1 / 56

Lecture 7 Recursive function calls

Lecture 7 Recursive function calls. Recursive programming Factorial Fibonacci number Matrix determinant Hanoi tower. Exercise I. Calculate the average of a series of numbers Use a while-loop to get numbers Allow the user to key in a number at each iteration

noelani-mcconnell
Télécharger la présentation

Lecture 7 Recursive function calls

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 7 Recursive function calls • Recursive programming • Factorial • Fibonacci number • Matrix determinant • Hanoi tower 軟體實作與計算實驗

  2. Exercise I • Calculate the average of a series of numbers • Use a while-loop to get numbers • Allow the user to key in a number at each iteration • Output the average of given numbers at each iteration • Halt when the given number equals zero • Draw a flow chart for problem solving • Write a MATLAB function to implement the flow chart 軟體實作與計算實驗

  3. Exercise #a#b • Generate four distinct digit characters within {’0’, ’1’, ’2’, ’3’, ’4’, ’5’,’6’,’7’,’8’,’9’} randomly • Use a while-loop to realize the game of #a#b • Allow a player to key-in a four-digit string • Response na’a’nb’b’ to the given string • Halt if the guess is scored as 4’a’ 軟體實作與計算實驗

  4. na denotes the number of characters in the guess that appear in the target with correct position • nb denotes the number of characters in the guess that appear in the target but stand at wrong position 軟體實作與計算實驗

  5. Example • Target : 6481, Guess : 1628 • Response : 0 ’a’ 3 ’b’ • Target : 6481, Guess : 1946 • Response : 0 ’a’ 3 ‘b’ • Target : 6481, Guess : 6283 • Response : 1 ‘a’ 1 ‘b’ • Target : 6481, Guess : 6481 • Response : 4 ’a’ 0 ‘b’ 軟體實作與計算實驗

  6. Exercise II • Draw a flow chart to realize the game of #a#b using a while-loop • Draw a flow chart to illustrate how to determine na for given target and guess • Draw a flow chart to illustrate how to determine nb for given target and guess 軟體實作與計算實驗

  7. Exercise III • Write MATLAB functions to implement flow charts in exercise II 軟體實作與計算實驗

  8. Recursive programming • Directly solve a task if problem size is small enough • Decompose a task of size n to subtasks of smaller sizes otherwise 軟體實作與計算實驗

  9. Problem size • Factorial • N! • problem size: N • Fibonacci number • Fn =Fn-1 + Fn-2 • problem size: n • Determinant of an n-by-n matrix A • det(A) • problem size : n 軟體實作與計算實驗

  10. Factorial • Definition • The factorial function is formally defined by • or recursively defined by 軟體實作與計算實驗

  11. Recursive definition factorial(n) n==0 T ans=1 ans= factoria(n-1) ans=n*ans 軟體實作與計算實驗

  12. MATLAB codes function ans=factorial(n) if n==0 ans=1; return else ans=n*factorial(n-1) end 軟體實作與計算實驗

  13. Fibonacci number • Definition • Fibonacci numbers is defined by • the recurrence relation • with seed values 軟體實作與計算實驗

  14. Recurrence relation fib(n) n==1 | n==0 ans=n ans=fib(n-1) ans=ans+fib(n-2) 軟體實作與計算實驗

  15. Termination condition • If n==1 or n==0 軟體實作與計算實驗

  16. Recurrence relation • if n > 1 • By the rule, the problem size is reduced from n to n-1 and n-2 • f(n) is decomposed to two sub-tasks, f(n-1), f(n-2) 軟體實作與計算實驗

  17. fib source codes function ans=fib(n) if n==1 | n==0 ans=1; return else ans = fib(n-1); ans = ans+fib(n-2); end 軟體實作與計算實驗

  18. return • MATLAB function halts whenever the ‘return’ statement is executed • The function exits after calling fib(1) or fib(0) 軟體實作與計算實驗

  19. Recursive call • The function fib calls itself when n > 1 • Recursive call • The function fib calls itself during execution • Problem sizes of two recursive calls are respectively n-1 and n-2 軟體實作與計算實驗

  20. A tree of recursive calls >> fib(3) execute fib 3 execute fib 2 execute fib 1 execute fib 0 execute fib 1 ans = 3 • Fib(3) • Fib(2) • Fib(1) • Fib(0) • Fib(1) 軟體實作與計算實驗

  21. >> fib(4) execute fib 4 execute fib 3 execute fib 2 execute fib 1 execute fib 0 execute fib 1 execute fib 2 execute fib 1 execute fib 0 ans = 5 Calling tree • Fib(4) • Fib(3) • Fib(2) • Fib(1) • Fib(0) • Fib(1) • Fib(2) • Fib(1) • Fib(0) 軟體實作與計算實驗

