Evenly spaced fitting points are often not well-suited for polynomial interpolation. A famous example is given...

 

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Evenly spaced fitting points are often not well-suited for polynomial interpolation. A famous example is given by Runge to interpolate f(x) = 1 1 + 25x on the interval [-1,1]. (A) Find a sixth-order approximation to f(x) using seven evenly spaced fitting points. Be sure to plot f(x) and po(x) on the same axes, and also plot the error f(x) - po(x). (B) Find a tenth-order approximation to (2) using eleven evenly spaced fitting points. Be sure to plot f(x) and pio(x) on the same axes, and also plot the error f(x) - Pio(x). (C) Find a 100th-order approximation to f(x) using one-hundred-and-one evenly space fitting points. Be sure to plot f(x) and pioo (x) on the same axes, and also plot the error f(x) - Pio(x). (D) Compare the results for these three approximations. Evenly spaced fitting points are often not well-suited for polynomial interpolation. A famous example is given by Runge to interpolate f(x) = 1 1 + 25x on the interval [-1,1]. (A) Find a sixth-order approximation to f(x) using seven evenly spaced fitting points. Be sure to plot f(x) and po(x) on the same axes, and also plot the error f(x) - po(x). (B) Find a tenth-order approximation to (2) using eleven evenly spaced fitting points. Be sure to plot f(x) and pio(x) on the same axes, and also plot the error f(x) - Pio(x). (C) Find a 100th-order approximation to f(x) using one-hundred-and-one evenly space fitting points. Be sure to plot f(x) and pioo (x) on the same axes, and also plot the error f(x) - Pio(x). (D) Compare the results for these three approximations.