Numpy functions¶

This section of the manual discusses those functions that were adapted from numpy.

all¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.all.html

The function takes one positional, and one keyword argument, the axis, with a default value of None, and tests, whether all array elements along the given axis evaluate to True. If the keyword argument is None, the flattened array is inspected.

Elements of an array evaluate to True, if they are not equal to zero, or the Boolean False. The return value if a Boolean ndarray.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12)).reshape((3, 4))

print('\na:\n', a)

b = np.all(a)
print('\nall of the flattened array:\n', b)

c = np.all(a, axis=0)
print('\nall of a along 0th axis:\n', c)

d = np.all(a, axis=1)
print('\nall of a along 1st axis:\n', d)

a:
 array([[0.0, 1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0]], dtype=float64)

all of the flattened array:
 False

all of a along 0th axis:
 array([False, True, True, True], dtype=bool)

all of a along 1st axis:
 array([False, True, True], dtype=bool)

any¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.any.html

The function takes one positional, and one keyword argument, the axis, with a default value of None, and tests, whether any array element along the given axis evaluates to True. If the keyword argument is None, the flattened array is inspected.

Elements of an array evaluate to True, if they are not equal to zero, or the Boolean False. The return value if a Boolean ndarray.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12)).reshape((3, 4))

print('\na:\n', a)

b = np.any(a)
print('\nany of the flattened array:\n', b)

c = np.any(a, axis=0)
print('\nany of a along 0th axis:\n', c)

d = np.any(a, axis=1)
print('\nany of a along 1st axis:\n', d)

a:
 array([[0.0, 1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0]], dtype=float64)

any of the flattened array:
 True

any of a along 0th axis:
 array([True, True, True, True], dtype=bool)

any of a along 1st axis:
 array([True, True, True], dtype=bool)

argmax¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html

See numpy.max.

argmin¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.html

See numpy.max.

argsort¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argsort.html

Similarly to sort, argsort takes a positional, and a keyword argument, and returns an unsigned short index array of type ndarray with the same dimensions as the input, or, if axis=None, as a row vector with length equal to the number of elements in the input (i.e., the flattened array). The indices in the output sort the input in ascending order. The routine in argsort is the same as in sort, therefore, the comments on computational expenses (time and RAM) also apply. In particular, since no copy of the original data is required, virtually no RAM beyond the output array is used.

Since the underlying container of the output array is of type uint16_t, neither of the output dimensions should be larger than 65535. If that happens to be the case, the function will bail out with a ValueError.

# code to be run in micropython

from ulab import numpy as np

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = np.argsort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

c = np.argsort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)

c = np.argsort(a, axis=None)
print('\nflattened a sorted:\n', c)

a:
 array([[1.0, 12.0, 3.0, 0.0],
       [5.0, 3.0, 4.0, 1.0],
       [9.0, 11.0, 1.0, 8.0],
       [7.0, 10.0, 0.0, 1.0]], dtype=float64)

a sorted along vertical axis:
 array([[0, 1, 3, 0],
       [1, 3, 2, 1],
       [3, 2, 0, 3],
       [2, 0, 1, 2]], dtype=uint16)

a sorted along horizontal axis:
 array([[3, 0, 2, 1],
       [3, 1, 2, 0],
       [2, 3, 0, 1],
       [2, 3, 0, 1]], dtype=uint16)

Traceback (most recent call last):
  File "/dev/shm/micropython.py", line 12, in <module>
NotImplementedError: argsort is not implemented for flattened arrays

Since during the sorting, only the indices are shuffled, argsort does not modify the input array, as one can verify this by the following example:

