Array math

Array math#

Basic mathematical functions operate elementwise on arrays, and are available both as operator overloads and as functions in the numpy module:

import numpy as np

a = np.array([0,1,2,3,4,5])
#a = np.arange(0,6)
print(a)

[0 1 2 3 4 5]

a + 2

array([2, 3, 4, 5, 6, 7])

a - 2

array([-2, -1,  0,  1,  2,  3])

a * 2

array([ 0,  2,  4,  6,  8, 10])

a / 2

array([0. , 0.5, 1. , 1.5, 2. , 2.5])

a + a

array([ 0,  2,  4,  6,  8, 10])

a * a

array([ 0,  1,  4,  9, 16, 25])

a - a

array([0, 0, 0, 0, 0, 0])

a/a # Warning on division by zero, but not an error!

<ipython-input-9-1b9ae9b69587>:1: RuntimeWarning: invalid value encountered in true_divide
  a/a # Warning on division by zero, but not an error!

array([nan,  1.,  1.,  1.,  1.,  1.])

1/a # Also warning, but not an error instead infinity

<ipython-input-10-6f47904706aa>:1: RuntimeWarning: divide by zero encountered in true_divide
  1/a # Also warning, but not an error instead infinity

array([       inf, 1.        , 0.5       , 0.33333333, 0.25      ,
       0.2       ])

b = np.array([1,0,1,0,1,0])
a + b

array([1, 1, 3, 3, 5, 5])

a ** 3

array([  0,   1,   8,  27,  64, 125], dtype=int32)

Universal Array Functions (Mathematical operations)#

#Square Roots
np.sqrt(a)

array([0.        , 1.        , 1.41421356, 1.73205081, 2.        ,
       2.23606798])

#exponential (e^)
np.exp(a)

array([  1.        ,   2.71828183,   7.3890561 ,  20.08553692,
        54.59815003, 148.4131591 ])

np.max(a) #same as arr.max()

5

np.min(a)

0

np.sin(a)

array([ 0.        ,  0.84147098,  0.90929743,  0.14112001, -0.7568025 ,
       -0.95892427])

np.cos(a)

array([ 1.        ,  0.54030231, -0.41614684, -0.9899925 , -0.65364362,
        0.28366219])

import numpy as np

x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)
print(x)
print(y)

# Elementwise sum; both produce the array
# [[ 6.0  8.0]
#  [10.0 12.0]]
print(x + y)
print(np.add(x, y))

# Elementwise difference; both produce the array
# [[-4.0 -4.0]
#  [-4.0 -4.0]]
print(x - y)
print(np.subtract(x, y))

# Elementwise product; both produce the array
# [[ 5.0 12.0]
#  [21.0 32.0]]
print(x * y)
print(np.multiply(x, y))

# Elementwise division; both produce the array
# [[ 0.2         0.33333333]
#  [ 0.42857143  0.5       ]]
print(x / y)
print(np.divide(x, y))

# Elementwise square root; produces the array
# [[ 1.          1.41421356]
#  [ 1.73205081  2.        ]]
print(np.sqrt(x))

[[1. 2.]
 [3. 4.]]
[[5. 6.]
 [7. 8.]]
[[ 6.  8.]
 [10. 12.]]
[[ 6.  8.]
 [10. 12.]]
[[-4. -4.]
 [-4. -4.]]
[[-4. -4.]
 [-4. -4.]]
[[ 5. 12.]
 [21. 32.]]
[[ 5. 12.]
 [21. 32.]]
[[0.2        0.33333333]
 [0.42857143 0.5       ]]
[[0.2        0.33333333]
 [0.42857143 0.5       ]]
[[1.         1.41421356]
 [1.73205081 2.        ]]

Note: that unlike MATLAB, * is elementwise multiplication, not matrix multiplication. We instead use the dot function to compute inner products of vectors, to multiply a vector by a matrix, and to multiply matrices. dot is available both as a function in the numpy module and as an instance method of array objects:

### Dot product: product of two arrays

f = np.array([1,2])
g = np.array([4,5])
### 1*4+2*5
np.dot(f, g)

