1. Computation for Physics 計算物理概論 Python Programming for Physicists

2. What is Python?

3. Python Programming Language Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python supports multiple programming paradigms, including object-oriented, imperative and functional programming styles. Python features a fully dynamic type system and automatic memory management,

4. PEP 20 “The Zen of Python” PEP=Python Enhancement Proposal Beautiful is better than ugly. Explicit is better than implicit. Simple is better than complex. Complex is better than complicated. Readability counts. Guido van Rossum Creator of python Benevolent Dictator for Life

5. Programming Language High-level language Low-level language

6. Compilation v.s. Interpretation SOURCE CODE SOURCE CODE INTERPRETER OUTPUT SOURCE CODE SOURCE CODE OUTPUT COMPILER OBJECT CODE OBJECT CODE EXECUTOR

7. Python: Interpreted Language Interactive mode – You type Python programs and the interpreter displays the result. Script mode – You store code in a file and use the interpreter to execute the contents of the file.

8. PYTHON DEVELOPMENT ENVIRONMENT

9. Python(x,y) Scientific-oriented Python Distribution based on Qt and Spyder Python(x,y) is a free scientific and engineering development software for numerical computations, data analysis and data visualization based on Python programming language, Qt graphical user interfaces and Spyder interactive scientific development environment. https://code.google.com/p/pythonxy/

10. Download Python(x,y) Download from one of the mirrors

11. Python(x,y) Plugins=Python Packages IDLE, IPython – Development environment NumPy – Multidimensional arrays support and basic operations SciPy – Advanced math, signal processing, optimization, statistics Matplotlib – 2D plotting library VPython – Creation of 3D interactive models of physical systems SymPy – Symbolic Mathematics Library

12. IDLE Integrated DeveLopment Environment IDLE is the Python IDE built with the Tkinter GUI toolkit. IDLE has the following features: – coded in 100% pure Python, using the Tkinter GUI toolkit – cross-platform: works on Windows and Unix – multi-window text editor with multiple undo, Python colorizing and many other features, e.g. smart indent and call tips – Python shell window (a.k.a. interactive interpreter) – debugger (not complete, but you can set breakpoints, view and step) It is a part of the standard library and is described in the reference manual: http://www.python.org/doc/current/lib/idle.html The source code for IDLE is included in the standard library. More information on developing IDLE can be found at http://docs.python.org/devguide/

13. IDLE Integrated development environment Python 2.7.3 (default, Apr 10 2012, 23:31:26) [MSC v.1500 32 bit (Intel)] on win32 Type "copyright", "credits" or "license()" for more information. >>>

14. Start IDLE 開始  所有程式  python(x,y)  IDLE

15. File  New Window

16. File  Save As

17. Run  Run Module

18. Python Shell

19. PYTHON PROGRAMMING LANGUAGE

20. What is a Program? A sequence of instructions that specifies how to perform a computation Basic instructions – Input – Output – Math – Conditional execution – Repetition

21. What is Debugging Programming is error-prone Programming errors=bugs The process of tracking bugs down=debugging Three kinds of errors Syntax errors Runtime errors Semantic errors

22. First Program=“Hello, World!” Traditionally, the first program you write in a new language is to display the words – “Hello, World” print(‘Hello, World!’) Case-sensitive

23. Variables, Expressions, Statements

24. Variables, Values, Stage Diagram State diagram – variables  value message  ’And now for something’ n  17 pi  3.1415926535 z  1+3i

25. Assignment Statement “=“ is an “assignment” message = ’And now for something’ n = 17 pi = 3.1415926535 z = 1+3j 2*x=y – Syntax Error

26. data types integer=integer number float=floating point number complex=complex floating point number str=string >>> type(17) >>> type(3.2) >>> type(1+3j) >>> type(‘Hello, World!’)

