cmlabs.integrate.test_simpsonq_error

cmlabs.integrate.test_simpsonq_error()[source]

Reach accuracy \(\epsilon\) error for Simpson’s rule (quadratic).

\[\begin{split}\begin{gather} |I_n - I_{2n}| \leq \epsilon, \\ |I_{2n} - I_{4n}| \leq \epsilon, \\ \ldots \\ |I_{2^k n} - I_{2^{k+1} n}| \leq \epsilon, \\ \end{gather}\end{split}\]

where \(I_n\) is the integral of the function \(f(x)\) over the interval \([a, b]\) using Simpson’s rule (quadratic) with \(n\) subintervals, and \(\epsilon\) is the desired accuracy.

Notes

\[\begin{split}\begin{gather} \int_a^b f(x) \, dx = \int_{0.5}^{1} x - \log_{10}(x + 2) \, dx \\ = \frac{1\ln{10} + 20\ln{5} - 24\ln{3} - 20\ln{2} + 4}{8\ln{10}} \\ \approx 0.1556335 \end{gather}\end{split}\]

Results

>>> # Test 4: Simpson's Rule (Quadratic) Error
>>> # - f(x) = x - lg(x + 2)
>>> # - X_n:  [0.5  0.75 1.  ]
>>> # - Y_n:  [0.102 0.311 0.523]
>>> # - X_2n:  [0.5   0.625 0.75  0.875 1.   ]
>>> # - Y_2n:  [0.102 0.206 0.311 0.416 0.523]
>>> simpsonq(X_n, Y_n)
>>> # - I_n:  0.15563399677393747
>>> simpsonq(X_2n, Y_2n)
>>> # - I_2n:  0.15563353007821978
>>> # |I_n - I_2n|:  4.6669571768243046e-07
>>> # 4.6669571768243046e-07 <= 0.0001
>>> # True