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/runge_kutta.py

@@ -0,0 +1,44 @@
+import numpy as np
+
+
+def runge_kutta(f, y0, x0, h, x_end):
+    """
+    Calculate the numeric solution at each step to the ODE f(x, y) using RK4
+
+    https://en.wikipedia.org/wiki/Runge-Kutta_methods
+
+    Arguments:
+    f -- The ode as a function of x and y
+    y0 -- the initial value for y
+    x0 -- the initial value for x
+    h -- the stepsize
+    x_end -- the end value for x
+
+    >>> # the exact solution is math.exp(x)
+    >>> def f(x, y):
+    ...     return y
+    >>> y0 = 1
+    >>> y = runge_kutta(f, y0, 0.0, 0.01, 5)
+    >>> y[-1]
+    148.41315904125113
+    """
+    n = int(np.ceil((x_end - x0) / h))
+    y = np.zeros((n + 1,))
+    y[0] = y0
+    x = x0
+
+    for k in range(n):
+        k1 = f(x, y[k])
+        k2 = f(x + 0.5 * h, y[k] + 0.5 * h * k1)
+        k3 = f(x + 0.5 * h, y[k] + 0.5 * h * k2)
+        k4 = f(x + h, y[k] + h * k3)
+        y[k + 1] = y[k] + (1 / 6) * h * (k1 + 2 * k2 + 2 * k3 + k4)
+        x += h
+
+    return y
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()