MathProblemsBank

14.8.2 Function Interpolation

A well-known function is given by the following table fragment: egin{tabular}{|c|c|c|c|c|} hline( x ) & 0 & ( 1 / 6 ) & ( 1 / 4 ) & ( 1 / 3 ) \ hline( f(x) ) & 1 & ( sqrt{3} / 2 ) & ( sqrt{2} / 2 ) & ( 1 / 2 ) \ hline end{tabular} Necessary: 1. Guess the function, 2. Calculate the estimate of the remainder term for the points (1/12) and (7/24), 3. Construct an interpolation polynomial in Langrange form and Newton form, 4. At points (1/12) and (7/24) calculate the value of the interpolation polynomial and the real interpolation errors (after all, the function is known!) and compare them with the estimate of the residual term, 5. Graphically depict the distribution of the estimate of the residual term and the real error over the entire interval.

At its core, Function Interpolation in numerical analysis illuminates the path between known data points, revealing the behavior of functions beyond the confines of the given data. Be it scientific experiments, engineering measurements, or financial records, real-world data is inherently incomplete and scattered. Function Interpolation comes to the rescue, offering techniques to construct approximations that smoothly connect the dots. The beauty of Function Interpolation lies in its myriad of methods that cater to diverse scenarios. From simple techniques like linear interpolation to more intricate approaches like polynomial, spline, and radial basis function interpolation, the toolbox of methods adapts to the complexity of the data and the desired level of smoothness. Each method offers a unique way to stitch together a coherent narrative from disjoint pieces.

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