maths.numerical_analysis.nth_root

Approximate the nth root of a real number using Newton’s Method.

The nth root of a real number R can be computed with Newton’s method, which starts with an initial guess x_0 and then iterates using the recurrence relation:

x_{k + 1} = x_k - ((x_k)**n - R)/(n*(x_k)**(n-1))

The recurrence relation can be rewritten for computational efficiency:

x_{k + 1} = (n-1)/n*x_k + R/(n*(x_k)**(n-1))

Given a tolerance TOL, a stopping criterion can be set as:

abs(x_{k + 1} - x_k) < TOL

References: - https://en.wikipedia.org/wiki/Nth_root#Using_Newton’s_method - Sauer, T. (2011): Numerical analysis.

USA. Addison-Wesley Publishing Company.

Functions

nth_root(→ float)

Approximate the nth root of the radicand for the given index

Module Contents

maths.numerical_analysis.nth_root.nth_root(radicand: float, index: int, tolerance: float = 0.0001) → float

Approximate the nth root of the radicand for the given index

Args:

radicand: number from which the root is taken index: positive integer which is the degree of the root tolerance: positive real number that establishes the stopping criterion

Returns:

new_approximation: approximation of the nth root of the radicand for the given index

Raises:

TypeError: radicand is not a real number TypeError: index is not an integer ValueError: index is not a positive integer TypeError: tolerance is not a real number ValueError: tolerance is not a positive real number ValueError: math domain error

>>> round(nth_root(9, 2),1)
3.0

>>> int(round(nth_root(-8, 3, 0.001)))
-2

>>> int(round(nth_root(256, 4, 0.001)))
4

>>> round(nth_root(2, 2), 5)
1.41421

>>> round(nth_root(0.25, 2, 0.00000001), 1)
0.5

>>> round(nth_root(-8/27, 3, 0.0000001), 5)
-0.66667

>>> nth_root(0, 2, 0.1)
0.0

>>> nth_root(0.0, 5)
0.0

>>> all(abs(nth_root(k, k, 0.00000001) - k**(1/k)) <= 1e-10 for k in range(1,10))
True

>>> nth_root('invalid input', 3, 0.0001)
Traceback (most recent call last):
    ...
TypeError: radicand must be a real number, not a str

>>> nth_root(4, 0.5, 0.0001)
Traceback (most recent call last):
    ...
TypeError: index must be an integer, not a float

>>> nth_root(16, -4, 0.001)
Traceback (most recent call last):
    ...
ValueError: index must be a positive integer, -4 <= 0

>>> nth_root(4, 2, '0.000001')
Traceback (most recent call last):
    ...
TypeError: tolerance must be a real number, not str

>>> nth_root(9, 2, -0.01)
Traceback (most recent call last):
    ...
ValueError: tolerance must be a positive real number, -0.01 <= 0

>>> nth_root(-256, 4, 0.0001)
Traceback (most recent call last):
    ...
ValueError: math domain error, radicand must be nonnegative for even index