cmlabs.integrate.test_midpoint_error

cmlabs.integrate.test_midpoint_error()[source]

Reach accuracy \(\epsilon\) error for midpoint 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 the midpoint 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 2: Midpoint Rule Error
>>> # - f(x) = x - lg(x + 2)
>>> # - X_n:  [0.5   0.625 0.75  0.875 1.   ]
>>> # - Y_n:  [0.102 0.206 0.311 0.416 0.523]
>>> # - X_2n:  [0.5   0.562 0.625 ... 0.875 0.938 1.   ]
>>> # - Y_2n:  [0.102 0.154 0.206 ... 0.416 0.47  0.523]
>>> midpoint(X_n, Y_n)
>>> # - I_n:  0.15555821080809373
>>> midpoint(X_2n, Y_2n)
>>> # - I_2n:  0.1556146558266645
>>> # |I_n - I_2n|:  5.644501857077211e-05
>>> # 5.644501857077211e-05 <= 0.001
>>> # True