Assignment:
Write a program or programs to interpolate the data given below at the specified points using Neville iteration, Newton’s interpolatory divided-difference formula, and a natural cubic spline.
Use your programs to do the following.
Q1. For x = 0.25, x = 0.5, and x = 0.95, construct the full Neville interpolation table and use this table to calculate the interpolated values using 2, 4, 6, and 11 nodes. These nodes should be the nodes closest to the given x. You will be calculating the values of the first, third, fifth and tenth degree interpolating polynomials at each given value of x.
Q2. Construct a divided difference table for this data set and use the table to compute Newton’s interpolatory divided-difference formula. Use this form of the interpolating polynomial to calculate the interpolated values for each x value.
Q3. Construct a natural cubic spline for this data set and use it to calculate interpolated values for each x value. Print the coefficients of each cubic polynomial in the cubic spline.
Q4. Compare your interpolated values with the values of the function f(x) = ex2 / 1 + 25x2 .
Q5. Graph the original function, the full interpolating polynomial, and the natural cubic spline on
a single plot.
Output:
a. Turn in a well-documented program or programs.
b. Print the full divided-difference table.
c. For each x value, print the full Neville interpolation table.
d. Print the interpolated values, clearly labeled, and the exact values.
e. Analyze your output and include a summary of your findings.
Data:
-1.0 0.104549
-0.8 0.111558
-0.6 0.143333
-0.4 0.234702
-0.2 0.520405
0.0 1.000000
0.2 0.520405
0.4 0.234702
0.6 0.143333
0.8 0.111558
1.0 0.104549
Provide complete and step by step solution for the question and show calculations and use formulas.