CSC 431

This week we finished chapter 5 (5.5, part of 5.6) on non-linear systems and chapter 6 (6.1, 6.2.1, 6.4) on optimization.

During class we saw some of Heath's simulations:

- Newton's Method
- Secant Method
- Multivariate Newton
- Golden Section Search
- Successive Parabolic Interpolation
- Newton's Method for finding local minimum
- Steepest Descent

and we discussed my solutions to homework 5, problem
1.

**Submission**: you can submit the homework by hardcopy
in class or by sending it to me as an email.

**Note**: for all of the problems on this homework you can use
the **computer**, but still explain your steps.

1. [n-th root, 15pt] In class we applied Newton's method to find a fixed-point function g(x) = (x+r/x)/2 to approximate the square root of r.

a) Find g(x) for the n-th root in place of square root, where n is a fixed number.

b) Run the method to find the first six significant decimal digits for the cube root of 2. How many iterations do you need starting at x = 1?

2. [Bernoulli's method, 10pt] Bernoulli's method applied to the equation
x*x + x - 1 = 0 gives an approximation for phi = (sqrt(5)+1)/2 as follows:
compute x_{n} = x_{n-1} + x_{n-2}, starting with x_{0}
= 0 and x_{1} = 1. For large enough values of n,
x_{n}/x_{n-1} will be
close to phi. Implement this procedure and test how fast the convergence
seems to be (at what rate does the number of significant digits increase in
each step?.

3. [Compound Interest, 20pt] Do Exercise 5.15 on page 251 of the book. (5.11, page 179 in the first edition).

4. [Accuracy of minimum, 5pt] Do Exercise 6.1 (a) on page 303 of the book (page 213 of the first edition).

5. [2-dimensional minimum finding, 20pt] Do Exercise 6.9, a) (page 304,
page 214 of first edition). You do not have to implement your own program,
you can use Health's simulations, however, for starting point x_{0}
= [-1 1]^{T} show the details of the calculations (Jacobian,
one-dimensional optimization problem) that are necessary to compute the next
approximation x_{1}.

Last updated: February 26th, 2010.