cmlabs.interpolate.test_lagrange_remainder_2

cmlabs.interpolate.test_lagrange_remainder_2()[source]

Estimate the derivative and remainder degree 2.

\[\begin{split}\begin{gather} R_2(x) = f(x) - L_2(x) \\ \\ f(x) = x - \log_{10}(x + 2) \\ f^{(3)}(x) = \frac{-2}{(x + 2)^3 \cdot \ln(10)} \\ \min{f^{(3)}_{[x_{i-1}, x_{i+1}]}} = \min_{x \in [x_{i-1}, x_{i+1}]} |f^{(3)}(x)| = \frac{-2}{(x_{i+1} + 2)^3 \cdot \ln(10)} \\ \max{f^{(3)}_{[x_{i-1}, x_{i+1}]}} = \max_{x \in [x_{i-1}, x_{i+1}]} |f^{(3)}(x)| = \frac{-2}{(x_{i-1} + 2)^3 \cdot \ln(10)} \\ \\ \left|\frac{\min{f^{(3)}(x)}}{6} \cdot \omega_3(x)\right| \leq |R_2(x)| \leq \left|\frac{\max{f^{(3)}(x)}}{6} \cdot \omega_3(x)\right|, \quad x \in [x_{i-1}, x_{i+1}] \end{gather}\end{split}\]

Results

>>> # Test 4: Estimating Remainder Degree 2 In Lagrange Interpolation Formula
>>> # - X:  [0.5   0.556 0.611 ... 0.889 0.944 1.   ]
>>> # - x* =  0.77
>>> # Nearest points: [0.722 0.778 0.833]
>>> # Nearest f(x) values: [0.287 0.334 0.381]
>>> # f'''(x) min: -0.03818750583802256
>>> # f'''(x) max: -0.04305698167361838
>>> f(0.77) - lagrange([0.722 0.778 0.833], [0.287 0.334 0.381], 0.77)
>>> # 1.5933154223768398e-07
>>> # R_min: -1.4979036069111843e-07
>>> # R_max: -1.6889086295708034e-07
>>> |-1.4979036069111843e-07| <= |1.5933154223768398e-07| <= |-1.688908629...|
>>> # True
>>> abs(-1.6889086295708034e-07) <= 1e-5
>>> # True