# Linearization and the Newton Iteration

### Assignment :

Finish Homework 7. Read Chapter 5. and the Web notes on DO loops. Study newton1.f and newton2.f, but remember that you are not yet responsible for the FORMAT statement.

### New Fortran:

none

The computer solution of a very large number of scientific and engineering problems has at it's heart the solution of a coupled set of non-linear equations. Frequently this solution is obtained with some variation of the Newton iteration. For this class you will only be held directly responsible for application of the Newton method to the solution of a single equation with one unknown. However, the method is very general, and can be applied to systems with any number of equations and a matching number of unknowns.

Use of HTML files with equations is very awkward at the present, so you need to be at a PC or Workstation with a Postscript viewer (any CAC lab machine). Read the contents of the file newton.ps. When you are done look at newton1.f for a Fortran implementation of the solution. As you think about the solution strategy, keep in mind that there are really two key steps. The first is to obtain a reasonable initial guess at the solution. This is often easier said than done, and side-stepped in the example. With a guess available you can attempt the second step of iterating to obtain a good approximation to the answer.

When you feel comfortable with the single equation solution method, take a look at the approach to solving more than one equation, illustrated in newton2d.ps.