Numerical Analysis
Numerical Analysis
10th Edition
ISBN: 9781305253667
Author: Richard L. Burden, J. Douglas Faires, Annette M. Burden
Publisher: Cengage Learning
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Chapter 2.2, Problem 7ES
To determine

The given equation x43x23=0,    p0=1 with fixed point iteration method when the accuracy is less than 102.

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Consider the function f(x) = 2x³-4x2-x+1. (a) Without doing a sketch, show that the cubic equation has at least one solution on the interval [0,1]. Use a theorem discussed in lectures, or see Section 1.8 of Calculus (7th ed) by Stewart. Ensure that the conditions of the theorem are satisfied (include this in your solution) (b) Now, by sketching the cubic (by hand or by computer), you should see that there is, in fact, exactly one zero in the interval [0,1]. Use Newton's method to find this zero accurate to 3 decimal places. You should include a sketch of the cubic, Newton's iteration formula, and the list of iterates. [Use a computer if possible, e.g., a spreadsheet or MatLab.]

Chapter 2 Solutions

Numerical Analysis

Ch. 2.1 - Let f(x) = (x + 2)(x + 1)x(x 1)3(x 2). To which...Ch. 2.1 - Find an approximation to 253 correct to within 104...Ch. 2.1 - Find an approximation to 3 correct to within 104...Ch. 2.1 - A trough of length L has a cross section in the...Ch. 2.1 - Use Theorem 2.1 to find a bound for the number of...Ch. 2.1 - Prob. 18ESCh. 2.1 - Prob. 19ESCh. 2.1 - Let f(x) = (x 1)10, p = 1, and pn = 1 + 1/n. Show...Ch. 2.1 - The function defined by f(x) = sin x has zeros at...Ch. 2.1 - Prob. 1DQCh. 2.1 - Prob. 2DQCh. 2.1 - Is the Bisection method sensitive to the starting...Ch. 2.2 - Use algebraic manipulation to show that each of...Ch. 2.2 - a. Perform four iterations, if possible, on each...Ch. 2.2 - Let f(x) = x3 2x + 1. To solve f(x) = 0, the...Ch. 2.2 - Let f(x) = x4 + 3x2 2. To solve f(x) = 0, the...Ch. 2.2 - The following four methods are proposed to compute...Ch. 2.2 - Prob. 6ESCh. 2.2 - Prob. 7ESCh. 2.2 - Prob. 8ESCh. 2.2 - Use Theorem 2.3 to show that g(x) = + 0.5...Ch. 2.2 - Use Theorem 2.3 to show that g(x) = 2x has a...Ch. 2.2 - Use a fixed-point iteration method to find an...Ch. 2.2 - Use a fixed-point iteration method to determine a...Ch. 2.2 - Use a fixed-point iteration method to determine a...Ch. 2.2 - Prob. 20ESCh. 2.2 - Prob. 21ESCh. 2.2 - a. Show that Theorem 2.3 is true if the inequality...Ch. 2.2 - a. Use Theorem 2.4 to show that the sequence...Ch. 2.2 - Prob. 24ESCh. 2.2 - Prob. 25ESCh. 2.2 - Suppose that g is continuously differentiable on...Ch. 2.3 - Let f(x) = x2 6 and p0 = 1. Use Newtons method to...Ch. 2.3 - Let f(x) = x3 cos x and p0 = 1. Use Newtons...Ch. 2.3 - Let f(x) = x2 6. With p0 = 3 and p1 = 2, find p3....Ch. 2.3 - Let f(x) = x3 cos x. With p0 = 1 and p1 = 0, find...Ch. 2.3 - Prob. 11ESCh. 2.3 - Prob. 12ESCh. 2.3 - The fourth-degree polynomial...Ch. 2.3 - Prob. 14ESCh. 2.3 - Prob. 15ESCh. 2.3 - Prob. 16ESCh. 2.3 - Prob. 22ESCh. 2.3 - Prob. 23ESCh. 2.3 - Prob. 24ESCh. 2.3 - Prob. 25ESCh. 2.3 - Prob. 27ESCh. 2.3 - A drug administered to a patient produces a...Ch. 2.3 - Prob. 30ESCh. 2.3 - Prob. 32ESCh. 2.3 - Prob. 1DQCh. 2.3 - Prob. 2DQCh. 2.3 - Prob. 3DQCh. 2.3 - Prob. 4DQCh. 2.4 - Prob. 6ESCh. 2.4 - a. Show that for any positive integer k, the...Ch. 2.4 - Prob. 8ESCh. 2.4 - a. Construct a sequence that converges to 0 of...Ch. 2.4 - Prob. 10ESCh. 2.4 - Prob. 11ESCh. 2.4 - Prob. 12ESCh. 2.4 - Prob. 13ESCh. 2.4 - Prob. 14ESCh. 2.4 - Prob. 1DQCh. 2.4 - Prob. 2DQCh. 2.4 - Prob. 4DQCh. 2.5 - Let g(x) = cos(x 1) and p0(0) = 2. Use...Ch. 2.5 - Prob. 4ESCh. 2.5 - Prob. 5ESCh. 2.5 - Prob. 6ESCh. 2.5 - Use Steffensens method to find, to an accuracy of...Ch. 2.5 - Prob. 8ESCh. 2.5 - Prob. 9ESCh. 2.5 - Use Steffensens method with p0 = 3 to compute an...Ch. 2.5 - Use Steffensens method to approximate the...Ch. 2.5 - Prob. 12ESCh. 2.5 - Prob. 13ESCh. 2.5 - Prob. 14ES
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