CSC151 2007S, Class 29: Numeric Recursion

Admin:
* Final reminder: Ian Young speaks at 4:30 about vim.
* We have a photographer in class today.
* Tomorrow's reading is not yet available.  Sorry.  Expect it tonight.
* No TA Today :-(
* What's worst?
  * Heck week: Week before fall or spring break
  * Hell week: Week 14
  * Purgatory week: Week 13
  * Finals week - No classes, makes it easier
  * Answer one: "All of 'em"
  * Answer two: "It depends on your schedule"
* Are there questions on assignment 11?
  * Testing permutations doesn't work
  * Homework 11 is extra credit.
* THe story of the sum 1 + 2 + 3 + ... + N

Overview:
* Recursion, Generalized.
* Thinking About Natural Numbers.
* Numeric Recursion.

Suppose we wanted to compute Gauss's sum, and we didn't know the formula.
What would we do?

* Define a procedure

(define sum-to-n 
  (lambda (n)
    ....)

Probably8 needs to be recursive, since we can't deal with "arbitrary"
long inputs without some simplification

(define sum-to-n 
  (lambda (n)
    (if (TEST_FOR_BASE_CASE)
        (BASE_COMPUTATION)
        (COMBINE n (sum-to-n (SIMPLIFY n))))))

A BAD BAD BAD SOLUTION

(define sum-to-n 
  (lambda (n)
    (if (null? n)
        0 or null or something
        (COMBINE n (sum-to-n (cdr n))))))

For numbers, natural base cases include "is n 1" or "is n 0"
For numbers, natural ways to simplify include "subtract 1" and "divide by 2"

(define sum-to-n 
  (lambda (n)
    (if (= n 0)
        0
        (+ n (sum-to-n (- n 1))))))

(define sum-to-n 
  (lambda (n)
    (if (= n 1)
        1
        (+ n (sum-to-n (- n 1))))))

(define sum-to-n 
  (lambda (n)
    (if (= n 2)
        3
        (+ n (sum-to-n (- n 1))))))

; What is (sum-to-n -1)
;  "Success"
;   We should check a precondition

What is (sum-to-n 5) using the following definition of sum-to-n?
(define sum-to-n 
  (lambda (n)
    (if (= n 1)
        1
        (+ n (sum-to-n (- n 1))))))

(sum-to-n 5)
=> (+ 5 (sum-to-n (- 5 1)))
=> (+ 5 (sum-to-n 4))
=> (+ 5 (+ 4 (sum-to-n (- 4 1))))
=> (+ 5 (+ 4 (sum-to-n 3)))
=> (+ 5 (+ 4 (+ 3 (sum-to-n (- 3 1)))))
=> (+ 5 (+ 4 (+ 3 (sum-to-n 2))))
=> (+ 5 (+ 4 (+ 3 (+ 2 (sum-to-n (- 2 1))))))
=> (+ 5 (+ 4 (+ 3 (+ 2 (sum-to-n 1)))))