27. data types integer=integer number float=floating point number complex=complex floating point number str=string >>> type(n) >>> type(pi) >>> type(z) >>> type(message)

28. Dynamically Typed >>> x=3 # x is of integer type >>> x=3.0 # x is of float type >>> x=‘blah’ # x is of string type >>> x=[3,3.0,’blah’] # x is of list type

29. Variable Names & Keywords Keywords – The interpreter uses keywords to recognize the structure of the program, and they cannot be used as variable names – and – if – else – while – …etc

30. Operators & Operand + – addition: x+y - – subtraction: x-y * – multiplication: x*y / – division: x/y ** – exponentizaton x**y // – the integer part of x/y: 14.5//3.6  4.0, 14.5//3.7  3.0 % – module=remainder after x/y: 1.5%0.4  0.3

31. Expression & Statement An expression is a combination of values, variables, and operators. – x – x+17 A statement is a unit of code that the Python interpreter can execute. – x=3 – print(‘Hello’)

32. Interactive & Script Mode Interactive mode Script mode Major difference – Python actually evaluates the expression, but it doesn’t display the value unless you tell it to

33. Order of Operations Rue of precedence Parentheses Exponentiation Multiplication and Division Left to right for operators with the same precedence

34. Comments It is a good idea to add notes to your programs to explain in natural language what the program is doing. Comments start with the # symbol # compute the velocity v=dx / dt v=dx / dt # compute the velocity

35. x=1 print(x) variable assignment statement

36. Variables Types Integer – x=1 Float – x=1.5 – x=1.0 – x=float(1) Complex – x=1.5+2.0j – x=1.5+0j – x=complex(1.5)

37. string String literals can be written enclosed in either two single or two double quotes x='This is a string' x="This is a string" y='1.234' # create a string with value ‘1.234’ y=1.234 # create a real number with value 1.234 z='x=1' # create a string with value ‘x=1’

38. type conversion functions string to integer >>> int(‘32’) 32 >>> int(3.999) 3 >>> int(-2.3) -2 >>> int('Hello') ValueError: invalid literal for int() with base 10: 'hello' >>> int('3.999') ValueError: invalid literal for int() with base 10: '3.999'

39. Help(int) >>>help(int) Help on class int in module __builtin__: class int(object) | int(x[, base]) -> integer | | Convert a string or number to an integer, if possible. A floating point | argument will be truncated towards zero (this does not include a string | representation of a floating point number!) When converting a string, use | the optional base. It is an error to supply a base when converting a | non-string. If base is zero, the proper base is guessed based on the | string content. If the argument is outside the integer range a | long object will be returned instead.

40. Type conversion functions string to float >>> float(‘32’) 32.0 >>> (‘3.14159’) 3.14159 >>> int(float(‘3.999’)) 3 class float(object) | float(x) -> floating point number | | Convert a string or number to a floating point number, if possible.

41. type conversion functions ‘blah’ to complex >>> complex(32) (32+0j) >>> complex(‘3.2’) (3.2+0j) >>> complex(3+j) NameError: name 'j' is not defined >>> complex(3+0j) (3+0j) >>> complex(3+1j) (3+1j) >>> complex(‘3+j’) (3+1j) >>> complex(‘3+0j’) (3+0j) >>> complex(‘3+1j’) (3+1j)

42. Output Statements >>> x = 1 >>> print(x) 1 >>> x=1 >>> y=2 >>> print(x,y) 1 2 >>> print("The value of x is", x, "and y=",y) The value of x is 1 and y=2

43. Input Statement >>> x = input("Enter the value of x:”) (The computer will stop and wait) >>> print("The value of x is ",x) >>> x = input("Enter the value of x:") (You enter "Hello") >>> print("The value of x is ",x) The value of x is Hello

44. Input Statement Convert input string to float temp = input("Enter the value of x: ") x=float(temp) print("The value of x is ",x) x = float(input("Enter the value of x: ")) print("The value of x is ",x)

45. Python 3 v.s. Python 2 Include following statement in your code from __future__ import division, print_function Division return a floating value – 3/2  1 # python 2 – 3/2  1.5 # python 3 – 4/2  2 # python 2 – 4/2  2.0 #python 3 Print is a statement – print "The energy is ", E # python 2 – print("The energy is ",E) # python 3

