Modified Euler's Method (#5258)

* Magnitude and Angle 

Core function to find Magnitude and Angle of two Given Vector

* Magnitude and Angle with Doctest

added Doctest to the functions

* Update linear_algebra/src/lib.py

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

* Update linear_algebra/src/lib.py

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

* Changes done 

and Magnitude and Angle Issues

* black

* Modified Euler's Method

Adding Modified Euler's method, which was the further change to a Euler method and known for better accuracy to the given value

* Modified Euler's Method (changed the typing of function)

Modified function is used for better accuracy

* Link added

Added link to an explanation as per Contributions Guidelines

* Resolving Pre-Commit error

* Pre-Commit Error Resolved

* Pre-Commit Error import statement Change

* Removed Import Math

* import math built issue

* adding space pre-commit error

* statement sorter for doc

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

maths/euler_modified.py

@@ -0,0 +1,54 @@
+from typing import Callable
+
+import numpy as np
+
+
+def euler_modified(
+    ode_func: Callable, y0: float, x0: float, step_size: float, x_end: float
+) -> np.array:
+    """
+    Calculate solution at each step to an ODE using Euler's Modified Method
+    The Euler is straightforward to implement, but can't give accurate solutions.
+    So, they Proposed some changes to improve the accuracy
+
+    https://en.wikipedia.org/wiki/Euler_method
+
+    Arguments:
+    ode_func -- The ode as a function of x and y
+    y0 -- the initial value for y
+    x0 -- the initial value for x
+    stepsize -- the increment value for x
+    x_end -- the end value for x
+
+    >>> # the exact solution is math.exp(x)
+    >>> def f1(x, y):
+    ...     return -2*x*(y**2)
+    >>> y = euler_modified(f1, 1.0, 0.0, 0.2, 1.0)
+    >>> y[-1]
+    0.503338255442106
+    >>> import math
+    >>> def f2(x, y):
+    ...     return -2*y + (x**3)*math.exp(-2*x)
+    >>> y = euler_modified(f2, 1.0, 0.0, 0.1, 0.3)
+    >>> y[-1]
+    0.5525976431951775
+    """
+    N = int(np.ceil((x_end - x0) / step_size))
+    y = np.zeros((N + 1,))
+    y[0] = y0
+    x = x0
+
+    for k in range(N):
+        y_get = y[k] + step_size * ode_func(x, y[k])
+        y[k + 1] = y[k] + (
+            (step_size / 2) * (ode_func(x, y[k]) + ode_func(x + step_size, y_get))
+        )
+        x += step_size
+
+    return y
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()