numerical analysis chapter

Numerical analysis I, Spring 2024 01:640:373 1 January 25, 2024 Homework assignment 2 Problem 12 (2.1) from the textbook[10 points] Let f (x) = (x + 2)(x + 1)2 x(x − 1)3 (x − 2). To which zero of f does the bisection method converge when applied on the following intervals? (a) [−1.5, 2.5], (b) [−0.5, 2.4]. In your solution, please justify your answers. 2 Problem 14 (2.1) from the textbook[10 points] √ Find an approximation to 3 correct to within 10−4 using the Bisection Algorithm. You can use the code from the lecture, your own program, or just a calculator. 3 Problem 18 (2.1) from the textbook[10 points] Use Theorem 2.1 to find a bound for the number of iterations needed to achieve an approximation with accuracy 10−3 to the solution of x3 + x − 4 = 0 lying in the interval [1, 4]. 4 Problem 10 (2.2) from the textbook[10 points] Use Theorem 2.3 to show that g(x) = 2−x has a unique fixed point on 31 , 1 . Use fixed-point iteration to find an approximation to the fixed point accurate to within 10−4 5 Problem (exam question)[10 points] Let g(x) = 1 (sin(x)6 + x4 ) + 1. 1000 (a) Define the fixed point of g. (b) Prove that g has an unique fixed point in the interval [1, 3]. 6 Programming exercise[10 points] Write down a program for solving cubic equations, more precisely: Input: Four numbers a, b, c, d. Output: x0 ∈ [−100, 100] such that f (x0 ) = ax30 + bx20 + cx0 + d = 0. You may assume that f (−100) and f (100) have opposite sign. Then, use it to find solution of 34×3 + 56×2 + 78x + 9 = 0 accurate to within 10−3 . 1 7 Problem 24 (2.2) from the textbook[10 points] Show that if A > 1 is any positive number, then the sequence defined by x0 = A and 1 A xn+1 = xn + 2 xn √ for n = 0, 1, 2, . . . , converges to A. 2