# code to be run in micropython

from ulab import numpy as np

a = np.array([0, 5, 1, 3, 2, 4], dtype=np.uint8)
print('\na:\n', a)
b = np.argsort(a, axis=0)
print('\nsorting indices:\n', b)
print('\nthe original array:\n', a)

a:
 array([0, 5, 1, 3, 2, 4], dtype=uint8)

sorting indices:
 array([0, 2, 4, 3, 5, 1], dtype=uint16)

the original array:
 array([0, 5, 1, 3, 2, 4], dtype=uint8)

clip¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.clip.html

Clips an array, i.e., values that are outside of an interval are clipped to the interval edges. The function is equivalent to maximum(a_min, minimum(a, a_max)) broadcasting takes place exactly as in minimum. If the arrays are of different dtype, the output is upcast as in Binary operators.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8)
print('a:\t\t', a)
print('clipped:\t', np.clip(a, 3, 7))

b = 3 * np.ones(len(a), dtype=np.float)
print('\na:\t\t', a)
print('b:\t\t', b)
print('clipped:\t', np.clip(a, b, 7))

a:           array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
clipped:     array([3, 3, 3, 3, 4, 5, 6, 7, 7], dtype=uint8)

a:           array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)
b:           array([3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0], dtype=float64)
clipped:     array([3.0, 3.0, 3.0, 3.0, 4.0, 5.0, 6.0, 7.0, 7.0], dtype=float64)

convolve¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.convolve.html

Returns the discrete, linear convolution of two one-dimensional arrays.

Only the full mode is supported, and the mode named parameter is not accepted. Note that all other modes can be had by slicing a full result.

# code to be run in micropython

from ulab import numpy as np

x = np.array((1, 2, 3))
y = np.array((1, 10, 100, 1000))

print(np.convolve(x, y))

array([1.0, 12.0, 123.0, 1230.0, 2300.0, 3000.0], dtype=float64)

diff¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.diff.html

The diff function returns the numerical derivative of the forward scheme, or more accurately, the differences of an ndarray along a given axis. The order of derivative can be stipulated with the n keyword argument, which should be between 0, and 9. Default is 1. If higher order derivatives are required, they can be gotten by repeated calls to the function. The axis keyword argument should be -1 (last axis, in ulab equivalent to the second axis, and this also happens to be the default value), 0, or 1.

Beyond the output array, the function requires only a couple of bytes of extra RAM for the differentiation stencil. (The stencil is an int8 array, one byte longer than n. This also explains, why the highest order is 9: the coefficients of a ninth-order stencil all fit in signed bytes, while 10 would require int16.) Note that as usual in numerical differentiation (and also in numpy), the length of the respective axis will be reduced by n after the operation. If n is larger than, or equal to the length of the axis, an empty array will be returned.

WARNING: the diff function does not implement the prepend and append keywords that can be found in numpy.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9), dtype=np.uint8)
a[3] = 10
print('a:\n', a)

print('\nfirst derivative:\n', np.diff(a, n=1))
print('\nsecond derivative:\n', np.diff(a, n=2))

c = np.array([[1, 2, 3, 4], [4, 3, 2, 1], [1, 4, 9, 16], [0, 0, 0, 0]])
print('\nc:\n', c)
print('\nfirst derivative, first axis:\n', np.diff(c, axis=0))
print('\nfirst derivative, second axis:\n', np.diff(c, axis=1))

a:
 array([0, 1, 2, 10, 4, 5, 6, 7, 8], dtype=uint8)

first derivative:
 array([1, 1, 8, 250, 1, 1, 1, 1], dtype=uint8)

second derivative:
 array([0, 249, 14, 249, 0, 0, 0], dtype=uint8)

c:
 array([[1.0, 2.0, 3.0, 4.0],
       [4.0, 3.0, 2.0, 1.0],
       [1.0, 4.0, 9.0, 16.0],
       [0.0, 0.0, 0.0, 0.0]], dtype=float64)

first derivative, first axis:
 array([[3.0, 1.0, -1.0, -3.0],
       [-3.0, 1.0, 7.0, 15.0],
       [-1.0, -4.0, -9.0, -16.0]], dtype=float64)

first derivative, second axis:
 array([[1.0, 1.0, 1.0],
       [-1.0, -1.0, -1.0],
       [3.0, 5.0, 7.0],
       [0.0, 0.0, 0.0]], dtype=float64)

dot¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html

WARNING: numpy applies upcasting rules for the multiplication of matrices, while ulab simply returns a float matrix.