46. Python 3 v.s. Python 2 Input returns a string – In Python 3 the input function always returns a string – x=input() – you enter ‘2.5’ – type(x)  # python 2 – type(x)  # python 3 – x=raw_input() – type(x)  # python 2

47. Python 3 v.s. Python 2 Iterators v.s. List – range(10)  [0,1,2,3,4,5,6,7,8,9] # python 2 – range(10)  range(0,10) # python 3 – list(range(10))  [0,1,2,3,4,5,6,7,8,9] # python 3 – x = [1,2,3] – map(log,x)  [0.0, 0.69…, 1.09…] # python 2 – map(log,x)  # python 3 – list(map(log,x))  [0.0, 0.69…, 1.09…] # python 3

48. “x=a+b” is an assignment statement! x = 1 – x  1 x = x+1 – x  x+1 – temp = x+1 – x=temp x = x**2 – 2 – x  x**2 - 2

49. Python Modifiers x += 1 – x = x + 1 x -= 4 – x = x - 4 x *= -2.6 – x = x*(-2.6) x /= 5*y – x = x / (5*y) x //= 3.4 – x = x // 3.4

50. Assignment of Multiple Variables x, y = 1, 2.5 – x = 1 – y = 2.5 x, y = 2*z+1, (x+y)/3 – temp1 = 2*z+1 – temp2 = (x+y)/3 – x = temp1 – y = temp2

51. Assignment of Multiple Variables You can interchange of values of two variables x, y = y, x – temp = y – y = x – x = temp

52. Try: A ball dropped from a tower Write a program to ask the user to enter h the height of the tower and a time interval t, then prints the height of the ball above the ground at time t after it is dropped.

53. A ball dropped from a tower Simple version h = float(input(“Enter the height of the tower: ")) t = float(input("Enter the time interval: ")) s = 9.81*t**2/2 print("The height of the ball is", h-s, "meters") To make it more readable g = 9.81 s = g*t**2/2

54. Another ball dropped from a tower A ball is dropped from a tower of height h with initial velocity zero. Write a program that asks the user to enter the height in meters and the calculates and prints the time the ball takes until it hits the ground.

55. Functions, Packages, & Modules >>> from math import log >>> x = log(2.5) log(...) log(x[, base]) Return the logarithm of x to the given base. If the base not specified, returns the natural logarithm (base e) of x.

56. math—Mathematical Functions from math import * Functions: log log10 exp sin, cos, tan asin, acos, atan sinh, cosh, tanh sqrt Constants: pi e

57. Try: Converting polar coordinates Get the user to enter the value of r and θ Convert those values to Cartesian coordinates Print out the results

58. Converting polar coordinates from math import sin, cos, pi r = float(raw_input("Enter r: ")) d = float(raw_input("Enter theta in degrees: ")) theta = d*pi/180 x = r*cos(theta) y = r*sin(theta) print("x= ",x, "y= ",y)

59. Comment Statements from math import sin, cos, pi # Ask the user for the values of the radius and angle r = float(input(“Enter r: “)) d = float(input(“Enter theta in degrees: “)) # Convert the angle to radians theta = d*pi/180 # Calculate the equivalent Cartesian coordinates x = r*cos(theta) y = r*sin(theta) print(“x= “,x,”y= “,y) # Print out the results

60. Multiline \ 1+2\ 3

61. help >>> help(sum) Help on built-in function sum in module __builtin__: sum(...) sum(sequence[, start]) -> value Returns the sum of a sequence of numbers (NOT strings) plus the value of parameter 'start' (which defaults to 0). When the sequence is empty, returns start.

