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Recursion

Recursion. Recursively defined sequences. To define a sequence recursively : give initial conditions , i.e., the values of the first few terms explicitly; give a recurrence relation , i.e., an equation that relates later terms in the sequence to earlier terms. Example:

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Recursion

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  1. Recursion

  2. Recursively defined sequences • To define asequence recursively: • give initial conditions, i.e., the values of the first few terms explicitly; • give a recurrence relation, i.e., an equation that relates later terms in the sequence to earlier terms. • Example: • Define the following sequence recursively: 1, 4, 7, 10, 13, … • Solution: a1 = 1, an = an-1 + 3 for n≥2

  3. Recursion is one of the central ideas of computer science To solve a problem recursively • Break it down into smaller subproblems each having the same form as the original problem; • When the process is repeated many times, the last of the subproblems are small and easy to solve; • The solutions of the subproblems can be woven together to form a solution to the original problem. • Example: The tower of Hanoi

  4. Tower of Hanoi: Move disks from left pole to right pole RULES: You may only move one disk at a time. A disk may only be moved to one of the three columns. You must never place a larger disk on top of a smaller disk. INITIAL STATE GOAL STATE Pole A Pole B Pole C Pole A Pole B Pole C

  5. The Tower of Hanoi • How to generalize the procedure to n disks? • How many moves are required? • Recursive procedure: • Transfer the top n-1 disks from pole A to pole B • Move the bottom disk from pole A to pole C • Transfer the top n-1 disks from pole B to pole C • Let ak denote the number of moves needed to transfer a tower of n disks from one pole to another using the above procedure • Then we have the following recursive formula for counting the moves: a1 = 1 an = 2an-1 + 1 for n≥2

  6. Recursive formula for Compound Interest • Suppose $10K is deposited in an account paying 3% interest compounded annually. • For each positive integer n, let a0 = the initial amount deposited; an = the amount on deposit at the end of year n. • Find a recursive relation for a0 , a1 , a2 ,… assuming no additional deposits or withdrawals. • We have the following recursive formula: a0 = 10,000 an = an-1 + 0.03*an-1 = 1.03*an-1 for n≥1

  7. Finding an explicit formula for a recursively defined sequence It is often helpful to know an explicit formula for the sequence, especially if you need • to compute terms with very large subscripts; • to examine general properties of the sequence. Examples • Recall the recursive formula for the compound interest example: a0 = 10,000 an = 1.03*an-1for n≥1 The explicit formula is an = 10000*(1.03)nfor n≥0 Note: this formula can be generalized to any geometric sequence.

  8. Finding an explicit formula for a recursively defined sequence Examples (cont.) • Suppose the sequence is given by the following recursive relation: a0 = 3 an = an-1 + 4 for n≥1 Then the explicit formula is an = 3 + 4*n for n≥0 • Note: this formula can be generalized to any arithmetic sequence.

  9. Finding an explicit formula for a recursively defined sequence Examples (cont.) • Recall the recursive formula for the Hanoi tower example: a1 = 1 an = 2an-1 + 1 for n≥2 • How to get explicit formula? • Compute the first few terms of this sequence: a1 = 1, a2 = 3, a3 = 7, a4 = 15, a5 = 31, a6 = 63 • Based on the pattern, an = 2n – 1 for n≥1 • But need a proof! • The proof by induction (blackboard)

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