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