lecture20 - LECTURE 20 Recursion Recursion vs Iteration E7,...

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1 LECTURE 20 Recursion Recursion vs Iteration E7, Fall 2008, M. Frenklach 1 RECURSION: x n = x n -1 * x function y = powerfun(x,n) % recursion: x^n = x^(n-1)*x if n == 1 y = x; else y = powerfun(x,n-1) * x; end 2 RECURSION: Fibonacci numbers 1 1 2 3 5 8 13 21 34 55 89 144 … F = F = 1 F 1 F 2 1 F n = F n-2 + F n-1 3
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2 function F = fib(n) % RECURSIVE version if < 3 F 1 = F 2 = 1; F n = F n-2 + F n-1 if n < 3 F = 1; else F = fib(n-2) + fib(n-1); end 4 F 1 = F 2 = 1; F n = F n-2 + F n-1 fib(5) =5 fib(3) =2 fib(4) =3 fib(1) =1 fib(2) =1 fib(2) =1 fib(3) =2 fib(1) =1 fib(2) =1 5 ITERATION – Repeated execution of a part of a code for. ..end while. ..end 6
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3 function F = fib_iter(n) % ITERATIVE version F = zeros(1,n); F 1 = F 2 = 1; F n = F n-2 + F n-1 F(1) = 1; F(2) = 1; for i = 3:n F(i) = F(i-2) + F(i-1); end 7 >> tic, fib(20); toc >> tic TIMING >> for i = 1:100 fib_iter(1000) >> end >> t = toc; t/100 8 % fib_ratio.m % ratio of fibonacci numbers n = 20; RATIO F(i+1)/F(i) f = fib_iter(n); r = f(2:n)./f(1:n-1); plot(r, '-o' ) 9
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4 RATIO F(i+1)/F(i) >> r' ans = 1 2 1.5 1.6667 1.6 1.625 1.6154 1 619 1.619 1.6176 1.6182 1.618 1.6181 1.618 1.618 1.618 1.618 1.618 1.618 1.618 10 ab a bb b bb ϕ ϕϕ + == + = GOLDEN RATIO = 2 1 10 15 1.6180339887 2 + = −−= + =≈ Leonardo Da Vinci's illustration from the Divina Proportione , Luca Pacioli 1509, Venice, depicting the golden proportion as it applies to the human face.
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lecture20 - LECTURE 20 Recursion Recursion vs Iteration E7,...

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