numerical analysis

Numerical analysis I, Spring 2024 01:640:373 1 February 9, 2024 Homework assignment 4 Problem 10 (3.1) from the textbook[10 points] √ Let f (x) = x − x2 and P2 (x) be the interpolation polynomial on x0 = 0, x1 and x2 = 1. Find the largest value of x1 in (0, 1) for which f (0.5) − P2 (0.5) = −0.25. 2 Exam question[10 points] Determine if the polynomial Q(x) = 4 + 4.5(x − 2) + 5.2(x − 2)(x − 3) + 0.6(x − 2)(x − 3)(x − 4) is the Lagrange interpolating polynomial P3 in Newton’s form for the function f (x) = 2x and the nodes x0 = 2, x1 = 3, x2 = 4, x3 = 5. Justify your answer. 3 Problem 6 (3.2) from the textbook[10 points] Neville’s method is used to approximate f (0.5), giving the following table. Determine P2 . 4 Problem 14 (3.3) from the textbook[10 points] For a function f , the Newton divided-difference formula gives the interpolating polynomial 16 x(x − 0.25)(x − 0.5) 3 on the nodes x0 = 0,×1 = 0.25, x2 = 0.5 and x3 = 0.75. Find f (0.75). P3 (x) = 1 + 4x + 4x(x − 0.25) + 5 Sample exam question[10 points] Let 1 . x+1 Find Newton’s form of the Lagrange interpolating polynomial for this function for nodes x0 = 1, x1 = 2, x2 = 4, i.e. consider the Lagrange Polynomial P2 , find a0 , a1 , a2 such that f (x) = P2 (x) = a0 + a1 (x − x0 ) + a2 (x − x0 )(x − x1 ). 1 6 Programming question[15 points] Let us consider a function f (x) = 1 . 25×2 + 1 For n ≥ 1, let Pn be the Lagrange interpolating polynomial for f the nodes of the form −1 + 2k n for k = 0, 1, . . . , n. Let Qn be the Lagrange interpolating polynomial for f and nodes of the form cos 2k−1 2n π for k = 0, 1, . . . , n. Use your favourite tool to generate the plots for the following 7 functions: f , P10 , Q10 , P20 , Q20 , P80 , Q80 . According to your results, answer the question: Which choice of the nodes seems to be better for f ? In your submission, please include all of the plots and the code which you used to generate it (e.g. screenshot). In the picture below you can see plots of f , P7 , and Q7 generated in Matlab. 7 Problem 14 (3.1) from the textbook[10 points] Construct the Lagrange interpolating polynomials for the following function, and find a bound for the absolute error on the interval [x0 , xn ]. f (x) = ex + e−x , x0 = −0.3, x1 = 0, x2 = 0.3, n = 2 Your solution will be accepted if your error estimate will be better than 0.03. 2