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.

Old exam problem

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.

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.

The course TMA4135 Matematikk 4D is almost identical to TMA4122/TMA4125.
2015-03-03, spreeman