Explanation Formula used: Newton’s method formula: p n + 1 = p n − f ( p n ) f ′ ( p n ) Calculation: In Newton Raphson method if the initial root is not close to the root, then Newton Raphson method may not converge or may converge to wrong root. Consider an example where Newton Raphson does not converge: Find a root of the function f ( x ) . f ( x ) = x − 2 sin x Use the Newton’s method, calculate the value of x n . x n = x n − 1 − f ( x n − 1 ) f ′ ( x n − 1 ) Now, differentiate the function f ( x ) = x − 2 sin x with respect to x : f ' ( x ) = 1 − 2 cos x Now, substitute the values f ( x ) = x − 2 sin x and f ' ( x ) = 1 − 2 cos x in x n = x n − 1 − f ( x n − 1 ) f ′ ( x n − 1 ) . x n = x n − 1 − x n − 1 − 2 sin x n − 1 1 − 2 cos x n − 1 = x n − 1 − 2 x n − 1 cos x n − 1 − x n − 1 + 2 sin x n − 1 1 − 2 cos x n − 1 = 2 sin x n − 1 − 2 x n − 1 cos x n − 1 1 − 2 cos x n − 1 Suppose the initial approximation is x 0 = 1.1 . Now, substitute n = 1 and x 0 = 1.1 in x n = 2 sin x n − 1 − 2 x n − 1 cos x n − 1 1 − 2 cos x n − 1

10th Edition

ISBN: 9781305253667

Author: Richard L. Burden, J. Douglas Faires, Annette M. Burden

Publisher: Cengage Learning

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

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The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 2 Solutions

Numerical Analysis

Ch. 2.1 - Use the Bisection method to find p3 for f(x)=xcosx...Ch. 2.1 - Let f(x) = 3(x +1)(x 12)(x 1) = 0. Use the...Ch. 2.1 - Use the Bisection method to find solutions...Ch. 2.1 - Use the Bisection method to find solutions...Ch. 2.1 - Use the Bisection method to find solutions...Ch. 2.1 - Prob. 7ES Ch. 2.1 - Prob. 8ES Ch. 2.1 - Prob. 9ES Ch. 2.1 - Prob. 10ES Ch. 2.1 - Prob. 11ES

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. 18ES Ch. 2.1 - Prob. 19ES Ch. 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. 1DQ Ch. 2.1 - Prob. 2DQ Ch. 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. 6ES Ch. 2.2 - Prob. 7ES Ch. 2.2 - Prob. 8ES Ch. 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. 20ES Ch. 2.2 - Prob. 21ES Ch. 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. 24ES Ch. 2.2 - Prob. 25ES Ch. 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. 11ES Ch. 2.3 - Prob. 12ES Ch. 2.3 - The fourth-degree polynomial...Ch. 2.3 - Prob. 14ES Ch. 2.3 - Prob. 15ES Ch. 2.3 - Prob. 16ES Ch. 2.3 - Prob. 22ES Ch. 2.3 - Prob. 23ES Ch. 2.3 - Prob. 24ES Ch. 2.3 - Prob. 25ES Ch. 2.3 - Prob. 27ES Ch. 2.3 - A drug administered to a patient produces a...Ch. 2.3 - Prob. 30ES Ch. 2.3 - Prob. 32ES Ch. 2.3 - Prob. 1DQ Ch. 2.3 - Prob. 2DQ Ch. 2.3 - Prob. 3DQ Ch. 2.3 - Prob. 4DQ Ch. 2.4 - Prob. 6ES Ch. 2.4 - a. Show that for any positive integer k, the...Ch. 2.4 - Prob. 8ES Ch. 2.4 - a. Construct a sequence that converges to 0 of...Ch. 2.4 - Prob. 10ES Ch. 2.4 - Prob. 11ES Ch. 2.4 - Prob. 12ES Ch. 2.4 - Prob. 13ES Ch. 2.4 - Prob. 14ES Ch. 2.4 - Prob. 1DQ Ch. 2.4 - Prob. 2DQ Ch. 2.4 - Prob. 4DQ Ch. 2.5 - Let g(x) = cos(x 1) and p0(0) = 2. Use...Ch. 2.5 - Prob. 4ES Ch. 2.5 - Prob. 5ES Ch. 2.5 - Prob. 6ES Ch. 2.5 - Use Steffensens method to find, to an accuracy of...Ch. 2.5 - Prob. 8ES Ch. 2.5 - Prob. 9ES Ch. 2.5 - Use Steffensens method with p0 = 3 to compute an...Ch. 2.5 - Use Steffensens method to approximate the...Ch. 2.5 - Prob. 12ES Ch. 2.5 - Prob. 13ES Ch. 2.5 - Prob. 14ES

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Solution Summary: The author explains Newton's method formula: if the initial root is not close to the root, then Newton Raphson method may not converge.

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Chapter 2 Solutions