Class 29: Numeric Recursion

Held: Tuesday, 11 March 2008

Summary: We visit a slightly different kind of recursion, numeric recursion. In this technique, we once again have procedures call themselves. However, the parameter that we simplify at every step is a number, rather than a list.

Related Pages:

• EBoard.
• Lab: Numeric Recursion.
• Reading: Numeric Recursion.

Notes:

• Sara Bebeau from the Fund for Public Interest Research plans to give an announcement at the start of class.
• I will reserve our normal start-of-class time for comments on campus events.
• Are there final questions on the exam?
• I head off to my conference after class today and am unlikely to have email access until Wednesday afternoon. Please send questions to Dr. Davis.
• Reading for tomorrow: Geometric Art.

Overview:

• Recursion, Generalized.
• Thinking About Natural Numbers.
• Numeric Recursion.

Patterns of Recursion

• While we've seen and written a variety of examples of direct recursion, they typically have the following form:

  (define recursive-proc
    (lambda (params)
      (if (base-case-test)
          (base-case params)
          (combine (partof params)
                   (recursive-proc (simplify params))))))
  

• In many cases, the combination ends up being a choice between two activities. In those cases, we might write:

  (define recursive-proc
    (lambda (params)
      (cond
        ((base-case-test) 
         (base-case params))
        ((special-case-test)
          (combine (partof params)
                   (recursive-proc (simplify params))))
        (else
          (recursive-proc (simplify params))))))
  

• For lists, the simplification was almost always take the cdr and the part-of was almost always take the car.

Recursion with Numbers

• While most of the recursion we've been doing has used lists as the structure to recurse over, you can recurse with many different kinds of values.
• It is fairly common to recurse using numbers.
• The natural base cases for integers are when you hit 0 or when you hit 1.
• The natural simplification step for recursive procedure using numbers calls typically involves subtracting 1 from the argument.
  • Other simplifications, such as dividing in half, are also possible.

Lab

• Do the lab.