Once you can invert a matrix, you might want to know, whether the inversion is correct. You can simply take the original matrix and its inverse, and multiply them by calling the dot function, which takes the two matrices as its arguments. If the matrix dimensions do not match, the function raises a ValueError. The result of the multiplication is expected to be the unit matrix, which is demonstrated below.

# code to be run in micropython

from ulab import numpy as np

m = np.array([[1, 2, 3], [4, 5, 6], [7, 10, 9]], dtype=np.uint8)
n = np.linalg.inv(m)
print("m:\n", m)
print("\nm^-1:\n", n)
# this should be the unit matrix
print("\nm*m^-1:\n", np.dot(m, n))

m:
 array([[1, 2, 3],
       [4, 5, 6],
       [7, 10, 9]], dtype=uint8)

m^-1:
 array([[-1.25, 1.0, -0.25],
       [0.4999999999999998, -1.0, 0.5],
       [0.4166666666666668, 0.3333333333333333, -0.25]], dtype=float64)

m*m^-1:
 array([[1.0, 0.0, 0.0],
       [4.440892098500626e-16, 1.0, 0.0],
       [8.881784197001252e-16, 0.0, 1.0]], dtype=float64)

Note that for matrix multiplication you don’t necessarily need square matrices, it is enough, if their dimensions are compatible (i.e., the the left-hand-side matrix has as many columns, as does the right-hand-side matrix rows):

# code to be run in micropython

from ulab import numpy as np

m = np.array([[1, 2, 3, 4], [5, 6, 7, 8]], dtype=np.uint8)
n = np.array([[1, 2], [3, 4], [5, 6], [7, 8]], dtype=np.uint8)
print(m)
print(n)
print(np.dot(m, n))

array([[1, 2, 3, 4],
       [5, 6, 7, 8]], dtype=uint8)
array([[1, 2],
       [3, 4],
       [5, 6],
       [7, 8]], dtype=uint8)
array([[50.0, 60.0],
       [114.0, 140.0]], dtype=float64)

equal¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.equal.html

numpy: https://numpy.org/doc/stable/reference/generated/numpy.not_equal.html

In micropython, equality of arrays or scalars can be established by utilising the ==, !=, <, >, <=, or => binary operators. In circuitpython, == and != will produce unexpected results. In order to avoid this discrepancy, and to maintain compatibility with numpy, ulab implements the equal and not_equal operators that return the same results, irrespective of the python implementation.

These two functions take two ndarrays, or scalars as their arguments. No keyword arguments are implemented.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(9))
b = np.zeros(9)

print('a: ', a)
print('b: ', b)
print('\na == b: ', np.equal(a, b))
print('a != b: ', np.not_equal(a, b))

# comparison with scalars
print('a == 2: ', np.equal(a, 2))

a:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
b:  array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], dtype=float64)

a == b:  array([True, False, False, False, False, False, False, False, False], dtype=bool)
a != b:  array([False, True, True, True, True, True, True, True, True], dtype=bool)
a == 2:  array([False, False, True, False, False, False, False, False, False], dtype=bool)

flip¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.flip.html

The flip function takes one positional, an ndarray, and one keyword argument, axis = None, and reverses the order of elements along the given axis. If the keyword argument is None, the matrix’ entries are flipped along all axes. flip returns a new copy of the array.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4, 5])
print("a: \t", a)
print("a flipped:\t", np.flip(a))

