Numerical Mathematics Problems (Matlab)
There are 5 problems. The answer should include Matlab code.
Question 1: Draw a free form curve on graph paper, making certain that the curve a function. Then read values of your function at a reasonable number 10-5, and compute the cubic spine that takes those value freely drawn curve to the graph of the cubic spline.
Question 2: Find the polynomial of degree 10 that interpolates the function arctan x at 11 equally spaced points in the interval (1, 6). Print the coefficients in the Newton form of the polynomial. Compute and print the difference between the polynomial and the function at 33 equally spaced points in the interval (0, 8). What conclusion can be drawn?
Question 3: Write routines analogous to Coef and Eval using the Lagrange form of the interplation ploynomial. Test on the exam polynomial. Test on the example given in this section at 20 points with h/2. Does the Lagrange form have any advantage over the Newton form?
Question 4: Use mathematical sofware such as Matlab, Maple, or Mathematica to find an interpolation polynomial for the points (0, 0), (1, 1), (2, 2.001), (3, 3), (4, 4), (5, 5). Evaluate the polynomial at the point x = 14 or x = 20 to show that slight roundoff errors in the data can lead to suspicious result in extrapolation.
Question 5: Find a rational function of the form
g (x) = a + bx/1 +cx
that interpolates the function f (x) = arctan (x) at the points x0 = 1, x1 = 2, and x2 = 3. On the same axes, plot the graphs of f and g, using dashed and dotted lines, respectively.