Dual Number Automatic Differentiation (TheAlgorithms#8760)

* Added dual_number_automatic_differentiation.py

* updating DIRECTORY.md

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* Update maths/dual_number_automatic_differentiation.py

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Diff for: DIRECTORY.md

@@ -549,6 +549,7 @@
   * [Dodecahedron](maths/dodecahedron.py)
   * [Double Factorial Iterative](maths/double_factorial_iterative.py)
   * [Double Factorial Recursive](maths/double_factorial_recursive.py)
+  * [Dual Number Automatic Differentiation](maths/dual_number_automatic_differentiation.py)
   * [Entropy](maths/entropy.py)
   * [Euclidean Distance](maths/euclidean_distance.py)
   * [Euclidean Gcd](maths/euclidean_gcd.py)

Diff for: maths/dual_number_automatic_differentiation.py

@@ -0,0 +1,141 @@
+from math import factorial
+
+"""
+https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers
+https://blog.jliszka.org/2013/10/24/exact-numeric-nth-derivatives.html
+
+Note this only works for basic functions, f(x) where the power of x is positive.
+"""
+
+
+class Dual:
+    def __init__(self, real, rank):
+        self.real = real
+        if isinstance(rank, int):
+            self.duals = [1] * rank
+        else:
+            self.duals = rank
+
+    def __repr__(self):
+        return (
+            f"{self.real}+"
+            f"{'+'.join(str(dual)+'E'+str(n+1)for n,dual in enumerate(self.duals))}"
+        )
+
+    def reduce(self):
+        cur = self.duals.copy()
+        while cur[-1] == 0:
+            cur.pop(-1)
+        return Dual(self.real, cur)
+
+    def __add__(self, other):
+        if not isinstance(other, Dual):
+            return Dual(self.real + other, self.duals)
+        s_dual = self.duals.copy()
+        o_dual = other.duals.copy()
+        if len(s_dual) > len(o_dual):
+            o_dual.extend([1] * (len(s_dual) - len(o_dual)))
+        elif len(s_dual) < len(o_dual):
+            s_dual.extend([1] * (len(o_dual) - len(s_dual)))
+        new_duals = []
+        for i in range(len(s_dual)):
+            new_duals.append(s_dual[i] + o_dual[i])
+        return Dual(self.real + other.real, new_duals)
+
+    __radd__ = __add__
+
+    def __sub__(self, other):
+        return self + other * -1
+
+    def __mul__(self, other):
+        if not isinstance(other, Dual):
+            new_duals = []
+            for i in self.duals:
+                new_duals.append(i * other)
+            return Dual(self.real * other, new_duals)
+        new_duals = [0] * (len(self.duals) + len(other.duals) + 1)
+        for i, item in enumerate(self.duals):
+            for j, jtem in enumerate(other.duals):
+                new_duals[i + j + 1] += item * jtem
+        for k in range(len(self.duals)):
+            new_duals[k] += self.duals[k] * other.real
+        for index in range(len(other.duals)):
+            new_duals[index] += other.duals[index] * self.real
+        return Dual(self.real * other.real, new_duals)
+
+    __rmul__ = __mul__
+
+    def __truediv__(self, other):
+        if not isinstance(other, Dual):
+            new_duals = []
+            for i in self.duals:
+                new_duals.append(i / other)
+            return Dual(self.real / other, new_duals)
+        raise ValueError()
+
+    def __floordiv__(self, other):
+        if not isinstance(other, Dual):
+            new_duals = []
+            for i in self.duals:
+                new_duals.append(i // other)
+            return Dual(self.real // other, new_duals)
+        raise ValueError()
+
+    def __pow__(self, n):
+        if n < 0 or isinstance(n, float):
+            raise ValueError("power must be a positive integer")
+        if n == 0:
+            return 1
+        if n == 1:
+            return self
+        x = self
+        for _ in range(n - 1):
+            x *= self
+        return x
+
+
+def differentiate(func, position, order):
+    """
+    >>> differentiate(lambda x: x**2, 2, 2)
+    2
+    >>> differentiate(lambda x: x**2 * x**4, 9, 2)
+    196830
+    >>> differentiate(lambda y: 0.5 * (y + 3) ** 6, 3.5, 4)
+    7605.0
+    >>> differentiate(lambda y: y ** 2, 4, 3)
+    0
+    >>> differentiate(8, 8, 8)
+    Traceback (most recent call last):
+        ...
+    ValueError: differentiate() requires a function as input for func
+    >>> differentiate(lambda x: x **2, "", 1)
+    Traceback (most recent call last):
+        ...
+    ValueError: differentiate() requires a float as input for position
+    >>> differentiate(lambda x: x**2, 3, "")
+    Traceback (most recent call last):
+        ...
+    ValueError: differentiate() requires an int as input for order
+    """
+    if not callable(func):
+        raise ValueError("differentiate() requires a function as input for func")
+    if not isinstance(position, (float, int)):
+        raise ValueError("differentiate() requires a float as input for position")
+    if not isinstance(order, int):
+        raise ValueError("differentiate() requires an int as input for order")
+    d = Dual(position, 1)
+    result = func(d)
+    if order == 0:
+        return result.real
+    return result.duals[order - 1] * factorial(order)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    def f(y):
+        return y**2 * y**4
+
+    print(differentiate(f, 9, 2))