/minimum

Secant method is great if local maxima/minima are not a concern