Polynomials

Functions in the polynomial sub-module can be invoked by importing the module first.

polyval

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

polyval takes two arguments, both arrays or other iterables.

# code to be run in micropython

import ulab as np
from ulab import poly

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

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

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

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, assuming uniform sampling.

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

# code to be run in micropython

import ulab as np
from ulab import poly

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', poly.polyfit(x, y, 2))

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

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

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

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 inv also apply. The example from above needs around 150 microseconds to return:

# code to be run in micropython

import ulab as np
from ulab import poly

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