62. Branching

63. Boolean expressions >>>5 == 5 True >>>5 == 6 False >>>type(True) >>>type(False)

64. Relational Operators x == y# x is equal to y x != y# x is not equal to y x > y# x is greater than y x < y# x is less than y x >= y# x is greater than or equal to y x <= y# x is less than or equal to y

65. Logical Operators and or not >>> 3>1 and 3>2 True >>> 3>1 or 3<2 Ture >> not (3>1) False

66. Numerical Values of True & False >>>int(True) 1 >>>int(False) 0 >>>0==False True >>>1==True True >>>2==True False

67. Conditional Execution if x > 0: print(‘x is positive’) print(‘Hello, World!’) Condition Compound statements

68. Alternative Execution if x % 2 ==0: print(‘x is even’) else: print(‘x is odd’)

69. Chained Conditionals if x < y: print(‘x is less than y’) elif x > y: print(‘x is greater than y’) else: print(‘x and y are equal’) elif=“else if”

70. Chained Conditionals if choice == ‘a’ draw_a() elif choice == ‘b’ draw_b() elif choice == ‘c’ draw_c() function call

71. Nested Conditionals if x == y: print(‘x and y are equal’) else: if x < y: print(‘x is less than y’) else: print(‘x is greater than y’)

72. Use Logic Operators to Simplify if 0 < x: if x < 10: print(‘x is ……’) if 0 < x and x < 10: print(‘x is ……’)

73. If (anything): 0 is interpreted as False Any other number is interpreted as True Empty lists and objects return False Non-empty lists and objects return True

74. if (anything): >>>if 0:... print(‘x’) >>>if 1:... print(‘x’) x >>>if 1.1:... print(‘x’) x >>>d=[] >>>if d:... print(‘x’) >>>d=[1,2] >>>if 1.1:... print(‘x’) x

75. Iteration

76. The ‘While’ Statement n=10 while n > 0: print(n) n = n-1 Loop You should change the value of one or more variables, otherwise it will become an infinite loop

77. Will the Loop Terminates for all n? n = 3 while n != 1: print(n) if n%2 == 0:# n is even n = n//2 else:# n is odd n = n*3+1 3,10,5,16,8,4,2,1

78. Break while True: line = raw_input(‘> ‘) if line == ‘done’: break print(line) print(‘finished’)

79. Try: Fibonacci Sequence Sequence of integers in which each is the sum of previous two, starting with 1,1 1, 1, 2, 3, 5, 8, 13, 21 f1 = 1 f2 = 1 next = f1 + f2 while f1 <= 1000: print(f1) f1 = f2 f2 = next next = f1 + f2

80. Fibonacci Sequence, Simplified f1, f2 = 1, 1 while f1 < 1000: print(f1) f1, f2 = f2, f1+f2

81. Try: Catalan Numbers

82. Try: Square Roots

83. Newton’s Method, Interactively >>>a=4.0 >>>x=3.0 >>>y=(x+a/x)/2 >>>print(y) 2.16666666667 >>>x=y >>>y=(x+a/x)/2 >>>print(y) 2.00641025641

84. Square Roots while True: print(x) y = (x + a/x ) / 2 if y == x: break x = y

85. Square Roots (Safer Check) epsilon = 0.0000001 while True: print(x) y = (x + a/x ) / 2 if abs(y-x) < epsilon: break x = y

86. User-Defined Functions

87. Factorial Function def factorial(n): f = 1.0 for k in range(1,n+1) f *= k return f a = factorial(10) b = factorial(r+2*s) Local variables Argument

88. Distance Function def distance(r, theta, z): x = r*cos(theta) y = r*sin(theta) d = sqrt(x**2+y**2+z**2) return d D = distance(2.0,0.1,-1.5) Multiple arguments

89. Use “def” Statement to Define a Function def add(arg1, arg2) x=arg1+arg2 return x def power(x,y) if x <=0: return 0 else return x**y

90. Return Arbitrary Type def cartesian(r,theta): x = r*cos(theta) y = r*sin(theta) position = [x,y] return position def f(z): # Some calculations here... return a,b x,y = f(1)

91. Return No Value def print_vector(r): print("(",r[0],r[1],r[2],")")

92. Map User-defined Function to a List def f(x): return 2*x-1 newlist = list(map(f,oldlist))

93. Try: Square Root Function Write your own square root function Input a and epsilon Output square root of a Compare the result with built-in function sqrt()

94. Lists

95. Build-in Type: List List=[element1, elemen2, element3,…] r = [ 1, 1, 2, 3, 5, 8, 13, 21] print(r) [ 1, 1, 2, 3, 5, 8, 13, 21] x = 1.0 y = 1.5 z = -2.2 r = [x,y,z] r = [ 2*x, x+y, z/sqrt(x**2+y**2) ]

