chore: import upstream snapshot with attribution

demo/taylor.py

@@ -0,0 +1,71 @@
+"""
+Interval arithmetic demo: estimating error of numerical Taylor series.
+
+This module can be run interactively with
+
+    python taylor.py
+
+"""
+from mpmath import mpi, exp, factorial, mpf
+
+def taylor(x, n):
+    print("-"*75)
+    t = x = mpi(x)
+    s = 1
+    print("adding 1")
+    print(s, "\n")
+    s += t
+    print("adding x")
+    print(s, "\n")
+    for k in range(2, n+1):
+        t = (t * x) / k
+        s += t
+        print("adding x^%i / %i! ~= %s" % (k, k, t.mid))
+        print(s, "\n")
+    print("-"*75)
+    return s
+
+# Note: this should really be computed using interval arithmetic too!
+def remainder(x, n):
+    xi = max(0, x)
+    r = exp(xi) / factorial(n+1)
+    r = r * x**(n+1)
+    return abs(r)
+
+def exponential(x, n):
+    """
+    Compute exp(x) using n terms of the Taylor series for exp using
+    intervals, and print detailed error analysis.
+    """
+    t = taylor(x, n)
+    r = remainder(x, n)
+    expx = exp(x)
+    print("Correct value of exp(x):    ", expx)
+    print()
+    print("Computed interval:          ")
+    print(t)
+    print()
+    print("Computed value (midpoint):  ", t.mid)
+    print()
+    print("Estimated rounding error:   ", t.delta)
+    print("Estimated truncation error: ", r)
+    print("Estimated total error:      ", t.delta + r)
+    print("Actual error                ", abs(expx - t.mid))
+    print()
+    u = t + mpi(-r, r)
+    print("Interval with est. truncation error added:")
+    print(u)
+    print()
+    print("Correct value contained in computed interval:", t.a <= expx <= t.b)
+    print("When accounting for truncation error:", u.a <= expx <= u.b)
+
+if __name__ == "__main__":
+    print("Interval arithmetic demo")
+    print()
+    print("This script sums the Taylor series for exp(x) using interval arithmetic,")
+    print("and then compares the numerical errors due to rounding and truncation.")
+    print()
+    x = mpf(input("Enter the value of x (e.g. 3.5): "))
+    n = int(input("Enter the number of terms n (e.g. 10): "))
+    print()
+    exponential(x, n)