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Introduction to NumPy
Lab Objective: NumPy is a powerful Python package for manipulating data with multi-dimensional vectors. Its versatility and speed makes Python an ideal language for applied and computational mathematics. In this lab we introduce basic NumPy data structures and operations as a rst step to numerical computing in Python.
Arrays
In many algorithms, data can be represented mathematically as a vector or a matrix. Conceptually, a vector is just a list of numbers and a matrix is a two-dimensional list of numbers (a list of lists). However, even basic linear algebra operations like matrix multiplication are cumbersome to implement and slow to execute when data is stored this way. The NumPy module^1 [Oli06, ADH+01, Oli07] oers a much better solution. The basic object in NumPy is the array, which is conceptually similar to a matrix. The NumPy array class is called ndarray (for n-dimensional array). The simplest way to explicitly create a 1-D ndarray is to de ne a list, then cast that list as an ndarray with NumPy's array() function.
import numpy as np
Create a 1-D array by passing a list into NumPy's array() function.
np.array([8, 4, 6, 0, 2]) array([8, 4, 6, 0, 2])
The string representation has no commas or an array() label.
print(np.array([1, 3, 5, 7, 9])) [1 3 5 7 9]
The alias np for numpy is standard in the Python community. An ndarray can have arbitrarily many dimensions. A 2-D array is a 1-D array of 1-D arrays (like a list of lists), a 3-D array is a 1-D array of 2-D arrays (a list of lists of lists), and, more generally, an n-dimensional array is a 1-D array of (n − 1)-dimensional arrays (a list of lists of lists of lists...). Each dimension is called an axis. For a 2-D array, the 0 -axis indexes the rows and the 1 -axis indexes the columns. Elements are accessed using brackets and indices, with the axes separated by commas. (^1) NumPy is not part of the standard library, but it is included in most Python distributions.
2 Lab 3. Introduction to NumPy
Create a 2-D array by passing a list of lists into array().
A = np.array( [ [1, 2, 3],[4, 5, 6] ] ) print(A) [[1 2 3] [4 5 6]]
Access elements of the array with brackets.
print(A[0, 1], A[1, 2]) 2 6
The elements of a 2-D array are 1-D arrays.
A[0] array([1, 2, 3])
Problem 1. There are two main ways to perform matrix multiplication in NumPy: with NumPy's dot() function (np.dot(A, B)), or with the @ operator (A @ B). Write a function that de nes the following matrices as NumPy arrays.
A =
[
]
B =
Return the matrix product AB. For examples of array initialization and matrix multiplication, use object introspection in IPython to look up the documentation for np.ndarray, np.array() and np.dot().
In [1]: import numpy as np
In [2]: np.array? # press 'enter'
Achtung!
The @ operator was not introduced until Python 3.5. It triggers the matmul() magic method,a^ which for the ndarray is essentially a wrapper around np.dot(). If you are using a previous version of Python, always use np.dot() to perform basic matrix multiplication. aSee the lab on Object Oriented Programming for an overview of magic methods.
Basic Array Operations
NumPy arrays behave dierently with respect to the binary arithmetic operators + and * than Python lists do. For lists, + concatenates two lists and * replicates a list by a scalar amount (strings also behave this way).
4 Lab 3. Introduction to NumPy
A = np.array([[1, 2, 3],[4, 5, 6]])
'A' is a 2-D array with 2 rows, 3 columns, and 6 entries.
print(A.ndim, A.shape, A.size) 2 (2, 3) 6
Note that ndim is the number of entries in shape, and that the size of the array is the product of the entries of shape.
Array Creation Routines
In addition to casting other structures as arrays via np.array(), NumPy provides e cient ways to create certain commonly-used arrays.
Function Returns arange() Array of sequential integers (like list(range())). eye() 2-D array with ones on the diagonal and zeros elsewhere. ones() Array of given shape and type, lled with ones. ones_like() Array of ones with the same shape and type as a given array. zeros() Array of given shape and type, lled with zeros. zeros_like() Array of zeros with the same shape and type as a given array. full() Array of given shape and type, lled with a speci ed value. full_like() Full array with the same shape and type as a given array.
Each of these functions accepts the keyword argument dtype to specify the data type. Common types include np.bool_, np.int64, np.float64, and np.complex128.
