NumPy. Computational Physics. NumPy

Transcription

1 NumPy Computational Physics NumPy

2 Outline Some Leftovers Get people on line! Write a function / Write a script NumPy NumPy Arrays; dexing; Iterating Creating Arrays Basic Operations Copying Linear Algebra

3 Calculate Something Basic python doesn't know much... We must import useful functions from the math module: > from math import sin imports the sin function To call the sin function: sin(theta) > from math import * imports all math functions To call the sin function: sin(theta) > import math imports the math module My Advice: To call the sin function: math.sin(theta) > import math as m imports math, calls it m To call the sin function: m.sin(theta) } USE THESE

4 Write a function 10]: import math 11]: def sind(theta):...: t = theta/180.*math.pi...: return(math.sin(t))...: Return when done with defi nition 12]: sind(90) Out12]: 1.0

5 Write a script Use Text Editor to create a fi le with python commands. this example: myscript.py # fi fi le: le: myscript.py myscript.py A = 'Hello 'Hello World!' World!' print print A To run the script enter: 13]: run myscript.py Out13]: Hello World!

6 Try some stuff out! Defne some numbers and operate on them (+,-,*,/) Build a long string by concatenating short strings Select substrings in your long string using the : operator. Write a degree input function to compute cosine. Write a fle with all trig functions ready to accept degree arguments (e.g. sind, cosd, tand, atand, atan2d)

7 Python Lists About NumPy Lists can have any type of data Locations of List items in memory is not predictable This limits mathematical use of Lists as arrays. NumPy Arrays All data have same type All data are together in memory List Data NumPy Array Data

8 Learning about NumPy See: wiki.scipy.org/tentative_numpy_tutorial NumPy creates a homogeneous, multidimensional, array. Elements all have same type. Each dimension of the array is called an axis. Number of axes is called the rank of the array. Positive numbers are used to index the data in the array, beginning with index 0, with one number for each axis of the array.

9 1]: 1]: import import numpy numpy as as np np 2]: 2]: a a = np.array(1,2,3],4,5,6],7,8,9]]) np.array(1,2,3],4,5,6],7,8,9]]) 3]: 3]: aa Out3]: Out3]: array(1, array(1, 2, 2, 3], 3], 4, 4, 5, 5, 6], 6], 7, 7, 8, 8, 9]]) 9]]) 4]: 4]: type(a) type(a) Out4]: Out4]: numpy.ndarray numpy.ndarray 5]: 5]: a.ndim a.ndim Out5]: Out5]: 22 6]: 6]: a.dtype a.dtype Out6]: Out6]: dtype('int64') dtype('int64') 7]: 7]: a.shape a.shape Out7]: Out7]: (3, (3, 3) 3) Example Import NumPy Create a 2D NumPy array Show the array Check type : it is a numpy.ndarray Check Dimensions Data Type Shape

10 NumPy Array Attributes Example Array: array(1, 2, 3], 4, 5, 6], 7, 8, 9]]) Attribute ndarray.ndim ndarray.shape ndarray.size ndarray.dtype Our Example Number of Dimensions 2 dimensions of array (3,3) total number of elements in array 9 type of the elements in the array int64

11 3]: 3]: aa Out3]: Out3]: Array(1, Array(1, 2, 2, 3], 3], 4, 4, 5, 5, 6], 6], 7, 7, 8, 8, 9]]) 9]]) 4]: 4]: a0,0] a0,0] Out4]: Out4]: 11 5]: 5]: a0,1] a0,1] Out5]: Out5]: 22 6]: 6]: a2,0] a2,0] Out6]: Out6]: 77 7]: 7]: a:,1] a:,1] Out7]: Out7]: array(2, array(2, 5, 5, 8]) 8]) 8]: 8]: a1,:] a1,:] Out8]: Out8]: array(4, array(4, 5, 5, 6]) 6]) 9]: 9]: a1,0:2] a1,0:2] Out9]: Out9]: array(4, array(4, 5]) 5]) dexing the Array Row index is First Column index is Second Use of colon operator to get a slice of the array.

