Delete arithmetic_analysis/ directory and relocate its contents (#10824)

* Remove eval from arithmetic_analysis/newton_raphson.py

* Relocate contents of arithmetic_analysis/

Delete the arithmetic_analysis/ directory and relocate its files because
the purpose of the directory was always ill-defined. "Arithmetic
analysis" isn't a field of math, and the directory's files contained
algorithms for linear algebra, numerical analysis, and physics.

Relocated the directory's linear algebra algorithms to linear_algebra/,
its numerical analysis algorithms to a new subdirectory called
maths/numerical_analysis/, and its single physics algorithm to physics/.

* updating DIRECTORY.md

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

maths/numerical_analysis/bisection.py

@@ -0,0 +1,55 @@
+from collections.abc import Callable
+
+
+def bisection(function: Callable[[float], float], a: float, b: float) -> float:
+    """
+    finds where function becomes 0 in [a,b] using bolzano
+    >>> bisection(lambda x: x ** 3 - 1, -5, 5)
+    1.0000000149011612
+    >>> bisection(lambda x: x ** 3 - 1, 2, 1000)
+    Traceback (most recent call last):
+        ...
+    ValueError: could not find root in given interval.
+    >>> bisection(lambda x: x ** 2 - 4 * x + 3, 0, 2)
+    1.0
+    >>> bisection(lambda x: x ** 2 - 4 * x + 3, 2, 4)
+    3.0
+    >>> bisection(lambda x: x ** 2 - 4 * x + 3, 4, 1000)
+    Traceback (most recent call last):
+        ...
+    ValueError: could not find root in given interval.
+    """
+    start: float = a
+    end: float = b
+    if function(a) == 0:  # one of the a or b is a root for the function
+        return a
+    elif function(b) == 0:
+        return b
+    elif (
+        function(a) * function(b) > 0
+    ):  # if none of these are root and they are both positive or negative,
+        # then this algorithm can't find the root
+        raise ValueError("could not find root in given interval.")
+    else:
+        mid: float = start + (end - start) / 2.0
+        while abs(start - mid) > 10**-7:  # until precisely equals to 10^-7
+            if function(mid) == 0:
+                return mid
+            elif function(mid) * function(start) < 0:
+                end = mid
+            else:
+                start = mid
+            mid = start + (end - start) / 2.0
+        return mid
+
+
+def f(x: float) -> float:
+    return x**3 - 2 * x - 5
+
+
+if __name__ == "__main__":
+    print(bisection(f, 1, 1000))
+
+    import doctest
+
+    doctest.testmod()

maths/numerical_analysis/intersection.py

@@ -0,0 +1,49 @@
+import math
+from collections.abc import Callable
+
+
+def intersection(function: Callable[[float], float], x0: float, x1: float) -> float:
+    """
+    function is the f we want to find its root
+    x0 and x1 are two random starting points
+    >>> intersection(lambda x: x ** 3 - 1, -5, 5)
+    0.9999999999954654
+    >>> intersection(lambda x: x ** 3 - 1, 5, 5)
+    Traceback (most recent call last):
+        ...
+    ZeroDivisionError: float division by zero, could not find root
+    >>> intersection(lambda x: x ** 3 - 1, 100, 200)
+    1.0000000000003888
+    >>> intersection(lambda x: x ** 2 - 4 * x + 3, 0, 2)
+    0.9999999998088019
+    >>> intersection(lambda x: x ** 2 - 4 * x + 3, 2, 4)
+    2.9999999998088023
+    >>> intersection(lambda x: x ** 2 - 4 * x + 3, 4, 1000)
+    3.0000000001786042
+    >>> intersection(math.sin, -math.pi, math.pi)
+    0.0
+    >>> intersection(math.cos, -math.pi, math.pi)
+    Traceback (most recent call last):
+        ...
+    ZeroDivisionError: float division by zero, could not find root
+    """
+    x_n: float = x0
+    x_n1: float = x1
+    while True:
+        if x_n == x_n1 or function(x_n1) == function(x_n):
+            raise ZeroDivisionError("float division by zero, could not find root")
+        x_n2: float = x_n1 - (
+            function(x_n1) / ((function(x_n1) - function(x_n)) / (x_n1 - x_n))
+        )
+        if abs(x_n2 - x_n1) < 10**-5:
+            return x_n2
+        x_n = x_n1
+        x_n1 = x_n2
+
+
+def f(x: float) -> float:
+    return math.pow(x, 3) - (2 * x) - 5
+
+
+if __name__ == "__main__":
+    print(intersection(f, 3, 3.5))

