14.8.2 Function Interpolation

The well -known function is set by the following fragment of the table: Egin {Tabular} {| 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 residual member for points (1/12) and (7/24), 3. Build interpolation in the form of a langgrand and in the form of a nyuton, 4. At points (1/12) and (7/24), the value of interpolation polynoma and the real interpolation errors (after all, the function is known!) And compare them with the assessment of the residual member, 5. Graphically depict the distribution of the assessment of the residual member and 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 Beond The Confines of the Given Data. Be it Scientific Experiments, Engineering Measurements, Or Financial Records, Real-World Data IS Inherently Incomplete and Scatted. 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 Interporation, The Toolbox of Methads Adapts to the Complexity of the Data and the Desired Level of Smoothness. Each Method Offers a Unique Way to Stitch Together a Coherent from Disjoint Pieces.