  22. Termination condition • If n==1 or n==0 軟體實作與計算實驗

  23. Binary to decimal representations Let b be a vector of binary bits Translate it to a decimal number such that • Problem statement 軟體實作與計算實驗

  24. Decomposition 軟體實作與計算實驗

  25. Halting condition 軟體實作與計算實驗

  26. Recurrence relation bin2dec(b,n) n==1 b1=b(n) b2=b(1:n-1) ans=bin2dec(b2,n-1) ans=2*ans+b1 ans=b 軟體實作與計算實驗

  27. function ans=bin2dec(b,n) if n==1 ans=b; return else b1=b(n); b2=b(1:n-1); ans=bin2dec(b2,n-1); ans=2*ans+b1; end 軟體實作與計算實驗

  28. Decimal to binary representations • Problem statement Let a denote a decimal number b = dec2bin(a) b denotes the binary representation of a 軟體實作與計算實驗

  29. Recurrence relation dec2bin(a) a<=1 a1=mod(a,2) a2=floor(a/2) ans=dec2bin(a2) ans=[ans a1] ans=a 軟體實作與計算實驗

  30. function ans=dec2bin(a) if a<=1 ans=a; return else a1=mod(a,2); a2=floor(a/2); ans=dec2bin(a2); ans=[ans a1]; end 軟體實作與計算實驗

  31. >> a=53; >> b=dec2bin(a) b = 1 1 0 1 0 1 >> bin2dec(b,6) ans = 53 軟體實作與計算實驗

  32. Determinant >> A=[1 2;3 4] A = 1 2 3 4 >> det(A) ans = -2 軟體實作與計算實驗

  33. Rule I • A is 2x2 軟體實作與計算實驗

  34. Flow chart • Let A be a 2-by-2 matrix [m,n]=size(A) m==2 & n==2 T ans=A(1,1)*A(2,2)-A(1,2)*A(2,1) 軟體實作與計算實驗

  35. 軟體實作與計算實驗

  36. n=3 • A is 3x3 • Ex n=3 • Calculating the determinant of a 3-by-3 matrix is translated to subtasks of calculating determinants of 2-by-2 matrices 軟體實作與計算實驗

  37. Rule II • A is nxn and n > 2 denotes a (n-1)x(n-1) sub-matrix of A. It is obtained by removing elements in the first row and the jth column of matrix A 軟體實作與計算實驗

  38. Recurrent relation for reduction of problem size • det(A) is decomposed to n sub-tasks • Each calculates determinant of an (n-1)-by-(n-1) matrix • The problem size is reduced from n to n-1 軟體實作與計算實驗

  39. Tree of recursive calls rule II . . . . . . rule I 軟體實作與計算實驗

  40. Append a=[1 2 3]’ A=[] A=[A a a]; 軟體實作與計算實驗

  41. B1=A(2:n,1:i-1) B2=A(2:n,i+1:n) B=[B1 B2] 軟體實作與計算實驗

  42. Flow chart ans=0 m==2 & n==2 function ans= mydet(A) for i=1:n T ans=A(1,1)*A(2,2)-A(1,2)*A(2,1) B=[A(2:n,1:i-1) A(2:n,i+1:n)] ai= (-1)^(i+1)*A(1,i)*mydet(B) ans = ans +ai 軟體實作與計算實驗

  43. MATLAB codes function ans=mydet(A) ans=0;[m,n]=size(A); if m==1 ans=A; return; end if m==2 ans=A(1,1)*A(2,2)-A(1,2)*A(2,1); return; else for i=1:n … end end 軟體實作與計算實驗

  44. function v=mydet(A) % Calculate determinant of A n=size(A,1);v=0; if n==2 v=A(1,1)*A(2,2)-A(1,2)*A(2,1); return end v=0; for i=1:n B=get_submatrix(A,i); det_B=mydet(B); s=(-1)^(i+1); v=v+s*A(1,i)*det_B; end return 軟體實作與計算實驗

  45. Hanoi Tower Problem 軟體實作與計算實驗

  46. Target 軟體實作與計算實驗

  47. Hanoi Tower PlayHanoiTower.rar 軟體實作與計算實驗

  48. Valid movement • A larger disk is inhibited to be placed on the top of a smaller disk. • Initial state: all disks on the first tower • Final state: all disks on the third tower 軟體實作與計算實驗

  49. 軟體實作與計算實驗

  50. Three steps for auto-play • Move n-1 objects from stack 1 to stack 2 • Move 1 object from stack 1 to stack 3 • Move n-1 objects from stack 2 to stack 3 軟體實作與計算實驗

More Related