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Math 551 Section 01

Summer 2023

Chapter 10 Homework due on Wednesday, July 5th at 11:59 PM

Notes: Try to answer all the questions by demonstrating all the steps of your cal-culations. Please submit your homework on Gradescope. This homework assignment covers Sections 10.1-10.5. Answer all the questions by demonstrating all the steps of your calculations. The MATLAB m-fifile for Problem 1 is posted on Blackboard.

1.  (15 points) Use the known values of the function f (x) = sin (x) (with y  = f (x)) at x = 0, π/6, π/4, π/3 and π/2 in order to derive an interpolating polynomial p(x) using a monomial basis. What is the degree of your polynomial? What is the inter- polation error magnitude Ip(1.2) - sin (1.2)I?  Make a plot of your data points and the underlying interpolating polynomial for x e [0, π/2] on the same graph.

2.  (50 points) Let f (x) = 1/x and data points x0  = 2, x1  = 3 and x2  = 4.  Note that you can use the abscissae to find the corresponding ordinates.

(a)  (20 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2 (x) of f (x) at the given points.  State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2  that you obtained [this is true due to the uniqueness of polynomial interpolation!]. Also, verify that p2 (xi ) = f (xi ) for i = 0, 1, 2.

(b)  (15 points) Use the Polynomial Interpolation Error theorem to find an upper

bound for the error      IIf - p2 II<  = max If (x) - p2 (x)I.

2≤x≤4

(c)  (15 points) Find the exact value of IIf - p2 II<  to at least 5 decimal places of accuracy. Of course, the answer should be less than or equal to the upper bound you found in part (b).

3.  (10 points)  Consider the data set {xi } containing (n + 1) distinct points and the corresponding Lagrange basis functions {Li (x)} . Then, prove that

n

Lj (x) = 1.

j=0

Hint :  Consider interpolating the function f (x) = 1 at the given points and use the Fundamental Theorem of Algebra.

4.  (25 points) For some function f , the divided difference table is given:

Fill in the unknown entries in the table.

Considering the table derive the interpolating polynomial p(x) using the Newton Divided Differences. What is the degree of your polynomial?