96. Access List Elements >>> r = [ 1.0, 1.5, -2.2] >>> r[0] 1.0 >>> r[4] Traceback (most recent call last): File " ", line 1, in IndexError: list index out of range >>> r[-1] -2.2 The index number starts at zero

97. Modify List Elements >>>r = [ 1.0, 1.5, -2.2] >>>r[0] = 3.5 1.0 >>>r [3.5, 1.5, -2.2]

98. List Operations >>> a = [1, 2, 3] >>> b = [4, 5, 6] >>> c = a + b >>> print(c) [1, 2, 3, 4, 5, 6] >>> [0] * 4 [0, 0, 0, 0] >>> [1, 2, 3] * 3 [1, 2, 3, 1, 2, 3, 1, 2, 3]

99. List Functions Functions that take ‘list’ as an argument >>> r = [ 1.0, 1.5, -2.2] >>> sum(r) 0.3 >>> sum(r)/len(r) 0.1

100. Meta-Function: map Apply functions to all elements of a list map(log, r) take the natural logarithm of each element of a list r from math import log r = [ 1.0, 1.5, 2.2 ] logr = list(map(log,r)) print(logr) [0.0, 0.4054651081, 0.7884573603]|

101. List Methods: Add an Element >>>r = [ 1.0, 1.5, -2.2 ] >>>x = 0.8 >>>r.append(2*x+1) >>>print(r) [1.0, 1.5, -2.2, 2.6] >>>t1 = [‘a’, ‘b’, ‘c’] >>>t2 = [‘d’, ‘e’] >>>t1.extend(t2) >>>print(t1) [‘a’, ‘b’, ‘c’, ‘d’, ‘e’]

102. Starts from an Empty List >>>r=[] >>>r.append(1.0) >>>r.append(1.5) >>>r.append(-2.2) >>>print(r) [1.0, 1.5, -2.2]

103. List Methods: Remove am Element >>>r = [ 1.0, 1.5, -2.2, 2.6 ] >>>r.pop() >>>print(r) [1.0, 1.5, -2.2] >>>t =[‘a’, ‘b’, ‘c’] >>>x = t.pop(1) >>>print(t) >>>print(x)

104. Remove a known Element >>>t = [‘a’, ‘b’, ‘c’] >>>t.remove(‘b’) >>>print(t) [‘a’, ‘c’]

105. Traveling a List: For Loop r = [ 1, 3, 5 ] for n in r: print(n) print(2*n) print("Finished") 1 2 3 6 5 10 Finished

106. Traveling a List: For Loop for i in [3, ‘Hello’, 9.5]: print i 3 Hello 9.5 for x in []: print(‘This never happens’)

107. Run Some Code a Number of Times r = [0, 1, 2, 3, 4 ] for i in r: blah print(‘Finished’) If we want to loop 10000000 times?

108. Built-in Function: range() range(5) gives [0, 1, 2, 3, 4] range(2,8) gives [2, 3, 4, 5, 6, 7] range(2,20,3) gives [2, 5, 8, 11, 14, 17] range(20,2,-3) gives [20, 17, 14, 11, 8, 5]

109. range(integer) p = 10 q = 2 range(p/q) (error!) range(p//q) (ok!)

110. Performing a Sum

111. List Slices: list[lower:upper:step] Inclusive lower element index (default=0) Exclusive upper element index (default=Length) >>>l=[1,2,3,4,5] >>>l[0:4] [1,2,3,4] >>>l[0:4:2] [1,3] >>>l[:4] [1,2,3,4] >>>l[2:] [3,4,5] >>>l[::2] [1,3,5]

112. List Comprehensions >>>[i*i for i in range(5)] [0, 1, 4, 9, 16] >>>[k*5 for k in [4,8,9]] [20, 40, 45] >>>[q**(0.5) for q in [4,6,9]] [2.0, 3.0, 4.0] >>>[ k%2 == 0 for k in range(5)] [True, False, True, False, True]

