library: add finite_difference.py

library/finite_difference.py

@@ -0,0 +1,69 @@
+#!/usr/bin/env python3
+# implementing: https://web.media.mit.edu/~crtaylor/calculator.html
+
+import numpy as np
+
+# from math import factorial
+_factorials = (  # all the factorials that fit into 53 bits
+    1,
+    1,
+    2,
+    6,
+    24,
+    120,
+    720,
+    5040,
+    40320,
+    362880,
+    3628800,
+    39916800,
+    479001600,
+    6227020800,
+    87178291200,
+    1307674368000,
+    20922789888000,
+    355687428096000,
+    6402373705728000,
+)
+
+__noisy = False
+
+
+def fd(offsets, degree):
+    if degree < 0:
+        raise RuntimeError("the degree must be greater than or equal to zero")
+    elif degree > 18:
+        raise RuntimeError("degrees greater than 18 are unsupported")
+    if len(offsets) <= degree:
+        raise RuntimeError("the number of offsets must be greater than the degree")
+    mat = np.array([[o**n for o in offsets] for n in range(len(offsets))])
+    vec = np.array([_factorials[i] if i == degree else 0 for i in range(len(offsets))])
+    if __noisy:
+        print("mat:", mat, "vec:", vec, sep="\n")
+    res = np.linalg.lstsq(mat, vec, rcond=None)[0]
+    return res
+
+
+if __name__ == "__main__":
+    __noisy = True
+    assert np.allclose(fd([-2, 0, 2], 1), [-1 / 4, 0, 1 / 4])
+
+    # + A * f(x - 2 * h)
+    # + B * f(x - 1 * h)
+    # + C * f(x)
+    # + D * f(x + 1 * h)
+    # + E * f(x + 2 * h)
+    assert np.allclose(fd([-2, -1, 0, 1, 2], 4), [1, -4, 6, -4, 1])
+
+    print("", "#" * 40, "", sep="\n")
+
+    ans = fd([-2, 7], 1)
+    print("ans:", ans)
+
+    ans = fd([-2, 0, 7], 1)
+    print("ans:", ans)
+
+    ans = fd([-0.2, 0.0, 0.7], 1)
+    print("ans:", ans)
+
+__all__ = ("fd",)