Polynomial (#6745)

* implement function to handle polynomial operations

* edit documentation

* fix type hint and linter errors

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* fix short variable name

* fix spelling

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

maths/polynomials/single_indeterminate_operations.py

@@ -0,0 +1,188 @@
+"""
+
+This module implements a single indeterminate polynomials class
+with some basic operations
+
+Reference: https://en.wikipedia.org/wiki/Polynomial
+
+"""
+
+from __future__ import annotations
+
+from collections.abc import MutableSequence
+
+
+class Polynomial:
+    def __init__(self, degree: int, coefficients: MutableSequence[float]) -> None:
+        """
+        The coefficients should be in order of degree, from smallest to largest.
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> p = Polynomial(2, [1, 2, 3, 4])
+        Traceback (most recent call last):
+        ...
+        ValueError: The number of coefficients should be equal to the degree + 1.
+
+        """
+        if len(coefficients) != degree + 1:
+            raise ValueError(
+                "The number of coefficients should be equal to the degree + 1."
+            )
+
+        self.coefficients: list[float] = list(coefficients)
+        self.degree = degree
+
+    def __add__(self, polynomial_2: Polynomial) -> Polynomial:
+        """
+        Polynomial addition
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> q = Polynomial(2, [1, 2, 3])
+        >>> p + q
+        6x^2 + 4x + 2
+        """
+
+        if self.degree > polynomial_2.degree:
+            coefficients = self.coefficients[:]
+            for i in range(polynomial_2.degree + 1):
+                coefficients[i] += polynomial_2.coefficients[i]
+            return Polynomial(self.degree, coefficients)
+        else:
+            coefficients = polynomial_2.coefficients[:]
+            for i in range(self.degree + 1):
+                coefficients[i] += self.coefficients[i]
+            return Polynomial(polynomial_2.degree, coefficients)
+
+    def __sub__(self, polynomial_2: Polynomial) -> Polynomial:
+        """
+        Polynomial subtraction
+        >>> p = Polynomial(2, [1, 2, 4])
+        >>> q = Polynomial(2, [1, 2, 3])
+        >>> p - q
+        1x^2
+        """
+        return self + polynomial_2 * Polynomial(0, [-1])
+
+    def __neg__(self) -> Polynomial:
+        """
+        Polynomial negation
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> -p
+         - 3x^2 - 2x - 1
+        """
+        return Polynomial(self.degree, [-c for c in self.coefficients])
+
+    def __mul__(self, polynomial_2: Polynomial) -> Polynomial:
+        """
+        Polynomial multiplication
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> q = Polynomial(2, [1, 2, 3])
+        >>> p * q
+        9x^4 + 12x^3 + 10x^2 + 4x + 1
+        """
+        coefficients: list[float] = [0] * (self.degree + polynomial_2.degree + 1)
+        for i in range(self.degree + 1):
+            for j in range(polynomial_2.degree + 1):
+                coefficients[i + j] += (
+                    self.coefficients[i] * polynomial_2.coefficients[j]
+                )
+
+        return Polynomial(self.degree + polynomial_2.degree, coefficients)
+
+    def evaluate(self, substitution: int | float) -> int | float:
+        """
+        Evaluates the polynomial at x.
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> p.evaluate(2)
+        17
+        """
+        result: int | float = 0
+        for i in range(self.degree + 1):
+            result += self.coefficients[i] * (substitution**i)
+        return result
+
+    def __str__(self) -> str:
+        """
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> print(p)
+        3x^2 + 2x + 1
+        """
+        polynomial = ""
+        for i in range(self.degree, -1, -1):
+            if self.coefficients[i] == 0:
+                continue
+            elif self.coefficients[i] > 0:
+                if polynomial:
+                    polynomial += " + "
+            else:
+                polynomial += " - "
+
+            if i == 0:
+                polynomial += str(abs(self.coefficients[i]))
+            elif i == 1:
+                polynomial += str(abs(self.coefficients[i])) + "x"
+            else:
+                polynomial += str(abs(self.coefficients[i])) + "x^" + str(i)
+
+        return polynomial
+
+    def __repr__(self) -> str:
+        """
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> p
+        3x^2 + 2x + 1
+        """
+        return self.__str__()
+
+    def derivative(self) -> Polynomial:
+        """
+        Returns the derivative of the polynomial.
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> p.derivative()
+        6x + 2
+        """
+        coefficients: list[float] = [0] * self.degree
+        for i in range(self.degree):
+            coefficients[i] = self.coefficients[i + 1] * (i + 1)
+        return Polynomial(self.degree - 1, coefficients)
+
+    def integral(self, constant: int | float = 0) -> Polynomial:
+        """
+        Returns the integral of the polynomial.
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> p.integral()
+        1.0x^3 + 1.0x^2 + 1.0x
+        """
+        coefficients: list[float] = [0] * (self.degree + 2)
+        coefficients[0] = constant
+        for i in range(self.degree + 1):
+            coefficients[i + 1] = self.coefficients[i] / (i + 1)
+        return Polynomial(self.degree + 1, coefficients)
+
+    def __eq__(self, polynomial_2: object) -> bool:
+        """
+        Checks if two polynomials are equal.
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> q = Polynomial(2, [1, 2, 3])
+        >>> p == q
+        True
+        """
+        if not isinstance(polynomial_2, Polynomial):
+            return False
+
+        if self.degree != polynomial_2.degree:
+            return False
+
+        for i in range(self.degree + 1):
+            if self.coefficients[i] != polynomial_2.coefficients[i]:
+                return False
+
+        return True
+
+    def __ne__(self, polynomial_2: object) -> bool:
+        """
+        Checks if two polynomials are not equal.
+        >>> p = Polynomial(2, [1, 2, 3])
+        >>> q = Polynomial(2, [1, 2, 3])
+        >>> p != q
+        False
+        """
+        return not self.__eq__(polynomial_2)