12 3]: 3]: a a Out3]: Out3]: array(0, array(0, 2, 2, 3, 3, 8, 8, 9]) 9]) 4]: 4]: for for x x in in a: a:...:...: print print xx...:...: ]: 5]: for for i i in in range(len(a)): range(len(a)):...:...: print print ai] ai]...:...: Iterating an Array our scripts, we will often deal with the elements of an array one at a time. Python provides ways to do this. A common way is the for loop. dented statements are carried out sequentially.

13 63]: 63]: a a = np.array(3., np.array(3., 4., 4., 0.],-4., 0.],-4., 5, 5, 0.], 0.], 9.]]) 9.]]) 64]: 64]: aa Out64]: Out64]: array( array( 3., 3., 4., 4., 0.], 0.], -4., -4., 5., 5., 0.], 0.], 9.]]) 9.]]) 65]: 65]: b b = np.arange(1.0, np.arange(1.0, 5.0, 5.0, 0.5) 0.5) 66]: 66]: bb Out66]: Out66]: array( array( 1. 1.,, 1.5, 1.5, 2. 2.,, 2.5, 2.5, 3. 3.,, 3.5, 3.5, 4. 4.,, 4.5]) 4.5]) 67]: 67]: c c = = np.linspace(1.0, np.linspace(1.0, 5.0, 5.0, 9) 9) 68]: 68]: cc Out68]: Out68]: array( array( 1. 1.,, 1.5, 1.5, 2. 2.,, 2.5, 2.5, 3. 3.,, 3.5, 3.5, 4. 4.,, 4.5, 4.5, 5.0]) 5.0]) 69]: 69]: d d = = np.zeros(5) np.zeros(5) 70]: 70]: dd Out70]: Out70]: array( array( 0.]) 0.]) Creating Arrays

14 Methods for Creating Arrays Method What it does numpy.array(...]) numpy.arange(first,last,step) numpy.linspace(first,last,n) enter array values directly creates array spaced by step beginning at first and ending when value is equal to or greater than last creates an evenly spaced array with n elements beginning at first and ending with last. numpy.zeros(n) numpy.zeros( (n,m) ) creates an array of n zeros creates a 2D array of zeros with n rows, m columns

15 Basic Operations Add (+), Subtract (-), Multiply (*),Divide (/), and exponentiation (**) all operate element by element they are NOT linear algebra operations. A = np.array(1,2]) B = np.array(3,4]) A+B = 4, 6] A * B = 3, 8] A**B = 1, 16] Use np.dot to do proper multiplication according to linear algebra rules: np.dot(a,b) = A0]*B0]+A1]*B1] = 11 NumPy does all the math operations on the array as well, e.g. np.sin(a) computes the sin of all the elements.

16 Makin' Copies If I have a 2D array a and set: c = a The new array c is NOT an independent copy of a. It is just another name for the same data. To copy an array do the following: c = a.copy() This applies also to slices, as in: c = a:,1].copy()

17 63]: 63]: a a = = np.array(3., np.array(3., 4., 4., 0.],-4., 0.],-4., 5, 5, 0.], 0.], 9.]]) 9.]]) 64]: 64]: a a Out64]: Out64]: array( array( 3., 3., 4., 4., 0.], 0.], -4., -4., 5., 5., 0.], 0.], 9.]]) 9.]]) 65]: 65]: import import numpy.linalg numpy.linalg as as linalg linalg 66]: 66]: b b = = linalg.inv(a) linalg.inv(a) 67]: 67]: b b Out67]: Out67]: array( array( , , , , ], ], , , , , ], ], 0. 0.,, 0. 0.,, ]]) ]]) 86]: 86]: np.dot(b,a) np.dot(b,a) Out86]: Out86]: array( array( 1., 1., 0.], 0.], 1., 1., 0.], 0.], 1.]]) 1.]]) 87]: 87]: d d Out87]: Out87]: array( array( 1., 1., 3., 3., 4.]) 4.]) 88]: 88]: np.linalg.solve(a,d) np.linalg.solve(a,d) Out88]: Out88]: array( , array( , , , ]) ]) Linear Algebra numpy.linalg Import linalg inv() inverts the 2D array solve(a,d) fi nds p, the solution to d = a p