A 1-D array of 5 zeros.
np.zeros(5) array([ 0., 0., 0., 0., 0.])
A 2x5 matrix (2-D array) of integer ones.
np.ones((2,5), dtype=np.int) # The shape is specified as a tuple. array([[1, 1, 1, 1, 1], [1, 1, 1, 1, 1]])
The 2x2 identity matrix.
I = np.eye(2) print(I) [[ 1. 0.] [ 0. 1.]]
Array of 3s the same size as 'I'.
np.full_like(I, 3) # Equivalent to np.full(I.shape, 3). array([[ 3., 3.], [ 3., 3.]])
Unlike native Python data structures, all elements of a NumPy array must be of the same data type. To change an existing array's data type, use the array's astype() method.
A list of integers becomes an array of integers.
x = np.array([0, 1, 2, 3, 4]) print(x) [0 1 2 3 4] x.dtype dtype('int64')
Change the data type to one of NumPy's float types.
x = x.astype(np.float64) # Equivalent to x = np.float64(x). print(x) [ 0. 1. 2. 3. 4.] # Floats are displayed with periods. x.dtype dtype('float64')
The following functions are for dealing with the diagonal, upper, or lower portion of an array.
Function Description diag() Extract a diagonal or construct a diagonal array. tril() Get the lower-triangular portion of an array by replacing entries above the diagonal with zeros. triu() Get the upper-triangular portion of an array by replacing entries below the diagonal with zeros.
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
Get only the upper triangular entries of 'A'.
np.triu(A) array([[1, 2, 3], [0, 5, 6], [0, 0, 9]])
Get the diagonal entries of 'A' as a 1-D array.
np.diag(A) array([1, 5, 9])
diag() can also be used to create a diagonal matrix from a 1-D array.
np.diag([1, 11, 111]) array([[ 1, 0, 0], [ 0, 11, 0], [ 0, 0, 111]])
See http://docs.scipy.org/doc/numpy/reference/routines.array-creation.html for the o cial documentation on NumPy's array creation routines.
Note
Indexing and slicing operations return a view of the array. Changing a view of an array also changes the original array. In other words, arrays are mutable. To create a copy of an array, use np.copy() or the array's copy() method. Changes to a copy of an array does not aect the original array, but copying an array uses more time and memory than getting a view.
Fancy Indexing
So-called fancy indexing is a second way to access or change the elements of an array. Instead of using slicing syntax, provide either an array of indices or an array of boolean values (called a mask ) to extract speci c elements.
x = np.arange(0, 50, 10) # The integers from 0 to 50 by tens. x array([ 0, 10, 20, 30, 40])
An array of integers extracts the entries of 'x' at the given indices.
index = np.array([3, 1, 4]) # Get the 3rd, 1st, and 4th elements. x[index] # Same as np.array([x[i] for i in index]). array([30, 10, 40])
A boolean array extracts the elements of 'x' at the same places as 'True'.
mask = np.array([True, False, False, True, False]) x[mask] # Get the 0th and 3rd entries. array([ 0, 30])
Fancy indexing is especially useful for extracting or changing the values of an array that meet some sort of criterion. Use comparison operators like < and == to create masks.
y = np.arange(10, 20, 2) # Every other integers from 10 to 20. y array([10, 12, 14, 16, 18])
Extract the values of 'y' larger than 15.
mask = y > 15 # Same as np.array([i > 15 for i in y]). mask array([False, False, False, True, True], dtype=bool) y[mask] # Same as y[y > 15] array([16, 18])
Change the values of 'y' that are larger than 15 to 100.
y[mask] = 100 print(y) [10 12 14 100 100]
8 Lab 3. Introduction to NumPy
While indexing and slicing always return a view, fancy indexing always returns a copy.
Problem 4. Write a function that accepts a single array as input. Make a copy of the array, then use fancy indexing to set all negative entries of the copy to 0. Return the copy.
Array Manipulation
Shaping
An array's shape attribute describes its dimensions. Use np.reshape() or the array's reshape() method to give an array a new shape. The total number of entries in the old array and the new array must be the same in order for the shaping to work correctly. Using a -1 in the new shape tuple makes the speci ed dimension as long as necessary.