14

import numpy as np

x = np.array([[1,2],[3,4]])
y = np.array([[5,6],[7,8]])

v = np.array([9,10])
w = np.array([11, 12])

# Inner product of vectors; both produce 219
print(v.dot(w))
print(np.dot(v, w))

# Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v))

# Matrix / matrix product; both produce the rank 2 array
# [[19 22]
#  [43 50]]
print(x.dot(y))
print(np.dot(x, y))

219
219
[29 67]
[29 67]
[[19 22]
 [43 50]]
[[19 22]
 [43 50]]

Numpy provides many useful functions for performing computations on arrays; one of the most useful is sum:

import numpy as np

x = np.array([[1,2],[3,4]])

print(np.sum(x))  # Compute sum of all elements; prints "10"
print(np.sum(x, axis=0))  # Compute sum of each column; prints "[4 6]"
print(np.sum(x, axis=1))  # Compute sum of each row; prints "[3 7]"

10
[4 6]
[3 7]

You can find the full list of mathematical functions provided by numpy in this documentation.

Apart from computing mathematical functions using arrays, we frequently need to reshape or otherwise manipulate data in arrays. The simplest example of this type of operation is transposing a matrix; to transpose a matrix, simply use the T attribute of an array object:

import numpy as np

x = np.array([[1,2], [3,4]])
print(x)    # Prints "[[1 2]
            #          [3 4]]"
print(x.T)  # Prints "[[1 3]
            #          [2 4]]"

# Note that taking the transpose of a rank 1 array does nothing:
v = np.array([1,2,3])
print(v)    # Prints "[1 2 3]"
print(v.T)  # Prints "[1 2 3]"

[[1 2]
 [3 4]]
[[1 3]
 [2 4]]
[1 2 3]
[1 2 3]

Numpy provides many more functions for manipulating arrays; you can see the full list in the documentation.

Matrix Multiplication#

The Numpy matmul() function is used to return the matrix product of 2 arrays.

a = np.ones((2,3))
print(a)

b = np.full((3,2), 2)
print(b)

np.matmul(a,b) # matmul() function is used to return the matrix product of 2 arrays. 

[[1. 1. 1.]
 [1. 1. 1.]]
[[2 2]
 [2 2]
 [2 2]]

array([[6., 6.],
       [6., 6.]])

### Matmul: matruc product of two arrays

h = [[1,2],[3,4]] 
i = [[5,6],[7,8]] 
### 1*5+2*7 = 19
np.matmul(h, i)

array([[19, 22],
       [43, 50]])

Determinant#

Last but not least, if you need to compute the determinant, you can use np.linalg.det().

Note: numpy takes care of the dimension.

# Find the determinant

c = np.identity(3)
np.linalg.det(c)

1.0

## Determinant 2*2 matrix

5*8-7*6
np.linalg.det(i)

-1.999999999999999

## Reference docs (https://docs.scipy.org/doc/numpy/reference/routines.linalg.html)

# Determinant
# Trace
# Singular Vector Decomposition
# Eigenvalues
# Matrix Norm
# Inverse
# Etc...

Broadcasting#

Broadcasting is a powerful mechanism that allows numpy to work with arrays of different shapes when performing arithmetic operations. Frequently we have a smaller array and a larger array, and we want to use the smaller array multiple times to perform some operation on the larger array.

For example, suppose that we want to add a constant vector to each row of a matrix. We could do it like this:

import numpy as np

# We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = np.empty_like(x)   # Create an empty matrix with the same shape as x

# Add the vector v to each row of the matrix x with an explicit loop
for i in range(4):
    y[i, :] = x[i, :] + v

# Now y is the following
# [[ 2  2  4]
#  [ 5  5  7]
#  [ 8  8 10]
#  [11 11 13]]
print(y)

[[ 2  2  4]
 [ 5  5  7]
 [ 8  8 10]
 [11 11 13]]