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print("\na flipped horizontally\n", np.flip(a, axis=1))
print("\na flipped vertically\n", np.flip(a, axis=0))
print("\na flipped horizontally+vertically\n", np.flip(a))

a:   array([1.0, 2.0, 3.0, 4.0, 5.0], dtype=float64)
a flipped:   array([5.0, 4.0, 3.0, 2.0, 1.0], dtype=float64)

a flipped horizontally
 array([[3, 2, 1],
       [6, 5, 4],
       [9, 8, 7]], dtype=uint8)

a flipped vertically
 array([[7, 8, 9],
       [4, 5, 6],
       [1, 2, 3]], dtype=uint8)

a flipped horizontally+vertically
 array([9, 8, 7, 6, 5, 4, 3, 2, 1], dtype=uint8)

interp¶

numpy: https://docs.scipy.org/doc/numpy/numpy.interp

The interp function returns the linearly interpolated values of a one-dimensional numerical array. It requires three positional arguments,x, at which the interpolated values are evaluated, xp, the array of the independent data variable, and fp, the array of the dependent values of the data. xp must be a monotonically increasing sequence of numbers.

Two keyword arguments, left, and right can also be supplied; these determine the return values, if x < xp[0], and x > xp[-1], respectively. If these arguments are not supplied, left, and right default to fp[0], and fp[-1], respectively.

# code to be run in micropython

from ulab import numpy as np

x = np.array([1, 2, 3, 4, 5]) - 0.2
xp = np.array([1, 2, 3, 4])
fp = np.array([1, 2, 3, 5])

print(x)
print(np.interp(x, xp, fp))
print(np.interp(x, xp, fp, left=0.0))
print(np.interp(x, xp, fp, right=10.0))

array([0.8, 1.8, 2.8, 3.8, 4.8], dtype=float64)
array([1.0, 1.8, 2.8, 4.6, 5.0], dtype=float64)
array([0.0, 1.8, 2.8, 4.6, 5.0], dtype=float64)
array([1.0, 1.8, 2.8, 4.6, 10.0], dtype=float64)

isfinite¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.isfinite.html

Returns a Boolean array of the same shape as the input, or a True/False, if the input is a scalar. In the return value, all elements are True at positions, where the input value was finite. Integer types are automatically finite, therefore, if the input is of integer type, the output will be the True tensor.

# code to be run in micropython

from ulab import numpy as np

print('isfinite(0): ', np.isfinite(0))

a = np.array([1, 2, np.nan])
print('\n' + '='*20)
print('a:\n', a)
print('\nisfinite(a):\n', np.isfinite(a))

b = np.array([1, 2, np.inf])
print('\n' + '='*20)
print('b:\n', b)
print('\nisfinite(b):\n', np.isfinite(b))

c = np.array([1, 2, 3], dtype=np.uint16)
print('\n' + '='*20)
print('c:\n', c)
print('\nisfinite(c):\n', np.isfinite(c))

isfinite(0):  True

====================
a:
 array([1.0, 2.0, nan], dtype=float64)

isfinite(a):
 array([True, True, False], dtype=bool)

====================
b:
 array([1.0, 2.0, inf], dtype=float64)

isfinite(b):
 array([True, True, False], dtype=bool)

====================
c:
 array([1, 2, 3], dtype=uint16)

isfinite(c):
 array([True, True, True], dtype=bool)

isinf¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.isinf.html

Similar to isfinite, but the output is True at positions, where the input is infinite. Integer types return the False tensor.

# code to be run in micropython

from ulab import numpy as np

print('isinf(0): ', np.isinf(0))

a = np.array([1, 2, np.nan])
print('\n' + '='*20)
print('a:\n', a)
print('\nisinf(a):\n', np.isinf(a))

b = np.array([1, 2, np.inf])
print('\n' + '='*20)
print('b:\n', b)
print('\nisinf(b):\n', np.isinf(b))

c = np.array([1, 2, 3], dtype=np.uint16)
print('\n' + '='*20)
print('c:\n', c)
print('\nisinf(c):\n', np.isinf(c))

isinf(0):  False

====================
a:
 array([1.0, 2.0, nan], dtype=float64)

isinf(a):
 array([False, False, False], dtype=bool)

====================
b:
 array([1.0, 2.0, inf], dtype=float64)

isinf(b):
 array([False, False, True], dtype=bool)

====================
c:
 array([1, 2, 3], dtype=uint16)

isinf(c):
 array([False, False, False], dtype=bool)

mean¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html

If the axis keyword is not specified, it assumes the default value of None, and returns the result of the computation for the flattened array. Otherwise, the calculation is along the given axis.

