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CS112 Scientific Computation
Department of Computer Science Wellesley College
Blowing a gasket
Classic recursive examples
Recursion 20-
Recursion
Recursion solves a problem by
solving a smaller instance of
the same problem
Think divide, conquer, and glue,
where all of the subproblems
have the same “shape” as the
original problem
Recursion 20-
Factorial
4! = 4 x (3 x 2 x 1)
= 4 x 3!
2! = 2 x (1)
= 2 x 1!
3! = 3 x (2 x 1)
= 3 x 2!
recursive
cases
base
case
function result = fact (n) if (n <= 1) % base case
result = 1; else % recursive step
result = n * fact(n-1); end
Recursion 20-
The recursive leap of faith
To solve a problem recursively:
If it is simple enough then solve it immediately Otherwise express solution in terms of smaller, similar problem(s)
Initially this will take some faith on your part
Recursion 20-
Fibonacci numbers, or multiplying rabbits?
Fibonacci numbers first appear around 500 B.C. in writings of a Sanscrit grammarian named Pingala who studied rhythm in language
Leonardo Fibonacci studied these numbers around 1200 in the context of multiplying rabbits
- In the first month, there’s one newborn pair
- Newborn pairs become fertile in their second month
- Each month, every fertile pair begets a new pair
- Rabbits never die
Recursion 20-
Fibonacci function
function result = fibonacci (n) if (n <= 2) % base case result = 1; else % recursive step result = fibonacci(n-1) + fibonacci(n-2); end
fib(5)
fib(4) fib(3)
Recursion 20-
Towers of Hanoi
Recursion 20-
A natural recursive solution
function towersOfHanoi (n, source, destination, spare) if (n == 1) % base case disp(['move disk 1 from ' source ' to ' destination]); else % recursive step towersOfHanoi(n-1, source, spare, destination); disp(['move disk ' num2str(n) ' from ' source ' to ' destination]); towersOfHanoi(n-1, spare, destination, source); end