maths/numerical_analysis/newton_forward_interpolation.py

@@ -0,0 +1,57 @@
+# https://www.geeksforgeeks.org/newton-forward-backward-interpolation/
+from __future__ import annotations
+
+import math
+
+
+# for calculating u value
+def ucal(u: float, p: int) -> float:
+    """
+    >>> ucal(1, 2)
+    0
+    >>> ucal(1.1, 2)
+    0.11000000000000011
+    >>> ucal(1.2, 2)
+    0.23999999999999994
+    """
+    temp = u
+    for i in range(1, p):
+        temp = temp * (u - i)
+    return temp
+
+
+def main() -> None:
+    n = int(input("enter the numbers of values: "))
+    y: list[list[float]] = []
+    for _ in range(n):
+        y.append([])
+    for i in range(n):
+        for j in range(n):
+            y[i].append(j)
+            y[i][j] = 0
+
+    print("enter the values of parameters in a list: ")
+    x = list(map(int, input().split()))
+
+    print("enter the values of corresponding parameters: ")
+    for i in range(n):
+        y[i][0] = float(input())
+
+    value = int(input("enter the value to interpolate: "))
+    u = (value - x[0]) / (x[1] - x[0])
+
+    # for calculating forward difference table
+
+    for i in range(1, n):
+        for j in range(n - i):
+            y[j][i] = y[j + 1][i - 1] - y[j][i - 1]
+
+    summ = y[0][0]
+    for i in range(1, n):
+        summ += (ucal(u, i) * y[0][i]) / math.factorial(i)
+
+    print(f"the value at {value} is {summ}")
+
+
+if __name__ == "__main__":
+    main()

maths/numerical_analysis/newton_method.py

@@ -0,0 +1,54 @@
+"""Newton's Method."""
+
+# Newton's Method - https://en.wikipedia.org/wiki/Newton%27s_method
+from collections.abc import Callable
+
+RealFunc = Callable[[float], float]  # type alias for a real -> real function
+
+
+# function is the f(x) and derivative is the f'(x)
+def newton(
+    function: RealFunc,
+    derivative: RealFunc,
+    starting_int: int,
+) -> float:
+    """
+    >>> newton(lambda x: x ** 3 - 2 * x - 5, lambda x: 3 * x ** 2 - 2, 3)
+    2.0945514815423474
+    >>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -2)
+    1.0
+    >>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -4)
+    1.0000000000000102
+    >>> import math
+    >>> newton(math.sin, math.cos, 1)
+    0.0
+    >>> newton(math.sin, math.cos, 2)
+    3.141592653589793
+    >>> newton(math.cos, lambda x: -math.sin(x), 2)
+    1.5707963267948966
+    >>> newton(math.cos, lambda x: -math.sin(x), 0)
+    Traceback (most recent call last):
+        ...
+    ZeroDivisionError: Could not find root
+    """
+    prev_guess = float(starting_int)
+    while True:
+        try:
+            next_guess = prev_guess - function(prev_guess) / derivative(prev_guess)
+        except ZeroDivisionError:
+            raise ZeroDivisionError("Could not find root") from None
+        if abs(prev_guess - next_guess) < 10**-5:
+            return next_guess
+        prev_guess = next_guess
+
+
+def f(x: float) -> float:
+    return (x**3) - (2 * x) - 5
+
+
+def f1(x: float) -> float:
+    return 3 * (x**2) - 2
+
+
+if __name__ == "__main__":
+    print(newton(f, f1, 3))

maths/numerical_analysis/newton_raphson.py

@@ -0,0 +1,45 @@
+# Implementing Newton Raphson method in Python
+# Author: Syed Haseeb Shah (github.com/QuantumNovice)
+# The Newton-Raphson method (also known as Newton's method) is a way to
+# quickly find a good approximation for the root of a real-valued function
+from __future__ import annotations
+
+from decimal import Decimal
+
+from sympy import diff, lambdify, symbols
+
+
+def newton_raphson(func: str, a: float | Decimal, precision: float = 1e-10) -> float:
+    """Finds root from the point 'a' onwards by Newton-Raphson method
+    >>> newton_raphson("sin(x)", 2)
+    3.1415926536808043
+    >>> newton_raphson("x**2 - 5*x + 2", 0.4)
+    0.4384471871911695
+    >>> newton_raphson("x**2 - 5", 0.1)
+    2.23606797749979
+    >>> newton_raphson("log(x) - 1", 2)
+    2.718281828458938
+    """
+    x = symbols("x")
+    f = lambdify(x, func, "math")
+    f_derivative = lambdify(x, diff(func), "math")
+    x_curr = a
+    while True:
+        x_curr = Decimal(x_curr) - Decimal(f(x_curr)) / Decimal(f_derivative(x_curr))
+        if abs(f(x_curr)) < precision:
+            return float(x_curr)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    # Find value of pi
+    print(f"The root of sin(x) = 0 is {newton_raphson('sin(x)', 2)}")
+    # Find root of polynomial
+    print(f"The root of x**2 - 5*x + 2 = 0 is {newton_raphson('x**2 - 5*x + 2', 0.4)}")
+    # Find value of e
+    print(f"The root of log(x) - 1 = 0 is {newton_raphson('log(x) - 1', 2)}")
+    # Find root of exponential function
+    print(f"The root of exp(x) - 1 = 0 is {newton_raphson('exp(x) - 1', 0)}")