A = np.arange(12) # The integers from 0 to 12 (exclusive). print(A) [ 0 1 2 3 4 5 6 7 8 9 10 11]
'A' has 12 entries, so it can be reshaped into a 3x4 matrix.
A.reshape((3,4)) # The new shape is specified as a tuple. y = np.arange(10, 20, 2) # Every other integers from 10 to 20. >>> y array([10, 12, 14, 16, 18]) # Extract the values of 'y' larger than 15. >>> mask = y > 15 # Same as np.array([i > 15 for i in y]). >>> mask array([False, False, False, True, True], dtype=bool) >>> y[mask] # Same as y[y > 15] array([16, 18]) # Change the values of 'y' that are larger than 15 to 100. >>> y[mask] = 100 >>> print(y) [10 12 14 100 100] 8 Lab 3. Introduction to NumPy While indexing and slicing always return a view, fancy indexing always returns a copy. Problem 4. Write a function that accepts a single array as input. Make a copy of the array, then use fancy indexing to set all negative entries of the copy to 0. Return the copy. ### Array Manipulation #### Shaping An array's shape attribute describes its dimensions. Use np.reshape() or the array's reshape() method to give an array a new shape. The total number of entries in the old array and the new array must be the same in order for the shaping to work correctly. Using a -1 in the new shape tuple makes the speci ed dimension as long as necessary. >>> A = np.arange(12) # The integers from 0 to 12 (exclusive). >>> print(A) [ 0 1 2 3 4 5 6 7 8 9 10 11] # 'A' has 12 entries, so it can be reshaped into a 3x4 matrix. >>> A.reshape((3,4)) # The new shape is specified as a tuple. array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]])
Reshape 'A' into an array with 2 rows and the appropriate number of columns.
A.reshape((2,-1)) 15 # Same as np.array([i > 15 for i in y]). >>> mask array([False, False, False, True, True], dtype=bool) >>> y[mask] # Same as y[y > 15] array([16, 18]) # Change the values of 'y' that are larger than 15 to 100. >>> y[mask] = 100 >>> print(y) [10 12 14 100 100] 8 Lab 3. Introduction to NumPy While indexing and slicing always return a view, fancy indexing always returns a copy. Problem 4. Write a function that accepts a single array as input. Make a copy of the array, then use fancy indexing to set all negative entries of the copy to 0. Return the copy. ### Array Manipulation #### Shaping An array's shape attribute describes its dimensions. Use np.reshape() or the array's reshape() method to give an array a new shape. The total number of entries in the old array and the new array must be the same in order for the shaping to work correctly. Using a -1 in the new shape tuple makes the speci ed dimension as long as necessary. >>> A = np.arange(12) # The integers from 0 to 12 (exclusive). >>> print(A) [ 0 1 2 3 4 5 6 7 8 9 10 11] # 'A' has 12 entries, so it can be reshaped into a 3x4 matrix. >>> A.reshape((3,4)) # The new shape is specified as a tuple. array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) # Reshape 'A' into an array with 2 rows and the appropriate number of columns. >>> A.reshape((2,-1)) array([[ 0, 1, 2, 3, 4, 5], [ 6, 7, 8, 9, 10, 11]])
Use np.ravel() to atten a multi-dimensional array into a 1-D array and np.transpose() or the T attribute to transpose a 2-D array in the matrix sense.
A = np.arange(12).reshape((3,4)) A array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]])
Flatten 'A' into a one-dimensional array.
np.ravel(A) # Equivalent to A.reshape(A.size) array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
Transpose the matrix 'A'.
A.T # Equivalent to np.transpose(A). array([[ 0, 4, 8],
10 Lab 3. Introduction to NumPy
Function Description concatenate() Join a sequence of arrays along an existing axis hstack() Stack arrays in sequence horizontally (column wise). vstack() Stack arrays in sequence vertically (row wise). column_stack() Stack 1-D arrays as columns into a 2-D array.
A = np.arange(6).reshape((2,3)) B = np.zeros((4,3))
vstack() stacks arrays vertically (row-wise).
np.vstack((A,B,A))
array([[ 0., 1., 2.], # A [ 3., 4., 5.], [ 0., 0., 0.], # B [ 0., 0., 0.], [ 0., 0., 0.], [ 0., 0., 0.], [ 0., 1., 2.], # A [ 3., 4., 5.]])
