Explanation Assume that, N ( h ) is an approximation to M for every h > 0 and that, M = N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + ⋯ for some constants K 1 , K 2 , K 3 , … . Replace h by h 3 in the equation M = N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + ⋯ and obtain, M = N ( h 3 ) + K 1 ( h 3 ) + K 2 ( h 3 ) 2 + K 3 ( h 3 ) 3 + ⋯ = N ( h 3 ) + K 1 ( h 3 ) + K 2 ( h 2 9 ) + K 3 ( h 3 27 ) + ⋯ ⋯ ⋯ ( 1 ) Add − 1 times the equation M = N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + ⋯ to 3 times the equation (1) as shown below. 2 M = 3 ( N ( h 3 ) + K 1 ( h 3 ) + K 2 ( h 2 9 ) + K 3 ( h 3 27 ) + ⋯ ) − ( N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + + ⋯ ) = 3 N ( h 3 ) + 3 K 1 ( h 3 ) + 3 K 2 ( h 2 9 ) + 3 K 3 ( h 3 27 ) − N ( h ) − K 1 h − K 2 h 2 − K 3 h 3 + ⋯ = 3 N ( h 3 ) − N ( h ) + K 1 h − K 1 h + ( 1 3 − 1 ) K 2 h 2 + ( 1 9 − 1 ) K 3 h 3 + ⋯ = 3 N ( h 3 ) − N ( h ) + ( − 2 3 ) K 2 h 2 + ( − 8 9 ) K 3 h 3 + ⋯ Rewrite 3 N ( h 3 ) as 2 N ( h 3 ) + N ( h 3 ) and obtain, 2 M = 2 N ( h 3 ) + N ( h 3 ) − N ( h ) + ( − 2 3 ) K 2 h 2 + ( − 8 9 ) K 3 h 3 + ⋯ = 2 N ( h 3 ) + N ( h 3 ) − N ( h ) + ( − 2 3 ) K 2 h 2 + ( − 8 9 ) K 3 h 3 + ⋯ M = N ( h 3 ) + 1 2 ( N ( h 3 ) − N ( h ) ) − 1 3 K 2 h 2 − 4 9 K 3 h 3 + ⋯ … … ( 2 ) Let N 2 ( h ) = N ( h 3 ) + 1 2 ( N ( h 3 ) − N ( h ) )

10th Edition

ISBN: 9781305253667

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

Publisher: Cengage Learning

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Chapter 4.2, Problem 9ES

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An O(h3) approximation to M using the values of N(h),N(h3), and N(h9).

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Chapter 4.2, Problem 8ES

Chapter 4.2, Problem 10ES

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Numerical Analysis

Ch. 4.1 - Use the forward-difference formulas and...Ch. 4.1 - The data in Exercise 1 were taken from the...Ch. 4.1 - Use the most accurate three-point formula to...Ch. 4.1 - Use the most accurate three-point formula to...Ch. 4.1 - The data in Exercise 5 were taken from the...Ch. 4.1 - The data in Exercise 6 were taken from the...Ch. 4.1 - Prob. 9ES Ch. 4.1 - Use the formulas given in this section to...Ch. 4.1 - The data in Exercise 9 were taken from the...Ch. 4.1 - Prob. 12ES