This works; however when the matrix x is very large, computing an explicit loop in Python could be slow. Note that adding the vector v to each row of the matrix x is equivalent to forming a matrix vv by stacking multiple copies of v vertically, then performing elementwise summation of x and vv. We could implement this approach like this:

import numpy as np

# We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
vv = np.tile(v, (4, 1))   # Stack 4 copies of v on top of each other
print(vv)                 # Prints "[[1 0 1]
                          #          [1 0 1]
                          #          [1 0 1]
                          #          [1 0 1]]"
y = x + vv  # Add x and vv elementwise
print(y)  # Prints "[[ 2  2  4
          #          [ 5  5  7]
          #          [ 8  8 10]
          #          [11 11 13]]"

[[1 0 1]
 [1 0 1]
 [1 0 1]
 [1 0 1]]
[[ 2  2  4]
 [ 5  5  7]
 [ 8  8 10]
 [11 11 13]]

Numpy broadcasting allows us to perform this computation without actually creating multiple copies of v. Consider this version, using broadcasting:

import numpy as np

# We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = x + v  # Add v to each row of x using broadcasting
print(y)  # Prints "[[ 2  2  4]
          #          [ 5  5  7]
          #          [ 8  8 10]
          #          [11 11 13]]"

[[ 2  2  4]
 [ 5  5  7]
 [ 8  8 10]
 [11 11 13]]

Explanation:

The line y = x + v works even though x has shape (4, 3) and v has shape (3,) due to broadcasting; this line works as if v actually had shape (4, 3), where each row was a copy of v, and the sum was performed elementwise.

Broadcasting two arrays together follows these rules:

  • If the arrays do not have the same rank, prepend the shape of the lower rank array with 1s until both shapes have the same length.

  • The two arrays are said to be compatible in a dimension if they have the same size in the dimension, or if one of the arrays has size 1 in that dimension.

  • The arrays can be broadcast together if they are compatible in all dimensions.

  • After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.

  • In any dimension where one array had size 1 and the other array had size greater than 1, the first array behaves as if it were copied along that dimension

If this explanation does not make sense, try reading the explanation from this documentation or this explanation.

Functions that support broadcasting are known as universal functions. You can find the list of all universal functions in this documentation.

Here are some applications of broadcasting:

import numpy as np

# Compute outer product of vectors
v = np.array([1,2,3])  # v has shape (3,)
w = np.array([4,5])    # w has shape (2,)
# To compute an outer product, we first reshape v to be a column
# vector of shape (3, 1); we can then broadcast it against w to yield
# an output of shape (3, 2), which is the outer product of v and w:
# [[ 4  5]
#  [ 8 10]
#  [12 15]]
print(np.reshape(v, (3, 1)) * w)

# Add a vector to each row of a matrix
x = np.array([[1,2,3], [4,5,6]])
# x has shape (2, 3) and v has shape (3,) so they broadcast to (2, 3),
# giving the following matrix:
# [[2 4 6]
#  [5 7 9]]
print(x + v)

# Add a vector to each column of a matrix
# x has shape (2, 3) and w has shape (2,).
# If we transpose x then it has shape (3, 2) and can be broadcast
# against w to yield a result of shape (3, 2); transposing this result
# yields the final result of shape (2, 3) which is the matrix x with
# the vector w added to each column. Gives the following matrix:
# [[ 5  6  7]
#  [ 9 10 11]]
print((x.T + w).T)
# Another solution is to reshape w to be a column vector of shape (2, 1);
# we can then broadcast it directly against x to produce the same
# output.
print(x + np.reshape(w, (2, 1)))

# Multiply a matrix by a constant:
# x has shape (2, 3). Numpy treats scalars as arrays of shape ();
# these can be broadcast together to shape (2, 3), producing the
# following array:
# [[ 2  4  6]
#  [ 8 10 12]]
print(x * 2)

[[ 4  5]
 [ 8 10]
 [12 15]]
[[2 4 6]
 [5 7 9]]
[[ 5  6  7]
 [ 9 10 11]]
[[ 5  6  7]
 [ 9 10 11]]
[[ 2  4  6]
 [ 8 10 12]]

  • Why is broadcasting useful?

    • Broadcasting typically makes your code more concise and faster, so you should strive to use it where possible. It lets you perform operations between arrays that are compatable, but not nescessarily identical, in shape. This makes your code cleaner.

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