1. We want to use a standard fixed point method to compute the real solution of f(x) = 3x³- 5 = 0. (Recall that two complex solutions also exist). A simple plot of f for x = [1.1, 1.3] shows that the real zero exists in this interval. 1.5 1.0 0.5 -0.5 -1.0 1.15 1.20 1.25 1.30 by solving 3 The standard fixed point method manipulates f(x): 0 into g(x): = =x so that the iterative scheme becomes xn+1 = g(xn). The iterative scheme will converge to the required solution if the root is in the interval defined by g'(x)| < 1. = (i) We begin by adding a to both sides of equation (1) to form x 3x3 3x3x 5. Will the root be found? - 5x such that g(x) = = == (ii) We now try adding -kx, k > 0 to both sides of equation (1) to form -kx 3x3 - 5 - kx such that g(x) (3x³ – kx – 5). What values of k will ensure the root is in the interval of convergence? = -1 k (iii) With a tolerance of 10-6 and letting k = 6.5, 10, 18, and 26, write a MATLAB livescript to com- pute the number of steps required for convergence. Plot the output for a sample of seed points, xo, from -5 to 5 in steps of 0.001. Compare and comment on the output.

1. We want to use a standard fixed point method to compute the real solution of f(x) = 3x³- 5 = 0. (Recall that two complex solutions also exist). A simple plot of f for x = [1.1, 1.3] shows that the real zero exists in this interval. 1.5 1.0 0.5 -0.5 -1.0 1.15 1.20 1.25 1.30 by solving 3 The standard fixed point method manipulates f(x): 0 into g(x): = =x so that the iterative scheme becomes xn+1 = g(xn). The iterative scheme will converge to the required solution if the root is in the interval defined by g'(x)| < 1. = (i) We begin by adding a to both sides of equation (1) to form x 3x3 3x3x 5. Will the root be found? - 5x such that g(x) = = == (ii) We now try adding -kx, k > 0 to both sides of equation (1) to form -kx 3x3 - 5 - kx such that g(x) (3x³ – kx – 5). What values of k will ensure the root is in the interval of convergence? = -1 k (iii) With a tolerance of 10-6 and letting k = 6.5, 10, 18, and 26, write a MATLAB livescript to com- pute the number of steps required for convergence. Plot the output for a sample of seed points, xo, from -5 to 5 in steps of 0.001. Compare and comment on the output.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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