Problem set 8

The textbook problems are from the 10th edition. See the scanned pages below if you do not have it.

• 19.1: 6 (scan)
• 19.2: 3, 7, 11, 12, 22 (scan 1, scan 2)

The following problem is adapted from the exam in TMA4135¹⁾ in the fall of 2008.

You are told that the cubic polynomial \(p(x) = x^3-x-1\) has a root in the interval \([1, 1.5]\). Which of the fixed-point iterations schemes

1. \(x_{n+1} = x_n^3-1\)
2. \(x_{n+1} = \frac{1}{x_n} + \frac{1}{x_n^2}\)
3. \(x_{n+1} = \sqrt[3]{x_n+1}\)

can be used to find that root? Justify your answer!

Use the iteration you chose with the starting value \(x_0=1\) to compute the root of the polynomial to three correct digits.

Make a Matlab program that solves equations of the form \(f(x) = 0\) on a given interval using the secant method (section 19.2) without using Matlab's built-in solvers. You can assume that your program will only be used for equations that has precisely one solution in the given interval.

Test your program by having it solve problem 19.2.21 in the textbook. (Problem 19.2.22, which you've done by hand, can also be a good test of your code).

Yurii suggests solving problem 6 from the August 2001 exam in Matematikk 4K as practice.