6.3. Array Broadcasting

Vector

Broadcasting

Matrix Multiplication

6.3.1. SetUp

>>> import numpy as np

6.3.2. Broadcasting Rules

• Source [1]

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, ...)

6.3.3. Addition

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a + a
array([[ 2,  4,  6],
       [ 8, 10, 12]])
>>>
>>> a + b
array([[ 5,  7,  9],
       [11, 13, 15]])
>>>
>>> a + c
array([[2, 4, 6],
       [5, 7, 9]])
>>>
>>> a + d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.4. Subtraction

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a - a
array([[0, 0, 0],
       [0, 0, 0]])
>>>
>>> a - b
array([[-3, -3, -3],
       [-3, -3, -3]])
>>>
>>> a - c
array([[0, 0, 0],
       [3, 3, 3]])
>>>
>>> a - d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.5. True Division

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a / a
array([[1., 1., 1.],
       [1., 1., 1.]])
>>>
>>> a / b
array([[0.25      , 0.4       , 0.5       ],
       [0.57142857, 0.625     , 0.66666667]])
>>>
>>> a / c
array([[1. , 1. , 1. ],
       [4. , 2.5, 2. ]])
>>>
>>> a / d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.6. Floor Division

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a // a
array([[1, 1, 1],
       [1, 1, 1]])
>>>
>>> a // b
array([[0, 0, 0],
       [0, 0, 0]])
>>>
>>> a // c
array([[1, 1, 1],
       [4, 2, 2]])
>>>
>>> a // d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.7. Modulo

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a % a
array([[0, 0, 0],
       [0, 0, 0]])
>>>
>>> a % b
array([[1, 2, 3],
       [4, 5, 6]])
>>>
>>> a % c
array([[0, 0, 0],
       [0, 1, 0]])
>>>
>>> a % d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.8. Power

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a ** a
array([[    1,     4,    27],
       [  256,  3125, 46656]])
>>>
>>> a ** b
array([[       1,       32,      729],
       [   16384,   390625, 10077696]])
>>>
>>> a ** c
array([[  1,   4,  27],
       [  4,  25, 216]])
>>>
>>> a ** d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.9. Root

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a ** (1/a)
array([[1.        , 1.41421356, 1.44224957],
       [1.41421356, 1.37972966, 1.34800615]])
>>>
>>> a ** (1/b)
array([[1.        , 1.14869835, 1.20093696],
       [1.21901365, 1.22284454, 1.22028494]])
>>>
>>> a ** (1/c)
array([[1.        , 1.41421356, 1.44224957],
       [4.        , 2.23606798, 1.81712059]])
>>>
>>> a ** (1/d)  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.10. Array Multiplication

• Multiplication * remains elementwise

• Does not correspond to matrix multiplication

>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> b = np.array([[4, 5, 6], [7, 8, 9]])
>>> c = np.array([1, 2, 3])
>>> d = np.array([4, 5])
>>>
>>> a * a
array([[ 1,  4,  9],
       [16, 25, 36]])
>>>
>>> a * b
array([[ 4, 10, 18],
       [28, 40, 54]])
>>>
>>> a * c
array([[ 1,  4,  9],
       [ 4, 10, 18]])
>>>
>>> a * d  
Traceback (most recent call last):
ValueError: operands could not be broadcast together with shapes (2,3) (2,)

6.3.11. Matrix Multiplication

>>> 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]])

>>> a = np.array([[1, 2, 3],
...               [4, 5, 6]])
>>>
>>> b = np.array([[4, 5, 6],
...               [7, 8, 9]])
>>>
>>> a @ b  
Traceback (most recent call last):
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)

6.3.12. Dot

• 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])

>>> a = np.array([1, 2, 3], float)
>>> b = np.array([0, 1, 1], float)
>>>
>>> np.dot(a, b)
5.0

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

6.3.13. References

[1]

Alex Chabot-Leclerc. Introduction to Numerical Computing with NumPy: Broadcasting Rules. Year: 2019. Retrieved: 2019-11-31. URL: https://youtu.be/ZB7BZMhfPgk?t=5142

6.3.14. Assignments

Code 6.62. Solution

"""
* Assignment: Numpy Broadcasting Arithmetic
* Complexity: easy
* Lines of code: 4 lines
* Time: 3 min