# code to be run in micropython

from ulab import numpy as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)
print('mean, flat: ', np.mean(a))
print('mean, horizontal: ', np.mean(a, axis=1))
print('mean, vertical: ', np.mean(a, axis=0))

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)
mean, flat:  5.0
mean, horizontal:  array([2.0, 5.0, 8.0], dtype=float64)
mean, vertical:  array([4.0, 5.0, 6.0], dtype=float64)

max¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.max.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.min.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.html

WARNING: Difference to numpy: the out keyword argument is not implemented.

These functions follow the same pattern, and work with generic iterables, and ndarrays. min, and max return the minimum or maximum of a sequence. If the input array is two-dimensional, the axis keyword argument can be supplied, in which case the minimum/maximum along the given axis will be returned. If axis=None (this is also the default value), the minimum/maximum of the flattened array will be determined.

argmin/argmax return the position (index) of the minimum/maximum in the sequence.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 0, 1, 10])
print('a:', a)
print('min of a:', np.min(a))
print('argmin of a:', np.argmin(a))

b = np.array([[1, 2, 0], [1, 10, -1]])
print('\nb:\n', b)
print('min of b (flattened):', np.min(b))
print('min of b (axis=0):', np.min(b, axis=0))
print('min of b (axis=1):', np.min(b, axis=1))

a: array([1.0, 2.0, 0.0, 1.0, 10.0], dtype=float64)
min of a: 0.0
argmin of a: 2

b:
 array([[1.0, 2.0, 0.0],
       [1.0, 10.0, -1.0]], dtype=float64)
min of b (flattened): -1.0
min of b (axis=0): array([1.0, 2.0, -1.0], dtype=float64)
min of b (axis=1): array([0.0, -1.0], dtype=float64)

median¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.median.html

The function computes the median along the specified axis, and returns the median of the array elements. If the axis keyword argument is None, the arrays is flattened first. The dtype of the results is always float.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12), dtype=np.int8).reshape((3, 4))
print('a:\n', a)
print('\nmedian of the flattened array: ', np.median(a))
print('\nmedian along the vertical axis: ', np.median(a, axis=0))
print('\nmedian along the horizontal axis: ', np.median(a, axis=1))

a:
 array([[0, 1, 2, 3],
       [4, 5, 6, 7],
       [8, 9, 10, 11]], dtype=int8)

median of the flattened array:  5.5

median along the vertical axis:  array([4.0, 5.0, 6.0, 7.0], dtype=float64)

median along the horizontal axis:  array([1.5, 5.5, 9.5], dtype=float64)

min¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.min.html

See numpy.max.

minimum¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.minimum.html

See numpy.maximum

maximum¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.maximum.html

Returns the maximum of two arrays, or two scalars, or an array, and a scalar. If the arrays are of different dtype, the output is upcast as in Binary operators. If both inputs are scalars, a scalar is returned. Only positional arguments are implemented.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4, 5], dtype=np.uint8)
b = np.array([5, 4, 3, 2, 1], dtype=np.float)
print('minimum of a, and b:')
print(np.minimum(a, b))

print('\nmaximum of a, and b:')
print(np.maximum(a, b))

print('\nmaximum of 1, and 5.5:')
print(np.maximum(1, 5.5))

minimum of a, and b:
array([1.0, 2.0, 3.0, 2.0, 1.0], dtype=float64)

maximum of a, and b:
array([5.0, 4.0, 3.0, 4.0, 5.0], dtype=float64)

maximum of 1, and 5.5:
5.5

not_equal¶

See numpy.equal.

polyfit¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html

polyfit takes two, or three arguments. The last one is the degree of the polynomial that will be fitted, the last but one is an array or iterable with the y (dependent) values, and the first one, an array or iterable with the x (independent) values, can be dropped. If that is the case, x will be generated in the function as range(len(y)).

