To determine To find which of the given fixed-point iterations converge to the cube root of 4 and rank the ones that converge from fastest to slowest. Explanation Given: The following equation is given: g ( x ) = 2 x Concept used: If g ( x ) is given then write in the form of x = g ( x ) . Now, label left side as x n + 1 and right side with x n . Then take x 1 and put it into the equation and repeat it with every new x until it converges. Calculation: Option (A) The following equation is given: g ( x ) = 2 x To find out whether cube root 4 is a fixed-point or not for the given equation. Applying fixed-point iteration, set x = g ( x ) So, x = 2 x Now, x x = 2 x 3 2 = 2 x 3 = 4 x = 4 3 So, we can see that cube root 4 is a fixed-point for the equation g ( x ) = 2 x . Assume initial guess x 0 = 1.0 Now, x n + 1 = 2 x n x 1 = 2 x 0 ( put x 0 = 1.0 ) x 1 = 2 Now, continue the process and we get: n x n 0 1.0 1 2 2 1.41421356 3 1.68179283 4 1.54221083 5 1.61049033 6 1.57598085 7 1.59314215 8 1.58453827 9 1.58883438 10 1.58668487 11 1.58775926 12 1.58722198 13 1.58749060 14 1.58735628 15 1.58742344 16 1.58740665 17 1.58739825 18 1.58740105 Therefore, from the table we can see that for g ( x ) = 2 x the iterative process converges to 1.58740105 that means cube root of 4. Thus, for the equation g ( x ) = 2 x fixed-point iteration converges to cube root of 4. Option (B) The following equation is given: g ( x ) = 3 x 4 + 1 x 2 To find out whether cube root 4 is a fixed-point or not for the given equation. Apply fixed-point iteration by setting x = g ( x ) . So, x = 3 x 4 + 1 x 2 Now, x = 3 x 4 + 1 x 2 4 x 3 = 3 x 3 + 4 x 3 = 4 x = 4 3 So, we can see that cube root 4 is a fixed-point for the equation g ( x ) = 3 x 4 + 1 x 2 . Assume initial guess x 0 = 1.0 Now, x n + 1 = 3 x n 4 + 1 x n 2 x 1 = 3 x 0 4 + 1 x 0 2 ( put x 0 = 1

Numerical Analysis, Books A La Carte Edition (3rd Edition)

3rd Edition

ISBN: 9780134697338

Author: Timothy Sauer

Publisher: PEARSON

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Chapter 1.2, Problem 16E

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To find which of the given fixed-point iterations converge to the cube root of 4 and rank the ones that converge from fastest to slowest.

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Chapter 1.2, Problem 15E

Chapter 1.2, Problem 17E

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Chapter 1 Solutions

Numerical Analysis, Books A La Carte Edition (3rd Edition)

Ch. 1.1 - Use the Intermediate Value Theorem to find an...Ch. 1.1 - Use the Intermediate Value Theorem to find an...Ch. 1.1 - Consider the equations in Exercise 1. Apply two...Ch. 1.1 - Consider the equations in Exercise 2. Apply two...Ch. 1.1 - Consider the equation x4=x3+10 . a. Find an...Ch. 1.1 - Suppose that the Bisection Method with starting...Ch. 1.1 - Prob. 1CP Ch. 1.1 - Use the Bisection Method to find the root to eight...Ch. 1.1 - Use the Bisection Method to locate all solutions...Ch. 1.1 - Prob. 4CP