English:
    1. Define `a: np.ndarray` with square root of each element in `A`
    2. Define `b: np.ndarray` with square root of each element in `B`
    3. Define `c: np.ndarray` with second power (square) of each element in `C`
    4. Add elements from `a` to `b`
    5. Multiply the result by `c`
    6. Run doctests - all must succeed

Polish:
    1. Zdefiniuj `a: np.ndarray` z pierwiastkiem kwadratowym każdego elementu `A`
    2. Zdefiniuj `b: np.ndarray` z pierwiastkiem kwadratowym każdego elementu `B`
    3. Zdefiniu `c: np.ndarray` z drugą potęgą (kwadratem) każdego z elementu w `C`
    4. Dodaj elementy z `a` do `b`
    5. Przemnóż wynik przez `c`
    6. Uruchom doctesty - wszystkie muszą się powieść

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> assert result is not Ellipsis, \
    'Assign result to variable: `result`'
    >>> assert type(result) is np.ndarray, \
    'Variable `result` has invalid type, expected: np.ndarray'

    >>> result
    array([[ 1.41421356,  2.73205081],
           [45.254834  ,  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)

# Square root of each element in `A` use np.pow()
# type: np.ndarray
a = ...

# Square root of each element in `B` use `**` operator
# type: np.ndarray
b = ...

# Second power (square) of each element in `C` use `**` operator
# type: np.ndarray
c = ...

# Add elements from `a` to `b` and then multiply by `c`
# Remember about the operator precedence
# type: np.ndarray
result = ...

Code 6.63. Solution

"""
* Assignment: Numpy Broadcasting Type Cast
* Complexity: easy
* Lines of code: 2 lines
* Time: 3 min

English:
    1. For given: `a: np.ndarray`, `b: np.ndarray` (see below)
    2. Add `a` and `b`
    3. Add `b` and `a`
    4. What happened?
    5. Run doctests - all must succeed

Polish:
    1. Dla danych: `a: np.ndarray`, `b: np.ndarray` (patrz sekcja input)
    2. Dodaj `a` i `b`
    3. Dodaj `b` i `a`
    4. Co się stało?
    5. Uruchom doctesty - wszystkie muszą się powieść

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> assert type(result_ab) is np.ndarray, \
    'Variable `result_ab` has invalid type, expected: np.ndarray'
    >>> assert type(result_ba) is np.ndarray, \
    'Variable `result_ba` has invalid type, expected: np.ndarray'

    >>> result_ab
    array([[5, 1],
           [2, 3]])

    >>> result_ba
    array([[5, 1],
           [2, 3]])
"""

import numpy as np


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


result_ab = ...
result_ba = ...

Code 6.64. Solution

"""
* Assignment: Numpy Broadcasting Matmul
* Complexity: easy
* Lines of code: 4 lines
* Time: 3 min

English:
    1. For given: `a: np.ndarray`, `b: np.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
    7. Run doctests - all must succeed

Polish:
    1. Dla danych: `a: np.ndarray`, `b: np.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
    7. Uruchom doctesty - wszystkie muszą się powieść

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> mul_ab(a, b)  # doctest: +NORMALIZE_WHITESPACE
    Traceback (most recent call last):
    ValueError: operands could not be broadcast together with shapes (4,4) (4,2)

    >>> matmul_ab(a, b)
    array([[ 9,  2],
           [ 7,  3],
           [21,  8],
           [28,  8]])

    >>> mul_ba(b, a)  # doctest: +NORMALIZE_WHITESPACE
    Traceback (most recent call last):
    ValueError: operands could not be broadcast together with shapes (4,2) (4,4)

    >>> matmul_ba(b, a)
    Traceback (most recent call last):
    ValueError: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 4 is different from 2)
"""

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]])


# Multiply `a` and `b` using scalar multiplication
# type: callable
def mul_ab(a, b):
    return ...

# Multiply `a` and `b` using matrix multiplication
# type: callable
def matmul_ab(a, b):
    return ...

# Multiply `b` and `a` using scalar multiplication
# type: callable
def mul_ba(b, a):
    return ...

# Multiply `b` and `a` using matrix multiplication
# type: callable
def matmul_ba(b, a):
    return ...