4.16. Array Arithmetic

Vectorized Operations

Single statement without a loop that explains a looping concept. Applies operation to each element.

import numpy as np


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

a ** 2
# array([1, 4, 9])

4.16.1. Broadcasting Rules

  1. Operations between multiple array objects are first checked for proper shape match

  2. Mathematical operators (+, -, *, /, exp, log, ...) apply element by element, on values

  3. Reduction operations (mean, std, skew, kurt, sum, prod, ...) apply to whole array, unless an axis is specified

  4. Missing values propagate, unless explicitly ignored (nanmean, nansum, ...)

4.16.2. Addition

import numpy as np


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

a + 2
# array([[3, 4, 5],
#        [6, 7, 8]])

a + a
# array([[ 2,  4,  6],
#        [ 8, 10, 12]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([4, 5])

a + b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

4.16.3. Subtraction

import numpy as np


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

a - 2
# array([[-1,  0,  1],
#        [ 2,  3,  4]])

a - a
# array([[0, 0, 0],
#        [0, 0, 0]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([4, 5])

a - b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

4.16.4. Division

import numpy as np


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

a / 2
# array([[0.5, 1. , 1.5],
#        [2. , 2.5, 3. ]])

a / a
# array([[1., 1., 1.],
#        [1., 1., 1.]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([4, 5])

a / b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

4.16.5. Square Root

import numpy as np


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

np.sqrt(a)
# array([[1., 1.41421356, 1.73205081],
#        [2., 2.23606798, 2.44948974]])

4.16.6. Modulo

import numpy as np


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

a % 2
# array([[1, 0, 1],
#        [0, 1, 0]])

a % a
# array([[0, 0, 0],
#        [0, 0, 0]])

a // a
# array([[1, 1, 1],
#        [1, 1, 1]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([4, 5])

a % b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

a // b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

4.16.7. Multiplication

import numpy as np


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

a * 2
# array([[ 2,  4,  6],
#        [ 8, 10, 12]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([4, 5])

a * b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

4.16.8. Power

import numpy as np


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

a ** 2
# array([[ 1,  4,  9],
#        [16, 25, 36]])

a * a
# array([[ 1,  4,  9],
#        [16, 25, 36]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([4, 5])

a ** b
# ValueError: operands could not be broadcast together with shapes (3,) (2,)

4.16.9. Array Multiplication

Warning

For two-dimensional arrays, multiplication * remains elementwise and does not correspond to matrix multiplication.

import numpy as np


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

b = np.array([[4, 5, 6],
              [7, 8, 9]])

a * b
# array([[ 4, 10, 18],
#        [ 7, 16, 27]])

4.16.10. Matrix Multiplication

../../_images/arithmetic-matmul.gif
../../_images/arithmetic-matmul.jpg
import numpy as np


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

b = np.array([[4, 5, 6],
              [7, 8, 9]])

a.dot(b)
# ValueError: shapes (3,) and (2,3) not aligned: 3 (dim 0) != 2 (dim 0)
import numpy as np


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

b = np.array([[1, 2],
              [3, 4],
              [5, 6]])

a.dot(b)
# array([[22, 28],
#        [49, 64]])
import numpy as np


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

b = np.array([[1, 2],
              [3, 4],
              [5, 6]])

a @ b
# array([[22, 28],
#        [49, 64]])
import numpy as np


a = np.array([1, 2, 3])
b = np.array([[4, 5, 6],
              [7, 8, 9]])

a @ b
# ValueError: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 2 is different from 3)

a.dot(b)
# ValueError: shapes (3,) and (2,3) not aligned: 3 (dim 0) != 2 (dim 0)
  • np.dot()

  • If either a or b is 0-D (scalar), it is equivalent to multiply and using numpy.multiply(a, b) or a * b is preferred.

  • If both a and b are 1-D arrays, it is inner product of vectors (without complex conjugation).

  • If both a and b are 2-D arrays, it is matrix multiplication, but using matmul or a @ b is preferred.

  • If a is an N-D array and b is a 1-D array, it is a sum product over the last axis of a and b.

  • If a is an N-D array and b is an M-D array (where M>=2), it is a sum product over the last axis of a and the second-to-last axis of b: dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])

import numpy as np


a = np.array([1, 2, 3], float)
b = np.array([0, 1, 1], float)

np.dot(a, b)
# 5.0
import numpy as np


a = np.array([[0, 1], [2, 3]], float)
b = np.array([2, 3], float)
c = np.array([[1, 1], [4, 0]], float)

a
# array([[ 0., 1.],
#        [ 2., 3.]])

np.dot(b, a)
# array([ 6., 11.])

np.dot(a, b)
# array([ 3., 13.])

np.dot(a, c)
# array([[ 4., 0.],
#        [ 14., 2.]])

np.dot(c, a)
# array([[ 2., 4.],
#        [ 0., 4.]])

4.16.11. Assignments

4.16.11.1. Arithmetic operations

English
  1. Use data from "Input" section (see below)

  2. For given: a: ndarray, b: ndarray, c: ndarray

  3. Calculate square root of each element in a and b

  4. Calculate second power (square) of each element in c

  5. Add elements from a to b

  6. Multiply the result by c

  7. Compare result with "Output" section (see below)

Polish
  1. Użyj danych z sekcji "Input" (patrz poniżej)

  2. Dla danych: a: ndarray, b: ndarray, c: ndarray

  3. Oblicz pierwiastek kwadratowy każdego z elementu w a i b

  4. Oblicz drugą potęgę (kwadrat) każdego z elementu w c

  5. Dodaj elementy z a do b

  6. Przemnóż wynik przez c

  7. Porównaj wyniki z sekcją "Output" (patrz poniżej)

Input
a = np.array([[0, 1], [2, 3]], float)
b = np.array([2, 3], float)
c = np.array([[1, 1], [4, 0]], float)
Output
array([[ 1.41421356,  2.73205081],
       [45.254834  ,  0.        ]])

4.16.11.2. Array Addition

English
  1. For given: a: ndarray, b: ndarray (see below)

  2. Add a and b

  3. Add b and a

  4. What happened?

Polish
  1. Dla danych: a: ndarray, b: ndarray (patrz sekcja input)

  2. Dodaj a i b

  3. Dodaj b i a

  4. Co się stało?

import numpy as np

a = np.array([[1, 0], [0, 1]])
b = [[4, 1], [2, 2]]

4.16.11.3. Array Multiplication

English
  1. For given: a: ndarray, b: ndarray (see below)

  2. Multiply a and b using scalar multiplication

  3. Multiply a and b using matrix multiplication

  4. Multiply b and a using scalar multiplication

  5. Multiply b and a using matrix multiplication

  6. Discuss results

Polish
  1. Dla danych: a: ndarray, b: ndarray (patrz sekcja input)

  2. Przemnóż a i b używając mnożenia skalarnego

  3. Przemnóż a i b używając mnożenia macierzowego

  4. Przemnóż b i a używając mnożenia skalarnego

  5. Przemnóż b i a używając mnożenia macierzowego

  6. Omów wyniki

import numpy as np

a = np.array([[1,0,1,0],
              [0,1,1,0],
              [3,2,1,0],
              [4,1,2,0]])

b = np.array([[4,1],
              [2,2],
              [5,1],
              [2,3]])