Newton raphson complex (#6545)

* Newton raphson better implementation

* flake8 test passed

* Update arithmetic_analysis/newton_raphson_new.py

Co-authored-by: Christian Clauss <cclauss@me.com>

* added multiline suggestions

Co-authored-by: Christian Clauss <cclauss@me.com>

arithmetic_analysis/newton_raphson_new.py

@@ -0,0 +1,84 @@
+# 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: F401, 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)}")