Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Textbook Question
Chapter 1.2, Problem 8E
Use Theorem 1.6 to determine whether Fixed-Point Iteration of
(a)
(b)
(c)
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4. Consider the function, f(x) = x³ = x² - 9x +9. Answer the following:
State the exact roots of f(x).
(a)
(b)
Construct three different fixed point functions g(x) such that f(x) = 0. (Make sure that one of the
g(x)'s that you constructed converges to at least a root).
(c)
(d)
Find the approximate root, x, of the above function using fixed point iterations up to 4 significant
figures within the error bound of 1 × 10-³ using xo = 0 and any fixed point function g(x) from part(b) that
converges to the root(s).
Find the convergence rate/ratio for g(x) constructed in previous part and also find which root it is
converging to?
Determine whether or not this converges or diverges
Let f (x) = x3 − 2x + 1, to solve the root finding problem f (x) = 0, derive each of the followingfixed-point functions and determine which, if any, will converge. Explain. Assume p0 = 1/2 for eachfunction
Chapter 1 Solutions
Numerical Analysis
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. 1CPCh. 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. 5CPCh. 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. 9CPCh. 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. 3ECh. 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. 21ECh. 1.2 - Prob. 22ECh. 1.2 - Assume that gx is continuously differentiable and...Ch. 1.2 - Assume that g is a continuously differentiable...Ch. 1.2 - Prob. 25ECh. 1.2 - Prove that a continuously differentiable function ...Ch. 1.2 - Prob. 27ECh. 1.2 - Prob. 28ECh. 1.2 - Prob. 29ECh. 1.2 - Prob. 30ECh. 1.2 - Prob. 31ECh. 1.2 - Find the set of all initial guesses for which the...Ch. 1.2 - Prob. 33ECh. 1.2 - Prob. 1CPCh. 1.2 - Prob. 2CPCh. 1.2 - Calculate the square roots of the following...Ch. 1.2 - Calculate the cube roots of the following numbers...Ch. 1.2 - Prob. 5CPCh. 1.2 - Prob. 6CPCh. 1.2 - Prob. 7CPCh. 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. 1CPCh. 1.3 - Carry' out Computer Problem 1 for fx=sinx3x3 .Ch. 1.3 - Prob. 3CPCh. 1.3 - Prob. 4CPCh. 1.3 - Prob. 5CPCh. 1.3 - Prob. 6CPCh. 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. 2CPCh. 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. 5CPCh. 1.4 - Prob. 6CPCh. 1.4 - Consider the function fx=esin3x+x62x4x31 on the...Ch. 1.4 - Prob. 8CPCh. 1.4 - Prob. 9CPCh. 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. 12CPCh. 1.4 - Prob. 13CPCh. 1.4 - Prob. 14CPCh. 1.4 - Prob. 15CPCh. 1.4 - Prob. 16CPCh. 1.4 - Consider the national population growth model...Ch. 1.5 - Prob. 1ECh. 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. 5ECh. 1.5 - If the Secant Method converges to, , and , then...Ch. 1.5 - Consider the following four methods for...Ch. 1.5 - Prob. 1CPCh. 1.5 - Use the Method of False Position to find the...Ch. 1.5 - Prob. 3CPCh. 1.5 - Prob. 4CPCh. 1.5 - Prob. 5CPCh. 1.5 - Prob. 6CPCh. 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. 5SACh. 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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- For each initial approximation, determine graphically whathappens if Newton’s method is used for the function whosegraph is shown. Does the sequence of approximations converge by using Newtons method? yes/noarrow_forwardNonearrow_forward7) The point p = 3 is a zero of the function f(x) = x³ – 7x? + 15x – 9, using Newton iteration to estimate the zero p = 3, Find the order of convergence R and the asymptotic error constant A .arrow_forward
- 2. Consider the function g(x) = 5 - 6/x a- Show that the function g(x) has a fixed point in the interval [2.5,3.5]. b- Find a root using Fixed Point Iteration. Use po = 2.6, to calculate the relative error.arrow_forwardNeed d, earrow_forwardf(2) = 2x² + x +; 3. a) Show that if you use the newton method on the function f(x) it not converges, instead is being cyclical if you start it with *o = 0 . b) Given function: f(x) = ax? +x+c with a ± 0 and c + 0 Which requirements need the coefficients a and c need to fulfill, in order for the newton method applied to the function f(x) to not converge, but instead being cyclical if you start it with Xo = 0.arrow_forward
- 2. Let 1 9(x) – 1+e * (a). of the fixed point theorem are satisfied. (b), (c). Show that g(x) has a unique fixed point on the interval [0, 1] by verifying the conditions If xo – 0, calculate r1, *2. Determine a bound for the error in the fixed point iteration after n steps.arrow_forward1,arrow_forward2. Write the formula for finding a root of f(x) = - x³ –- cos (x), where x is a real positive number, by Newton- Raphson's method. Then, compute the second iteration approximation. Discuss the order of convergence of Newton-Raphson's method.arrow_forward
- a) show whether or not fn(x) =sin(x/n) converges poinwise b) show whether or not fn(x) =sin(x/n) converces uniformly . c) show whether or not fn(x) =(1-x)xn converges uniformlyarrow_forwardQ9.7 Consider the function f(t) defined as f(t)= 2t^(2) + 8t +9 for the range - < t ≤, and extended periodically. Calculate the following values: X Find f(1.5m) 2(1.5pi)^2 + 8(1.5pi) +9_ Find f(-2π)= 2(-2pi)^2 + 8(-2pi) +9_ Find f(4)= 73 X Find f(10m)=_200pi^(2)+80pi+9_ X Xarrow_forwardWrite down the iterative map corresponding to the Newton's method solution of the equation In(x) 1/2 = 0. Use the fixed-point theorem to show that the iteration converges to its unique fixed-point for every initial guess o € [1, 2].arrow_forward
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