maths/numerical_analysis/newton_raphson_new.py

@@ -0,0 +1,83 @@
+# Implementing Newton Raphson method in Python
+# Author: Saksham Gupta
+#
+# The Newton-Raphson method (also known as Newton's method) is a way to
+# quickly find a good approximation for the root of a functreal-valued ion
+# The method can also be extended to complex functions
+#
+# Newton's Method - https://en.wikipedia.org/wiki/Newton's_method
+
+from sympy import diff, lambdify, symbols
+from sympy.functions import *  # noqa: F403
+
+
+def newton_raphson(
+    function: str,
+    starting_point: complex,
+    variable: str = "x",
+    precision: float = 10**-10,
+    multiplicity: int = 1,
+) -> complex:
+    """Finds root from the 'starting_point' onwards by Newton-Raphson method
+    Refer to https://docs.sympy.org/latest/modules/functions/index.html
+    for usable mathematical functions
+
+    >>> newton_raphson("sin(x)", 2)
+    3.141592653589793
+    >>> newton_raphson("x**4 -5", 0.4 + 5j)
+    (-7.52316384526264e-37+1.4953487812212207j)
+    >>> newton_raphson('log(y) - 1', 2, variable='y')
+    2.7182818284590455
+    >>> newton_raphson('exp(x) - 1', 10, precision=0.005)
+    1.2186556186174883e-10
+    >>> newton_raphson('cos(x)', 0)
+    Traceback (most recent call last):
+        ...
+    ZeroDivisionError: Could not find root
+    """
+
+    x = symbols(variable)
+    func = lambdify(x, function)
+    diff_function = lambdify(x, diff(function, x))
+
+    prev_guess = starting_point
+
+    while True:
+        if diff_function(prev_guess) != 0:
+            next_guess = prev_guess - multiplicity * func(prev_guess) / diff_function(
+                prev_guess
+            )
+        else:
+            raise ZeroDivisionError("Could not find root") from None
+
+        # Precision is checked by comparing the difference of consecutive guesses
+        if abs(next_guess - prev_guess) < precision:
+            return next_guess
+
+        prev_guess = next_guess
+
+
+# Let's Execute
+if __name__ == "__main__":
+    # Find root of trigonometric function
+    # Find value of pi
+    print(f"The root of sin(x) = 0 is {newton_raphson('sin(x)', 2)}")
+
+    # Find root of polynomial
+    # Find fourth Root of 5
+    print(f"The root of x**4 - 5 = 0 is {newton_raphson('x**4 -5', 0.4 +5j)}")
+
+    # Find value of e
+    print(
+        "The root of log(y) - 1 = 0 is ",
+        f"{newton_raphson('log(y) - 1', 2, variable='y')}",
+    )
+
+    # Exponential Roots
+    print(
+        "The root of exp(x) - 1 = 0 is",
+        f"{newton_raphson('exp(x) - 1', 10, precision=0.005)}",
+    )
+
+    # Find root of cos(x)
+    print(f"The root of cos(x) = 0 is {newton_raphson('cos(x)', 0)}")

maths/numerical_analysis/secant_method.py

@@ -0,0 +1,29 @@
+"""
+Implementing Secant method in Python
+Author: dimgrichr
+"""
+from math import exp
+
+
+def f(x: float) -> float:
+    """
+    >>> f(5)
+    39.98652410600183
+    """
+    return 8 * x - 2 * exp(-x)
+
+
+def secant_method(lower_bound: float, upper_bound: float, repeats: int) -> float:
+    """
+    >>> secant_method(1, 3, 2)
+    0.2139409276214589
+    """
+    x0 = lower_bound
+    x1 = upper_bound
+    for _ in range(repeats):
+        x0, x1 = x1, x1 - (f(x1) * (x1 - x0)) / (f(x1) - f(x0))
+    return x1
+
+
+if __name__ == "__main__":
+    print(f"Example: {secant_method(1, 3, 2)}")