Ch. 1.1 - Prob. 5CP Ch. 1.1 - Use the Bisection Method to calculate the solution...Ch. 1.1 - Use the Bisection Method to find the two real...Ch. 1.1 - The Hilbert matrix is the nn matrix whose ijth...Ch. 1.1 - Prob. 9CP Ch. 1.1 - A planet orbiting the sun traverses an ellipse....Ch. 1.2 - Find all fixed points of the following gx . a. 3x...Ch. 1.2 - Find all fixed points of the following gx . x+63x2...Ch. 1.2 - Prob. 3E Ch. 1.2 - Show that -1, 0, and 1 are fixed points of the...Ch. 1.2 - For which of the following gx is r=3 a fixed...Ch. 1.2 - For which of the following is a fixed...Ch. 1.2 - Use Theorem 1.6 to determine whether Fixed-Point...Ch. 1.2 - Use Theorem 1.6 to determine whether Fixed-Point...Ch. 1.2 - Find each fixed point and decide whether...Ch. 1.2 - Find each fixed point and decide whether...Ch. 1.2 - Express each equation as a fixed-point problem...Ch. 1.2 - Consider the Fixed-Point Iteration xgx=x20.24 ....Ch. 1.2 - (a) Find all fixed points of. (b) To which of the...Ch. 1.2 - Which of the following three Fixed-Point...Ch. 1.2 - Which of the following three Fixed-Point...Ch. 1.2 - Which of the following three Fixed-Point...Ch. 1.2 - Check that and -1 are roots of. Isolate the term...Ch. 1.2 - Prove that the method of Example 1.6 will...Ch. 1.2 - Explore the idea of Example 1.6 for cube roots. Lf...Ch. 1.2 - Improve the cube root algorithm of Exercise 19 by...Ch. 1.2 - Prob. 21E Ch. 1.2 - Prob. 22E Ch. 1.2 - Assume that gx is continuously differentiable and...Ch. 1.2 - Assume that g is a continuously differentiable...Ch. 1.2 - Prob. 25E Ch. 1.2 - Prove that a continuously differentiable function ...Ch. 1.2 - Prob. 27E Ch. 1.2 - Prob. 28E Ch. 1.2 - Prob. 29E Ch. 1.2 - Prob. 30E Ch. 1.2 - Prob. 31E Ch. 1.2 - Find the set of all initial guesses for which the...Ch. 1.2 - Prob. 33E Ch. 1.2 - Prob. 1CP Ch. 1.2 - Prob. 2CP Ch. 1.2 - Calculate the square roots of the following...Ch. 1.2 - Calculate the cube roots of the following numbers...Ch. 1.2 - Prob. 5CP Ch. 1.2 - Prob. 6CP Ch. 1.2 - Prob. 7CP Ch. 1.3 - Find the forward and backward error for the...Ch. 1.3 - Find the forward and backward error for the...Ch. 1.3 - (a) Find the multiplicity of the root r=0 of...Ch. 1.3 - (a) Find the multiplicity of the root of. (b)...Ch. 1.3 - Find the relation between forward and backward...Ch. 1.3 - Let be a positive integer. The equation defining...Ch. 1.3 - Let be the Wilkinson polynomial. (a) Prove that ...Ch. 1.3 - Let fx=xnaxn1 , and set gx=xn . (a) Use the...Ch. 1.3 - Prob. 1CP Ch. 1.3 - Carry' out Computer Problem 1 for fx=sinx3x3 .Ch. 1.3 - Prob. 3CP Ch. 1.3 - Prob. 4CP Ch. 1.3 - Prob. 5CP Ch. 1.3 - Prob. 6CP Ch. 1.4 - Apply two steps of Newton’s Method with initial...Ch. 1.4 - Apply two steps of Newton’s Method with initial...Ch. 1.4 - Use Theorem 1.11 or 1.12 to estimate the error...Ch. 1.4 - Estimate as in Exercise 3. (a) ; , (b) ; , Ch. 1.4 - Consider the equation 8x412x3+6x2x=0 . For each of...Ch. 1.4 - Sketch a function f and initial guess for which...Ch. 1.4 - Let fx=x47x3+18x220x+8 . Does Newton’s Method...Ch. 1.4 - Prove that Newton’s Method applied to fx=ax+b...Ch. 1.4 - Show that applying Newton’s Method to fx=x2A...Ch. 1.4 - Find the Fixed-Point Iteration produced by...Ch. 1.4 - Use Newton’s Method to produce a quadratically...Ch. 1.4 - Suppose Newton’s Method is applied to the...Ch. 1.4 - (a) The function has a root at . If the error ...Ch. 1.4 - Let denote the Newton’s Method iteration for the...Ch. 1.4 - Each equation has one root. Use Newton’s Method to...Ch. 1.4 - Prob. 2CP Ch. 1.4 - Apply Newton’s Method to find the only root to as...Ch. 1.4 - Carry out the steps of Computer Problem 3 for (a)...Ch. 1.4 - Prob. 5CP Ch. 1.4 - Prob. 6CP Ch. 1.4 - Consider the function fx=esin3x+x62x4x31 on the...Ch. 1.4 - Prob. 8CP Ch. 1.4 - Prob. 9CP Ch. 1.4 - Set fx=54x6+45x5102x469x3+35x2+16x4 . Plot the...Ch. 1.4 - The ideal gas law for a gas at low temperature and...Ch. 1.4 - Prob. 12CP Ch. 1.4 - Prob. 13CP Ch. 1.4 - Prob. 14CP Ch. 1.4 - Prob. 15CP Ch. 1.4 - Prob. 16CP Ch. 1.4 - Consider the national population growth model...Ch. 1.5 - Prob. 1E Ch. 1.5 - Apply two steps of the Method of False Position...Ch. 1.5 - Apply two steps of Inverse Quadratic Interpolation...Ch. 1.5 - A commercial fisher wants to set the net at a...Ch. 1.5 - Prob. 5E Ch. 1.5 - If the Secant Method converges to, , and , then...Ch. 1.5 - Consider the following four methods for...Ch. 1.5 - Prob. 1CP Ch. 1.5 - Use the Method of False Position to find the...Ch. 1.5 - Prob. 3CP Ch. 1.5 - Prob. 4CP Ch. 1.5 - Prob. 5CP Ch. 1.5 - Prob. 6CP Ch. 1.5 - Write a MATLAB function file for f . The...Ch. 1.5 - Plot f on , . You may use the @ symbol as...Ch. 1.5 - Reproduce Figure 1.15. The MATLAB commands and...Ch. 1.5 - Solve the forward kinematics problem for the...Ch. 1.5 - Prob. 5SA Ch. 1.5 - Find a strut length p2 , with the rest of the...Ch. 1.5 - Calculate the intervals in p2 , with the rest of...Ch. 1.5 - Prob. 8SA

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