Computation for Physics 計算物理概論

1. Computation for Physics計算物理概論 Introduction to NumPy and SciPy

2. List v.s. Array (numpy package) • List • Variable length, you can add or remove elements. • Elements can be of different types. • Flexible. • Array • The number of elements is fixed. • The elements must be of the same type. • Behave roughly like vectors and matrices. • Faster.

3. 1D Array >>>from numpy import zeros >>>a = zeros(4, float) >>>print(a) [0. 0. 0. 0.] >>>a = zeros(5, float) >>>print(a) [0 0 0 0 0] >>>a = zeros(3,complex) >>>print(a)

4. 2D Array >>>a = zeros([3,4],float) >>>print(a) [[ 0. 0. 0. 0.] [ 0. 0. 0. 0.] [ 0. 0. 0. 0.]]

5. Array Initialization in NumPy • zeros • Create an array with all elements equal to zero. • ones • Create an array with all elements equal to one. • empty • Create an empty array. (Faster.) • In practice the aren't empty. (Random stuff in the memory).

6. List  Array  List >>>r = [1.0, 1.5, -2.2] >>>a = array(r,float) >>>a = array([1.0, 1.5, -2.2], float) >>>print(a[0]) 1.0 >>>a = array([[1,2,3],[4,5,6]], int) >>>print(a) [[ 1 2 3] [ 4 5 6]] >>>print(a) >>>r = list(a)

7. Accessing and Modifying Elements >>>a = array([1.0, 1.5, -2.2], float) >>>a[0] 1.0 >>>a[2]=4 (4 will be converted into float) >>>a = array([[1,2,3],[4,5,6]], int) >>>a[0][1] 2 >>>a[0][1]=0 >>>print(a) [[ 1 0 3] [ 4 5 6]]

8. Reading a 1D Array from a File values.txt 1.0 1.5 -2.2 2.6 >>>from numpy import loadtxt >>>a = loadtxt("values.txt", float) >>>print(a) [ 1. 1.5 -2.2 2.6]

9. Reading a 2D Array from a File values.txt 1 2 3 4 3 4 5 6 5 6 7 8 >>>from numpy import loadtxt >>>a = loadtxt("values.txt", float) >>>print(a) [[ 1. 2. 3. 4.] [ 3. 4. 5. 6.] [ 5. 6. 7. 8.]]

10. Arithmetic with Arrays >>>a[0] = a[1] + 1 >>>x = a[2]**2 -2*a[3]/y >>>r = [1,2,3,4] >>>a = array(r, int) >>>s = 2*r >>>b = 2*a >>>print(s) [1,2,3,4,1,2,3,4] >>>print(b) [2 4 6 8]

11. Add or Subtract Two Arrays >>>a = array([1,2,3,4],int) >>>b = array([2,4,6,8],int) >>>print(a+b) [3 6 9 12] >>>print(b-a) [1 2 3 4] (Two arrays must have the same size) >>>>print(a+1)

12. Multiply Two Arrays ≠ Dot Product >>>a = array([1,2,3,4],int) >>>b = array([2,4,6,8],int) >>>print(a*b) [2 8 18 32] >>>print(b/a) [2.0 2.0 2.0 2.0]

13. "Dot Product" of Two 1D Arrays="Inner Product" >>>from numpy import array, dot >>>a = array([1,2,3,4],int) >>>b = array([2,4,5,8],int) >>>print(dot(a,b)) 60

14. "Dot Product" of Two 2D Arrays = "Matrix Product" >>>a = array([[1,3],[2,4]],int) >>>b = array([[4,-2],[-3,1]],int) >>>c = array([[1,2],[2,1]],int) >>>print(dot(a,b)+2*c) [[-3 5] [ 0 2]]

15. Array Functions • Most list functions can be applied to 1D array • sum • max • min • len • You can also apply math functions >>>a = array([1,2,3,4],float) >>>print(sin(a)) [0.84147098 0.90929743 0.14112001 -0.7568025]

16. Array Methods >>>a = array([1,2,3,4],float) >>>a.size 4 >>>a.shape (4,) >>>a = array([[1,2,3],[4,5,6],int) >>>a.size 6 >>>a.shape (2,3)

17. Try: Average of a Set of Values in a File • A set of numbers stored in a file values.txt. • We don't know how many numbers there are. • Want to calculate their mean.

18. Mean and Mean-Square from numpy import loadtxt values = loadtxt("values.txt",float) mean = sum(values)/len(values) mean_sqr = sum(values*values)/len(values) print(mean) print(mean_sqr)

19. Geometric Mean from numpy import loadtxt,log,exp from math import log,exp values = loadtxt("values.txt",float) geometric=exp(sum(log(values))/len(values)) print(geometric)

20. numpy.arange • arange([start],stop[,step],dtype=None) • Return evenly spaced values within a given interval. • Values are generated within the half-open interval [start, stop) (in other words, the interval including start but excluding stop). For integer arguments the function is equivalent to the Python built-in range function, but returns an ndarray rather than a list. • When using a non-integer step, such as 0.1, the results will often not be consistent. It is better to use linspace for these cases. >>> np.arange(3) array([0, 1, 2]) >>> np.arange(3.0) array([ 0., 1., 2.]) >>> np.arange(3,7) array([3, 4, 5, 6]) >>> np.arange(3,7,2) array([3, 5])

21. numpy.linspace • Return evenly spaced numbers over a specified interval. • Return evenly spaced numbers over a specified interval. • Returns num evenly spaced samples, calculated over the interval [start, stop ]. • The endpoint of the interval can optionally be excluded. >>>linspace(2.0, 3.0, num=5) array([ 2., 2.25, 2.5, 2.75, 3. ]) >>>linspace(2.0, 3.0, num=5, endpoint=False) array([ 2. , 2.2, 2.4, 2.6, 2.8]) >>>linspace(2.0, 3.0, num=5, retstep=True) (array([ 2. , 2.25, 2.5 , 2.75, 3. ]), 0.25)

22. numpy.random • import numpy.random • numpy.random.random() • from numpy.random import random • random() • from numpy.random import * • random()

23. numpy.random • seed([seed]) • get_state() • set_state(state) • random(size)=random_sample(size)=ranf([size])=sample([size]) • Return random floats in the half-open interval [0.0, 1.0). • randint(low,high=None,size=None) • Return random integers from low (inclusive) to high (exclusive). • choice(a[,size,replace,p]) • shuffle(X) • permutation(X)

24. Try: Random Number • Input N. • Generate N random numbers. • Output the mean and the standard deviation.