# Problem set 8

## Problems from the textbook

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

**19.1:**6 (scan)

## Old exam problem

The following problem is adapted from the exam in TMA4135^{1)} 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

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

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.

## Programming problem

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).

## Optional extra problem

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

^{1)}