cmlabs.integrate.test_newton_cotes_error

cmlabs.integrate.test_newton_cotes_error()[source]

Reach accuracy \(\epsilon\) error for Newton-Cotes rule.

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

where \(I_n\) is the integral of the function \(f(x)\) over the interval \([a, b]\) using Newton-Cotes rule 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 7: Newton-Cotes Rule Error
>>> # - f(x) = x - lg(x + 2)
>>> # - X_n:  [0.5 0.6 0.7 0.8 0.9 1. ]
>>> # - Y_n:  [0.102 0.185 0.269 0.353 0.438 0.523]
>>> # - Coef:  [0.066 0.26  0.174 0.174 0.26  0.066]
>>> newton_cotes(X_n, Y_n, df['coef'])
>>> # - I_n:  0.1556334987504589
>>> # - I_2n:  0.1556334984772168
>>> # |I_n - I_2n|:  2.7324209561641055e-10
>>> # 2.7324209561641055e-10 <= 1e-06
>>> # True