Chapter 3: Iteration and Invariants

• Two Ways of Computing Factorial
• Recursive vs. Iterative
• Accumulating Partial Products
• Writing a Procedure for this Iterative Process
• Writing a Procedure for Actual Iterative Factorial
• Comparing the Recursive and Iterative Processes
• Another Procedure for Making Paper Chains
• Summary
• Exercise 3.1, p. 50
• Exercise 3.1
• Tracing num-times-range
• Section 3.2: Using Invariants
• Iteration and Invariants
• Iteration and Verification
• Example: Fermat Numbers
• Computing Fermat Numbers
• Writing repeatedly-square
• Another Fact
• repeatedly-square Again
• Fermat Numbers Again
• Number of Digits in Fermat Numbers
• Procedure repeatedly-double
• Internal Definitions
• Perfect Numbers
• Using Iteration to Sum Up Divisors
• Procedure sum-from-plus: Base Case
• Procedure sum-from-plus: Returned Value
• Procedure sum-from-plus: Recursive Call
• The divides? Predicate
• Using perfect? to Search for Perfect Numbers
• Automating the Search
• The let Expression
• The Search for Perfect Numbers
• Approximation Through Iterative Improvement
• Approximating Square Root: Newton's Method
• A Procedure for Approximation Through Iterative Improvement
• The Golden Ratio
• Approximating the Golden Ratio
• Logical Expressions
• Logical Expression Details
• Logical Expression Examples
• Example
• Using and and or for Control