Polynomials

Accompanying notebook centered around learning Chapter 1 of the Programmer’s introduction to Mathematics.

A single variable polynomial with real coefficients is a function of the form \(f\) that takes a real number as input, produces a real number as output, and has the form:

\[f(x)=a_0+a_1x+a_2x^2+...+a_nx^n\]

where \(a_0,a_1,...,a_n\) are real numbers. The number \(a_i\) are called the coefficients of f. The degree of the polynomial is the integer \(n\).

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Polynomial

 Polynomial (function_name:str='f', variable_name:str='x',
             degree:Union[int,str]='n', coefficient_name:str='a')

A class used to represent a general polynomial, which is a function that both takes and returns real numbers. Currently only supports single-variable polynomial functions. Polynomials require: - coefficients - degree

	Type	Default	Details
function_name	str	f	name of function
variable_name	str	x	name of variable
degree	typing.Union[int, str]	n	degree of polynomial
coefficient_name	str	a	str for name of coefficients

Polynomial().equality()

\(\displaystyle f{\left(x \right)} = \sum_{i=0}^{n} x^{i} {a}_{i}\)

print(Polynomial())

f(x) = a[0] * x**0 + a[1] * x**1 + ... + a[n] * x**n

test_eq(Polynomial().right_hand_side(), Polynomial().equality().rhs)

x = variable('x')
n = variable('n')
i = variable('i')
a = sp.IndexedBase('a')

f_x = sp.Sum(a[i] * x**i, (i, 0, n))

f_x

test_eq(Polynomial().right_hand_side(), f_x)

f_x

\(\displaystyle \sum_{i=0}^{n} x^{i} {a}_{i}\)

# create g(t) = 2 + 0t + 4t^2 - t^3
g = Polynomial(variable_name='t', function_name='g', degree='n', coefficient_name='a')

g.set_degree(3)

\(\displaystyle g{\left(t \right)} = t^{3} {a}_{3} + t^{2} {a}_{2} + t {a}_{1} + {a}_{0}\)

g.set_coefficients([2, 0, 4, -1])

\(\displaystyle g{\left(t \right)} = - t^{3} + 4 t^{2} + 2\)

If you want to set a specific polynomial all at once, here’s a helper function for that:

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polynomial

 polynomial (degree:int, coefficients:list)

Returns a polynomial function with the given degree and coefficients

	Type	Details
degree	int	degree of polynomial
coefficients	list	list of coefficients starting with the lowest indexed coefficient

Sometimes it might be better to directly generate a polynomial with the equation function.

For example, to create the above polynomial as a one liner, see below. It’s up to you to decide which is clearer.

equation('g', variable('t'), 2 + (0*variable('t')) + (4*variable('t')**2) + (-1*variable('t')**3))

\(\displaystyle g{\left(t \right)} = - t^{3} + 4 t^{2} + 2\)

Existence and Uniqueness for Polynomials

For any integer \(n\geq 0\), and any list of \(n+1\) points \((x_0,y_0),(x_1,y_1),...,(x_n,y_n)\) in R^2 there is a unique polynomial \(f\) of degree at most \(n\) such that \(f(x_i)=y_i\) for all \(i=0,1,...,n\).

polynomial_to_test = polynomial(3, [2, 0, 4, -1])
simple_polynomial = polynomial(1, [1, 1])

polynomial_to_test.polynomial

\(\displaystyle f{\left(x \right)} = - x^{3} + 4 x^{2} + 2\)

simple_polynomial.polynomial

\(\displaystyle f{\left(x \right)} = x + 1\)

sample_pairs = make_examples('real', 6, 5, positive_only=True, increasing_only=True)
for pair in sample_pairs:
    print(pair)
    print('bigger:', polynomial_to_test.get_output(pair[0]))
    print('simple:', simple_polynomial.get_output(pair[0]))

(97, 5)
bigger: Eq(f(97), -875035)
simple: Eq(f(97), 98)
(187, 5)
bigger: Eq(f(187), -6399325)
simple: Eq(f(187), 188)
(261, 5)
bigger: Eq(f(261), -17507095)
simple: Eq(f(261), 262)
(315, 5)
bigger: Eq(f(315), -30858973)
simple: Eq(f(315), 316)
(338, 5)
bigger: Eq(f(338), -38157494)
simple: Eq(f(338), 339)
(382, 5)
bigger: Eq(f(382), -55159270)
simple: Eq(f(382), 383)