If the lengths of x, and y are not the same, the function raises a ValueError.

# code to be run in micropython

from ulab import numpy as np

x = np.array([0, 1, 2, 3, 4, 5, 6])
y = np.array([9, 4, 1, 0, 1, 4, 9])
print('independent values:\t', x)
print('dependent values:\t', y)
print('fitted values:\t\t', np.polyfit(x, y, 2))

# the same with missing x
print('\ndependent values:\t', y)
print('fitted values:\t\t', np.polyfit(y, 2))

independent values:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0], dtype=float64)
dependent values:    array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float64)
fitted values:               array([1.0, -6.0, 9.000000000000004], dtype=float64)

dependent values:    array([9.0, 4.0, 1.0, 0.0, 1.0, 4.0, 9.0], dtype=float64)
fitted values:               array([1.0, -6.0, 9.000000000000004], dtype=float64)

Execution time¶

polyfit is based on the inversion of a matrix (there is more on the background in https://en.wikipedia.org/wiki/Polynomial_regression), and it requires the intermediate storage of 2*N*(deg+1) floats, where N is the number of entries in the input array, and deg is the fit’s degree. The additional computation costs of the matrix inversion discussed in linalg.inv also apply. The example from above needs around 150 microseconds to return:

# code to be run in micropython

from ulab import numpy as np

@timeit
def time_polyfit(x, y, n):
    return np.polyfit(x, y, n)

x = np.array([0, 1, 2, 3, 4, 5, 6])
y = np.array([9, 4, 1, 0, 1, 4, 9])

time_polyfit(x, y, 2)

execution time:  153  us

polyval¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyval.html

polyval takes two arguments, both arrays or generic micropython iterables returning scalars.

# code to be run in micropython

from ulab import numpy as np

p = [1, 1, 1, 0]
x = [0, 1, 2, 3, 4]
print('coefficients: ', p)
print('independent values: ', x)
print('\nvalues of p(x): ', np.polyval(p, x))

# the same works with one-dimensional ndarrays
a = np.array(x)
print('\nndarray (a): ', a)
print('value of p(a): ', np.polyval(p, a))

coefficients:  [1, 1, 1, 0]
independent values:  [0, 1, 2, 3, 4]

values of p(x):  array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float64)

ndarray (a):  array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=float64)
value of p(a):  array([0.0, 3.0, 14.0, 39.0, 84.0], dtype=float64)

roll¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.roll.html

The roll function shifts the content of a vector by the positions given as the second argument. If the axis keyword is supplied, the shift is applied to the given axis.

# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3, 4, 5, 6, 7, 8])
print("a:\t\t\t", a)

a = np.roll(a, 2)
print("a rolled to the left:\t", a)

# this should be the original vector
a = np.roll(a, -2)
print("a rolled to the right:\t", a)

a:                   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
a rolled to the left:        array([7.0, 8.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0], dtype=float64)
a rolled to the right:       array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)

Rolling works with matrices, too. If the axis keyword is 0, the matrix is rolled along its vertical axis, otherwise, horizontally.

Horizontal rolls are faster, because they require fewer steps, and larger memory chunks are copied, however, they also require more RAM: basically the whole row must be stored internally. Most expensive are the None keyword values, because with axis = None, the array is flattened first, hence the row’s length is the size of the whole matrix.

Vertical rolls require two internal copies of single columns.