A = A.T B = np.ones((3,4))
hstack() stacks arrays horizontally (column-wise).
np.hstack((A,B,A)) array([[ 0., 3., 1., 1., 1., 1., 0., 3.], [ 1., 4., 1., 1., 1., 1., 1., 4.], [ 2., 5., 1., 1., 1., 1., 2., 5.]])
column_stack() stacks arrays horizontally, including 1-D arrays.
np.column_stack((A, np.zeros(3), np.ones(3), np.full(3, 2))) array([[ 0., 3., 0., 1., 2.], [ 1., 4., 0., 1., 2.], [ 2., 5., 0., 1., 2.]])
See http://docs.scipy.org/doc/numpy-1.10.1/reference/routines.array-manipulation.html for more array manipulation routines and documentation.
Problem 5. Write a function that de nes the following matrices as NumPy arrays.
A =
[
]
B =
C =
Use NumPy's stacking functions to create and return the block matrix:
0 AT^ I
A 0 0
B 0 C
where I is the 3 × 3 identity matrix and each 0 is a matrix of all zeros of appropriate size. A block matrix of this form is used in the interior point method for linear optimization.
Array Broadcasting
Many matrix operations make sense only when the two operands have the same shape, such as element-wise addition. Array broadcasting extends such operations to accept some (but not all) operands with dierent shapes, and occurs automatically whenever possible. Suppose, for example, that we would like to add dierent values to the columns of an m × n matrix A. Adding a 1-D array x with the n entries to A will automatically do this correctly. To add dierent values to the dierent rows of A, rst reshape a 1-D array of m values into a column array. Broadcasting then correctly takes care of the operation. Broadcasting can also occur between two 1-D arrays, once they are reshaped appropriately.
A = np.arange(12).reshape((4,3)) x = np.arange(3) A array([[ 0, 1, 2], [ 3, 4, 5], [ 6, 7, 8], [ 9, 10, 11]]) x array([0, 1, 2])
Add the entries of 'x' to the corresponding columns of 'A'.
A + x array([[ 0, 2, 4],
[ 3, 5, 7], [ 6, 8, 10], [ 9, 11, 13]])
y = np.arange(0, 40, 10).reshape((4,1)) y array([[ 0], [10], [20], [30]])
Add the entries of 'y' to the corresponding rows of 'A'.
A + y array([[ 0, 1, 2], [13, 14, 15], [26, 27, 28],
x = np.arange(-2, 3) np.tan(x) array([ 2.18503986, -1.55740772, 0. , 1.55740772, -2.18503986]) math.tan(x) Traceback (most recent call last): File "", line 1, in TypeError: only length-1 arrays can be converted to Python scalars
Always use universal NumPy functions, not the math module, when working with arrays.
Other Array Methods
The np.ndarray class itself has many useful methods for numerical computations.
Method Returns all() True if all elements evaluate to True. any() True if any elements evaluate to True. argmax() Index of the maximum value. argmin() Index of the minimum value. argsort() Indices that would sort the array. clip() restrict values in an array to t within a given range max() The maximum element of the array. mean() The average value of the array. min() The minimum element of the array. sort() Return nothing; sort the array in-place. std() The standard deviation of the array. sum() The sum of the elements of the array. var() The variance of the array.
Each of these np.ndarray methods has an equivalent NumPy function. For example, A.max() and np.max(A) operate the same way. The one exception is the sort() function: np.sort() returns a sorted copy of the array, while A.sort() sorts the array in-place and returns nothing. Every method listed can operate along an axis via the keyword argument axis. If axis is speci ed for a method on an n-D array, the return value is an (n − 1)-D array, the speci ed axis having been collapsed in the evaluation process. If axis is not speci ed, the return value is usually a scalar. Refer to the NumPy Visual Guide in the appendix for more visual examples.
A = np.arange(9).reshape((3,3)) A array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
Find the maximum value in the entire array.
A.max() 8
Find the minimum value of each column.
14 Lab 3. Introduction to NumPy
A.min(axis=0) # np.array([min(A[:,i]) for i in range(3)]) array([0, 1, 2])
Compute the sum of each row.