113. Examples

114. Try: Emission Line of Hydrogen R = 1.097e-2 for m in [1,2,3]: print("Series for m =", m) for k in [1,2,3,4,5]: n=m+k invlambda = R*(1/m**2-1/n**2) print(" ",1/invlambda," nm")

115. Use range() Function to Simplify R = 1.097e-2 for m in range(1,4): print("Series for m =",m) for n in range(m+1,m+6): invlambda = R*(1/m**2-1/n**2) print(" ",1/invlambda," nm")

116. Try: The Madelung Constant In condensed matter physics the Madelung constant gives the total electric potential felt by an atom in a solid. It depends on the charges on the other atoms nearby and their locations. Consider for instance solid sodium chloride—table salt. The sodium chloride crystal has atoms arranged on a cubic lattice, but with alternating sodium and chlorine atoms, the sodium ones having a single positive charge +e and the chlorine ones a single negative charge −e, where e is the charge on the electron. If we label each position on the lattice by three integer coordinates (i, j, k), then the sodium atoms fall at positions where i + j + k is even, and the chlorine atoms at positions where i + j + k is odd.

117. Try: The Madelung Constant Consider a sodium atom at the origin, i = j = k = 0, the spacing of atoms on the lattice is a. The distance from the origin to the atom at position (i, j, k) is The potential at the origin created by such an atom is The sign of the expression depending on whether i + j + k is even or odd. The total potential felt by the sodium atom is the sum of this quantity over all other atoms. Assume a cubic box around the sodium at the origin, with L atoms in all directions. M=Madelung constant is the value of M when L → ∞, but one can get a good approximation just by using a large value of L.

118. Try: The Madelung Constant Write a program to calculate and print the Madelung constant for sodium chloride. Use as large a value of L as you can, while still having your program run in reasonable time— say in a minute or less.

119. The Semi-Empirical Mass Formula

120. Try: Prime Factors and Prime Numbers Suppose we have an integer n and we want to know its prime factors. The prime factors can be calculated relatively easily by dividing repeatedly by all integers from 2 up to n and checking to see if the remainder is zero.

121. Try: Factor List Input an integer n Output the factor list Input=28 Output=[2,2,7]

122. Try: Factorlist def factor(n): factorlist = [] k = 2 while k <= n: while n%k == 0: factorlist.append(k) n //= k #n=n//k k += 1 return factorlist

123. Try: Optimization: Find All the Primes Finds all the primes up to n. Create a list to store the primes, which starts out with just the one prime number 2 in it. Look over n=2 to 10000 Call factor(n) If the factorlist contains only one element, the n is a prime. Add n to the prime list. Slow and stupid, any better way?

124. Try: Optimization: Find All the Primes Finds all the primes up to n. Create a list to store the primes, which starts out with just the one prime number 2 in it. For each number n from 3 to 10000 check whether the number is divisible by any of the primes in the list up to and including. As soon as you find a single prime factor you can stop checking the rest of them—you know n is not a prime. If you find no prime factors or less then n is prime and you should add it to the list. You can print out the list all in one go at the end of the program, or you can print out the individual numbers as you find them.

125. Time your code! from time import * start_time=time() blah end_time=time() used_time=end_time-start_time

126. Good Programming Style

127. Include comments in your programs. Use meaningful variable names. Use the right types of variables. Import functions first. Give your constants names. Employ user-defined functions, where appropriate. Print out partial results and updates throughout your program. Lay out your programs clearly. Don’t make your programs unnecessarily complicated.

128. Dictionary

129. Dictionary {Key:Value} >>>eng2sp = dict() >>>print(eng2sp) {} >>>eng2sp[‘one’]='uno' >>>print(eng2sp) {'one':'uno'} >>>eng2sp={'one':'uno','two':'dos','three':'tres'} >>>print(eng2sp} {'one':'uno','three':'tres','two':'dos'} >>>print(eng2sp['two']) dos >>>print(eng2sp['four']) KeyError: 'four' >>>len(eng2sp) 3

130. Something as a Key or a Value? >>>'one' in eng2sp True >>>'uno' in eng2sp False >>>vals = eng2sp.values() >>>'uno' in vals True Python2's "has_key()" is moved in Python3

131. Dictionary as a Set of Counters Want to count how many times each letter appears in a given string. Create 26 variables, one for each letter of the alphabet. Create a list with 26 elements. Convert each character to a number which is used as an index. Create a a dictionary with characters as key and counter as values.