# code to be run in micropython

from ulab import numpy as np

a = np.array(range(12)).reshape((3, 4))
print("a:\n", a)
a = np.roll(a, 2, axis=0)
print("\na rolled up:\n", a)

a = np.array(range(12)).reshape((3, 4))
print("a:\n", a)
a = np.roll(a, -1, axis=1)
print("\na rolled to the left:\n", a)

a = np.array(range(12)).reshape((3, 4))
print("a:\n", a)
a = np.roll(a, 1, axis=None)
print("\na rolled with None:\n", a)

a:
 array([[0.0, 1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0]], dtype=float64)

a rolled up:
 array([[4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0],
       [0.0, 1.0, 2.0, 3.0]], dtype=float64)
a:
 array([[0.0, 1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0]], dtype=float64)

a rolled to the left:
 array([[1.0, 2.0, 3.0, 0.0],
       [5.0, 6.0, 7.0, 4.0],
       [9.0, 10.0, 11.0, 8.0]], dtype=float64)
a:
 array([[0.0, 1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0, 7.0],
       [8.0, 9.0, 10.0, 11.0]], dtype=float64)

a rolled with None:
 array([[11.0, 0.0, 1.0, 2.0],
       [3.0, 4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0, 10.0]], dtype=float64)

sort¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sort.html

The sort function takes an ndarray, and sorts its elements in ascending order along the specified axis using a heap sort algorithm. As opposed to the .sort() method discussed earlier, this function creates a copy of its input before sorting, and at the end, returns this copy. Sorting takes place in place, without auxiliary storage. The axis keyword argument takes on the possible values of -1 (the last axis, in ulab equivalent to the second axis, and this also happens to be the default value), 0, 1, or None. The first three cases are identical to those in diff, while the last one flattens the array before sorting.

If descending order is required, the result can simply be flipped, see flip.

WARNING: numpy defines the kind, and order keyword arguments that are not implemented here. The function in ulab always uses heap sort, and since ulab does not have the concept of data fields, the order keyword argument would have no meaning.

# code to be run in micropython

from ulab import numpy as np

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = np.sort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

c = np.sort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)

c = np.sort(a, axis=None)
print('\nflattened a sorted:\n', c)

a:
 array([[1.0, 12.0, 3.0, 0.0],
       [5.0, 3.0, 4.0, 1.0],
       [9.0, 11.0, 1.0, 8.0],
       [7.0, 10.0, 0.0, 1.0]], dtype=float64)

a sorted along vertical axis:
 array([[1.0, 3.0, 0.0, 0.0],
       [5.0, 10.0, 1.0, 1.0],
       [7.0, 11.0, 3.0, 1.0],
       [9.0, 12.0, 4.0, 8.0]], dtype=float64)

a sorted along horizontal axis:
 array([[0.0, 1.0, 3.0, 12.0],
       [1.0, 3.0, 4.0, 5.0],
       [1.0, 8.0, 9.0, 11.0],
       [0.0, 1.0, 7.0, 10.0]], dtype=float64)

flattened a sorted:
 array([0.0, 0.0, 1.0, ..., 10.0, 11.0, 12.0], dtype=float64)

Heap sort requires \(\sim N\log N\) operations, and notably, the worst case costs only 20% more time than the average. In order to get an order-of-magnitude estimate, we will take the sine of 1000 uniformly spaced numbers between 0, and two pi, and sort them:

# code to be run in micropython

import ulab as np
from ulab import vector
from ulab import numerical

@timeit
def sort_time(array):
    return numerical.sort(array)

b = vector.sin(np.linspace(0, 6.28, num=1000))
print('b: ', b)
sort_time(b)
print('\nb sorted:\n', b)

std¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html

If the axis keyword is not specified, it assumes the default value of None, and returns the result of the computation for the flattened array. Otherwise, the calculation is along the given axis.

# code to be run in micropython

from ulab import numpy as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)
print('sum, flat array: ', np.std(a))
print('std, vertical: ', np.std(a, axis=0))
print('std, horizonal: ', np.std(a, axis=1))

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)
sum, flat array:  2.581988897471611
std, vertical:  array([2.449489742783178, 2.449489742783178, 2.449489742783178], dtype=float64)
std, horizonal:  array([0.8164965809277261, 0.8164965809277261, 0.8164965809277261], dtype=float64)

sum¶

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html

If the axis keyword is not specified, it assumes the default value of None, and returns the result of the computation for the flattened array. Otherwise, the calculation is along the given axis.