Ch. 4.1 - Use the following data and the knowledge that the...Ch. 4.1 - Prob. 14ES Ch. 4.1 - Prob. 15ES Ch. 4.1 - Prob. 16ES Ch. 4.1 - Prob. 17ES Ch. 4.1 - Prob. 18ES Ch. 4.1 - Prob. 19ES Ch. 4.1 - Prob. 20ES Ch. 4.1 - Prob. 21ES Ch. 4.1 - In a circuit with impressed voltage (t) and...Ch. 4.1 - In Exercise 9 of Section 3.4, data were given...Ch. 4.1 - Derive an O(h4) five-point formula to approximate...Ch. 4.1 - Use the formula derived in Exercise 24 and the...Ch. 4.1 - a. Analyze the round-off errors, as in Example 4,...Ch. 4.1 - Derive a method for approximating f (x0) whose...Ch. 4.1 - Consider the function e(h)=h+h26M, where M is a...Ch. 4.1 - Prob. 1DQ Ch. 4.1 - Prob. 2DQ Ch. 4.2 - Apply the extrapolation process described in...Ch. 4.2 - Add another line to the extrapolation table in...Ch. 4.2 - The following data give approximations to the...Ch. 4.2 - Prob. 6ES Ch. 4.2 - Prob. 7ES Ch. 4.2 - The forward-difference formula can be expressed as...Ch. 4.2 - Prob. 9ES Ch. 4.2 - Prob. 10ES Ch. 4.2 - Prob. 11ES Ch. 4.2 - Prob. 12ES Ch. 4.2 - Prob. 13ES Ch. 4.3 - Approximate the following integrals using the...Ch. 4.3 - Approximate the following integrals using the...Ch. 4.3 - Find a bound for the error in Exercise 1 using the...Ch. 4.3 - Prob. 4ES Ch. 4.3 - Repeat Exercise 1 using Simpsons rule. 1....Ch. 4.3 - Prob. 6ES Ch. 4.3 - Prob. 7ES Ch. 4.3 - Prob. 8ES Ch. 4.3 - Prob. 9ES Ch. 4.3 - Prob. 10ES Ch. 4.3 - Prob. 11ES Ch. 4.3 - Prob. 12ES Ch. 4.3 - The Trapezoidal rule applied to 02f(x)dx gives the...Ch. 4.3 - Prob. 14ES Ch. 4.3 - Approximate the following integrals using formulas...Ch. 4.3 - Prob. 17ES Ch. 4.3 - Suppose that the data of Exercise 17 have...Ch. 4.3 - Prob. 19ES Ch. 4.3 - Prob. 20ES Ch. 4.3 - The quadrature formula...Ch. 4.3 - The quadrature formula...Ch. 4.3 - Find the constants c0, c1, and x1 so that the...Ch. 4.3 - Find the constants x0, x1, and c1 so that the...Ch. 4.3 - Prob. 25ES Ch. 4.3 - Prob. 26ES Ch. 4.3 - Prob. 27ES Ch. 4.3 - Derive Simpsons Three-Eighths rule (the closed...Ch. 4.3 - Prob. 1DQ Ch. 4.3 - Prob. 2DQ Ch. 4.4 - Use the Composite Trapezoidal rule with the...Ch. 4.4 - Prob. 2ES Ch. 4.4 - Use the Composite Simpsons rule to approximate the...Ch. 4.4 - Prob. 4ES Ch. 4.4 - Prob. 5ES Ch. 4.4 - Prob. 6ES Ch. 4.4 - Prob. 7ES Ch. 4.4 - Prob. 8ES Ch. 4.4 - Prob. 9ES Ch. 4.4 - Prob. 10ES Ch. 4.4 - Determine the values of n and h required to...Ch. 4.4 - Repeat Exercise 11 for the integral 0x2cosxdx. 11....Ch. 4.4 - Determine the values of n and h required to...Ch. 4.4 - Repeat Exercise 13 for the integral 12xlnxdx. 13....Ch. 4.4 - Prob. 15ES Ch. 4.4 - Prob. 17ES Ch. 4.4 - A car laps a race track in 84 seconds. The speed...Ch. 4.4 - Prob. 19ES Ch. 4.4 - Prob. 20ES Ch. 4.4 - Prob. 21ES Ch. 4.4 - Prob. 23ES Ch. 4.4 - Prob. 24ES Ch. 4.4 - Prob. 25ES Ch. 4.4 - Prob. 26ES Ch. 4.4 - Prob. 1DQ Ch. 4.4 - Prob. 2DQ Ch. 4.5 - Use Romberg integration to compute R3, 3 for the...Ch. 4.5 - Use Romberg integration to compute R3, 3 for the...Ch. 4.5 - Prob. 3ES Ch. 4.5 - Prob. 4ES Ch. 4.5 - Use the following data to approximate 15f(x)dx as...Ch. 4.5 - Prob. 9ES Ch. 4.5 - Prob. 10ES Ch. 4.5 - Prob. 11ES Ch. 4.5 - Romberg integration for approximating 01f(x)dx...Ch. 4.5 - Prob. 15ES Ch. 4.5 - Prob. 18ES Ch. 4.5 - Prob. 19ES Ch. 4.5 - Prob. 1DQ Ch. 4.5 - Prob. 4DQ Ch. 4.6 - Prob. 1ES Ch. 4.6 - Prob. 2ES Ch. 4.6 - Prob. 11ES Ch. 4.6 - Prob. 12ES Ch. 4.6 - Could Romberg integration replace Simpsons rule in...Ch. 4.7 - Approximate the following integrals using Gaussian...Ch. 4.7 - Approximate the following integrals using Gaussian...Ch. 4.7 - Repeat Exercise 1 with n = 3. 1. Approximate the...Ch. 4.7 - Repeat Exercise 2 with n = 3. 2. Approximate the...Ch. 4.7 - Repeat Exercise 1 with n = 4. 1. Approximate the...Ch. 4.7 - Repeat Exercise 2 with n = 4. 2. Approximate the...Ch. 4.7 - Repeat Exercise 1 with n = 5. 1. Approximate the...Ch. 4.7 - Repeat Exercise 2 with n = 5. 2. Approximate the...Ch. 4.7 - Describe the differences and similarities between...Ch. 4.7 - Prob. 2DQ Ch. 4.8 - Prob. 1DQ Ch. 4.8 - Prob. 2DQ Ch. 4.8 - Prob. 3DQ Ch. 4.8 - Prob. 4DQ Ch. 4.9 - Suppose a body of mass m is traveling vertically...Ch. 4.9 - The Laguerre polynomials {L0(x), L1(x) ...} form...Ch. 4.9 - Prob. 7ES Ch. 4.9 - Prob. 8ES Ch. 4.9 - Prob. 9ES Ch. 4.9 - Prob. 1DQ Ch. 4.9 - Prob. 2DQ

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An O ( h 3 ) approximation to M using the values of N ( h ) , N ( h 3 ) , and N ( h 9 ) .

Solution Summary: The author explains the O(h3) approximation to M using the values of M, N, and N.

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