132. Dictionary as a Set of Counters def histogram(s): d = dict() for c in s: if c not in d: d[c] = 1 else: d[c] += 1 return d >>>h = histogram('brontosaurus') >>>print(h) {'a':1, 'b':1, 'o':2, 'n':1, 's':2, 'r':2, 'u':2, 't':1 }

133. Dictionaries and Lists def invert_dict(d): inverse = dict() for key in d: val = d[key] if val not in inverse: inverse[val] = [key] else: inverse[val].append(key) return inverse >>> hist = histogram('parrot') >>> print hist {'a': 1, 'p': 1, 'r': 2, 't': 1, 'o': 1} >>> inverse = invert_dict(hist) >>> print inverse {1: ['a', 'p', 't', 'o'], 2: ['r']}

134. Fibonacci Function fibonacci(0)=0 fibonacci(1)=1 fibonacci(n)=ibonacci(n-1)+ibonacci(n-2) def fibonacci (n): if n == 0: return 0 elif n == 1: return 1 else: return fibonacci(n-1) + fibonacci(n-2)

135. Keep Tracks of the Values known = {0:0, 1:1} def fibonacci(n): if n in known: return known[n] res = fibonacci(n-1) + fibonacci(n-2) known[n] = res return res

136. Tuples

137. Tuples are Immutable A tuple is a sequence of values. The values can be any type, and they are indexed by integers, so in that respect tuples are a lot like lists. The important difference is that tuples are immutable.

138. tuple=a comma-separated list of values >>>t = 'a', 'b', 'c', 'd', 'e' >>>t = ('a', 'b', 'c', 'd', 'e') >>>t1 = 'a', >>>type(t1) >>>t2 = ('a') >>>type(t2)

139. Built-in Function: tuple >>>t = tuple() >>>print(t) () >>>t = tuple('lupins') >>>print(t) ('l', 'u', 'p', 'i', 'n', 's') >>>t = tuple(['l','u','p','i','n',s']) >>>print(t) ('l', 'u', 'p', 'i', 'n', 's')

140. Most List Operators also Work on Tuples >>> t = ('a', 'b', 'c', 'd', 'e') >>> print t[0] 'a' >>> print t[1:3] ('b', 'c') >>> t[0] = 'A' TypeError: object doesn't support item assignment >>> t = ('A',) + t[1:] >>> print t ('A', 'b', 'c', 'd', 'e')

141. Tuple Assignment >>>a, b = b, a >>>a, b = 1, 2, 3 ValueError: too many values to unpack >>>addr = 'monty@python.org' >>>uname,domain = addr.split('@') >>>print(uname) monty >>>print(domain) python.org

142. Tuples ad Return Values >>>t = divmod(7,3) >>>print(t) (2, 1) >>>quot, rem = divmod(7,3) >>>print(quot) 2 >>>print(rem) 1 def min_max(t): return min(t), max(t)

143. Variable-length argument tuples Functions can take a variable number of arguments A parameter name that begins with * gathers arguments into a tuple def printall(*agr): print args >>>printall(1, 2.0, '3') (1, 2.0, '3')

144. Scatter divmod takes exactly two arguments; it doesn't work with a tuple But you can scatter the tuple >>>t = (7,3) >>>divmod(t) TypeError: divmod expected 2 arguments, got 1 >>>divmod(*t) (2, 1)

145. Lists and Tuples zip is a built-in function that takes two or more sequences and "zip" them into a list/iterator of tuples where each tuple contains one element from each sequence. >>>s = 'abc' >>>t = [0,1,2] >>>zip(s, t) [('a',0),('b',1),('c',2)] >>>zip('Anne', 'Elk') [('A','E'),('n','l'),('n','k')]