Algorithms

Premise: What is an algorithm?

Iteration

Let's examine some interesting mathematical algorithms that lend themselves to iterative implementation.

• Powers: `x^n` where `x in R, n in W`. (Q: Is there a faster way?)
• Area Under (and Between) a Curve (Q: How Long is a Curve?)
• Fibonacci Sequence: `1, 1,2,3,5,8,...`
• Golden Ratio: `phi`
• Transcendentals (p. 21) (More Examples)

  `pi = 4 sum_{n=0}^{oo} (-1)^n/{2n+1} = 4(1-1/3+1/5-1/7+...)`

  `sin x=sum_{n=0}^{oo}(-1)^n x^{2n+1}/{(2n+1)!}= x-x^3/{3!}+x^5/{5!}-x^7/{7!}+...`

• Feedback (NAND Gate Oscillator) Approximation of the Square Root of `a`: `x larr 1/2(x+a/x)`
• Binomial Theorem (p. 3): Pascal's Triangle
• Euclid's Greatest Common Divisor (GCD)
• Base Conversion