Explanation Calculation: Error formula for Newton Raphson method: In this method the n th approximation to the root is: p n = p n − 1 − f ( p n − 1 ) f ′ ( p n − 1 ) Now, suppose p n = φ ( p n − 1 ) . Substitute the value p n = φ ( p n − 1 ) in p n = p n − 1 − f ( p n − 1 ) f ′ ( p n − 1 ) . φ ( p n − 1 ) = p n − 1 − f ( p n − 1 ) f ′ ( p n − 1 ) Let e n − 1 be the error in the value of the root in the n th iteration and p is the actual root of the equation. Now, write e n − 1 as: e n − 1 = p n − 1 − p p n − 1 = e n − 1 + p Substitute the value p n − 1 = e n − 1 + p in p n = φ ( p n − 1 ) . p n = φ ( e n − 1 + p ) Now, expand the above equation by using Taylor’s theorem. p n = p + e n − 1 φ ′ ( p ) 1 ! + ( e n − 1 ) 2 φ ″ ( p ) 2 ! + ⋅ ⋅ ⋅ p n − p = e n − 1 φ ′ ( p ) 1 ! + ( e n − 1 ) 2 φ ″ ( p ) 2 ! + ⋅ ⋅ ⋅ Now from the equation e n − 1 = p n − 1 − p , e n that is the error in n th can be written as e n = p n − p Substitute the value e n = p n − p in p n − p = e n − 1 φ ′ ( p ) 1 ! + ( e n − 1 ) 2 φ ″ ( p ) 2 ! + ⋅ ⋅ ⋅ . e n = e n − 1 φ ′ ( p ) 1 ! + ( e n − 1 ) 2 φ ″ ( p ) 2 ! + ⋅ ⋅ ⋅ Since, p n = p n − 1 − f ( p n − 1 ) f ′ ( p n − 1 ) . So, the value of φ ( p n − 1 ) at p that is: φ ( p ) = p − f ( p ) f ′ ( p ) Differentiate the above equation with respect to f ′ ( p ) : φ ′ ( p ) = f ( p ) f ′ ′ ( p ) [ f ′ ( p ) ] 2 All p value of function is 0 because if the exact root of equation

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Chapter 2 Solutions

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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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To derive: The error formula for Newton’s method.

Solution Summary: The author explains the error formula for Newton's method. The nth approximation to the root is p_n=phi

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Chapter 2 Solutions