MA3525 ELEMENTARY NUMERICAL METHODS: HW1
The first 3 are pencil-paper problems and the last 2 are matlab problems.
Due midnight of Sunday October 6. 0.5 points will be deducted for every 24 hours of late submission.
(1) Write down the Newton’s method formula for x − 3 sin x = 0, and compute an approximation with 2 iterations with starting value x0 = 2. (two decimal places)
(2) Let s(x) be a cubic spline
(a) Find b and c in the cubic spline
(b) This spline satisfies “clamped” endpoint conditions. What are the values of the two clamps?
(3) Consider 3 points (x, y): (0, 1), (2, 2), (3, 4). Use these 3 points to construct an interpolation polynomial of degree 2, and use it to compute an approximation of y at x = 1 .
(4) Write MATLAB codes of the Newton’s method and Secant method to find the root of
x3 − 9x2 + 23x − 16 = 0
in [0, 2] with error less than 10−10. The derivative needed in the newton’s method can be calculated by hand and written as a function. The true solution is 1.1391941468830
In the report, please make a table to list the value, the error. Also discuss and show the convergence rate α numerically, with |en+1| = C|en|α
(5) MATLAB problem. On [−6, 6], for the function f(x) = x2 + 1/1 , first use 21 equally-distributed nodes to find an interpolation polynomial p(x) of degree 20. Then use 21 Chebyshev interpolation nodes to construct a q(x) of of degree 20 (you can use the same program with different nodes).
Draw f(x), p(x), and q(x), and find the maximum difference of f and p, f and q on [−6, 6]. (no need to write out p(x) and q(x) explicitly.)