A.sum(axis=1) # np.array([sum(A[i,:]) for i in range(3)]) array([3, 12, 21])
See http://docs.scipy.org/doc/numpy/reference/generated/numpy.ndarray.html for a more comprehensive list of array methods.
Problem 6. A matrix is called row-stochastica^ if its rows each sum to 1. Stochastic matrices are fundamentally important for nite discrete random processes and some machine learning algorithms. Write a function than accepts a matrix (as a 2-D array). Divide each row of the matrix by the row sum and return the new row-stochastic matrix. Use array broadcasting and the axis argument instead of a loop. aSimilarly, a matrix is called column-stochastic if its columns each sum to 1.
Problem 7. This problem comes from https://projecteuler.net. In the 20 × 20 grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08 49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00 81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65 52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91 22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80 24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50 32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70 67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21 24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72 21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95 78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92 16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57 86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58 19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40 04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66 88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69 04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36 20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16 20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54 01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48
The product of these numbers is 26 × 63 × 78 × 14 = 1788696. Write a function that returns the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the grid.
16 Lab 3. Introduction to NumPy
Additional Material
Random Sampling
The submodule np.random holds many functions for creating arrays of random values chosen from probability distributions such as the uniform, normal, and multinomial distributions. It also contains some utility functions for getting non-distributional random samples, such as random integers or random samples from a given array.
Function Description choice() Take random samples from a 1-D array. random() Uniformly distributed oats over [0, 1). randint() Random integers over a half-open interval. random_integers() Random integers over a closed interval. randn() Sample from the standard normal distribution. permutation() Randomly permute a sequence / generate a random sequence.
Function Distribution beta() Beta distribution over [0, 1]. binomial() Binomial distribution. exponential() Exponential distribution. gamma() Gamma distribution. geometric() Geometric distribution. multinomial() Multivariate generalization of the binomial distribution. multivariate_normal() Multivariate generalization of the normal distribution. normal() Normal / Gaussian distribution. poisson() Poisson distribution. uniform() Uniform distribution.
Note that many of these functions have counterparts in the standard library's random module. These NumPy functions, however, are much better suited for working with large collections of random samples.
5 uniformly distributed values in the interval [0, 1).
np.random.random(5) array([ 0.21845499, 0.73352537, 0.28064456, 0.66878454, 0.44138609])
A 2x5 matrix (2-D array) of integers in the interval [10, 20).
np.random.randint(10, 20, (2,5)) array([[17, 12, 13, 13, 18], [16, 10, 12, 18, 12]])
Saving and Loading Arrays
It is often useful to save an array as a le for later use. NumPy provides several easy methods for saving and loading array data.
Function Description save() Save a single array to a .npy le. savez() Save multiple arrays to a .npz le. savetxt() Save a single array to a .txt le. load() Load and return an array or arrays from a .npy or .npz le. loadtxt() Load and return an array from a text le.
Save a 100x100 matrix of uniformly distributed random values.
x = np.random.random((100,100)) np.save("uniform.npy", x) # Or np.savetxt("uniform.txt", x).
Read the array from the file and check that it matches the original.
y = np.load("uniform.npy") # Or np.loadtxt("uniform.txt"). np.allclose(x, y) # Check that x and y are close entry-wise. True
To save several arrays to a single le, specify a keyword argument for each array in np.savez(). Then np.load() will return a dictionary-like object with the keyword parameter names from the save command as the keys.
Save two 100x100 matrices of normally distributed random values.
x = np.random.randn(100,100) y = np.random.randn(100,100) np.savez("normal.npz", first=x, second=y)
Read the arrays from the file and check that they match the original.
arrays = np.load("normal.npz") np.allclose(x, arrays["first"]) True np.allclose(y, arrays["second"]) True
Bibliography
[ADH+01] David Ascher, Paul F Dubois, Konrad Hinsen, Jim Hugunin, Travis Oliphant, et al. Numerical python, 2001. [1]
[Oli06] Travis E Oliphant. A guide to NumPy, volume 1. Trelgol Publishing USA, 2006. [1]
[Oli07] Travis E Oliphant. Python for scienti c computing. Computing in Science & Engineering, 9(3), 2007. [1]