# code to be run in micropython

from ulab import numpy as np

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)

print('sum, flat array: ', np.sum(a))
print('sum, horizontal: ', np.sum(a, axis=1))
print('std, vertical: ', np.sum(a, axis=0))

a:
 array([[1.0, 2.0, 3.0],
       [4.0, 5.0, 6.0],
       [7.0, 8.0, 9.0]], dtype=float64)
sum, flat array:  45.0
sum, horizontal:  array([6.0, 15.0, 24.0], dtype=float64)
std, vertical:  array([12.0, 15.0, 18.0], dtype=float64)

trace¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.trace.html

The trace function returns the sum of the diagonal elements of a square matrix. If the input argument is not a square matrix, an exception will be raised.

The scalar so returned will inherit the type of the input array, i.e., integer arrays have integer trace, and floating point arrays a floating point trace.

# code to be run in micropython

from ulab import numpy as np

a = np.array([[25, 15, -5], [15, 18,  0], [-5,  0, 11]], dtype=np.int8)
print('a: ', a)
print('\ntrace of a: ', np.trace(a))

b = np.array([[25, 15, -5], [15, 18,  0], [-5,  0, 11]], dtype=np.float)

print('='*20 + '\nb: ', b)
print('\ntrace of b: ', np.trace(b))

a:  array([[25, 15, -5],
       [15, 18, 0],
       [-5, 0, 11]], dtype=int8)

trace of a:  54
====================
b:  array([[25.0, 15.0, -5.0],
       [15.0, 18.0, 0.0],
       [-5.0, 0.0, 11.0]], dtype=float64)

trace of b:  54.0

trapz¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.trapz.html

The function takes one or two one-dimensional ndarrays, and integrates the dependent values (y) using the trapezoidal rule. If the independent variable (x) is given, that is taken as the sample points corresponding to y.

# code to be run in micropython

from ulab import numpy as np

x = np.linspace(0, 9, num=10)
y = x*x

print('x: ',  x)
print('y: ',  y)
print('============================')
print('integral of y: ', np.trapz(y))
print('integral of y at x: ', np.trapz(y, x=x))

x:  array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0], dtype=float64)
y:  array([0.0, 1.0, 4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0], dtype=float64)
============================
integral of y:  244.5
integral of y at x:  244.5

where¶

numpy: https://numpy.org/doc/stable/reference/generated/numpy.where.html

The function takes three positional arguments, condition, x, and y, and returns a new ndarray, whose values are taken from either x, or y, depending on the truthness of condition. The three arguments are broadcast together, and the function raises a ValueError exception, if broadcasting is not possible.

The function is implemented for ndarrays only: other iterable types can be passed after casting them to an ndarray by calling the array constructor.

If the dtypes of x, and y differ, the output is upcast as discussed earlier.

Note that the condition is expanded into an Boolean ndarray. This means that the storage required to hold the condition should be taken into account, whenever the function is called.

The following example returns an ndarray of length 4, with 1 at positions, where condition is smaller than 3, and with -1 otherwise.

# code to be run in micropython

from ulab import numpy as np

condition = np.array([1, 2, 3, 4], dtype=np.uint8)
print(np.where(condition < 3, 1, -1))

array([1, 1, -1, -1], dtype=int16)

The next snippet shows, how values from two arrays can be fed into the output:

# code to be run in micropython


from ulab import numpy as np

condition = np.array([1, 2, 3, 4], dtype=np.uint8)
x = np.array([11, 22, 33, 44], dtype=np.uint8)
y = np.array([1, 2, 3, 4], dtype=np.uint8)
print(np.where(condition < 3, x, y))

array([11, 22, 3, 4], dtype=uint8)