• Read Chapter 5, Simulation
• and Chapter 6, Software Engineering
• Quiz on Wednesday

• Function invokes itself

• Raw multiplications
  • X^N = X * X * X * ... * X
  • N-1 multiplications
• Save work; clump things together
  • SX = X * X
  • X^N = SX * SX * ... * SX
  • roughly N/2 times ...
• Refine this latter approach

• Base case: X^0 = 1.0
• N is even case: X^N = X^(N/2) * X^(N/2)
• N is odd case: X^N = X * X^(N/2) * X^(N/2)

Any nontrivial computer language that one can invent is apparently capable of computing no more and no fewer functions that all the other nontrivial programming languages.

This part of Java lets you solve all kinds of algorithms

• Putting things in a row (countable sets)

• Sets mapping integers to integers uncountable

• Same as proof that irrational numbers are uncountable

• build many different models; choose best
• simulate, using program: use loops

Build Dog Lot.                     x    |
Put fence next to wall.         ********|
Have one roll of fencing.       *       |
                                *       |
                             y  *       |
                                *       |
                                *       |
                                *       |
                                ********|
                                   x    |