Description

Assignment 4
Note: Please use Matlab, or a public domain approximation to it in this assignment. The code must compile on one of the lab machines with your instructions. Document your code thoroughly!
Instructions
1.Write a Newton’s method code to solve the pair of equations given by
f x x1 1( , 2)   x13 x2 1
f x x2( 1, 2)   x23 x1 1
Use appropriate criteria to measure convergence and to terminate the iteration. Use
starting values of x1 1.0,x2  0.1 .Perform a few experiments with different starting values and count the number of iterations it takes to get to convergence.
2. Repeat the process with the potentially difficult system of equations given by
f x x1( 1, 2)    x12 x22 2 0 f2(x x1, 2)  e(x11)   x22 1.0 0
Using the top equation to eliminate x2from the bottom equation. Find the solution either by using. Newton’s method for a single equation or by guesswork. What happens if the last term in the second equation is -2.0 instead of -1.0?

Solve the full problem with values x1 1.1,x2 1.1, x1  2.0,x2  0.5 , x1  3.0,x2  5.0 x1 0.7,x2 1.14as starting values. In which cases do you get a solution?.
Verify your results by using the top equation to eliminate x2from the bottom equation. Find the solution either by using. Newton’s method for a single equation or by guesswork.
Does the matlab routine fsolve compute a solution in all these cases?

3. LORAN LOng RAnge Navigation calculates the position of a boat at sea using signals from fixed transmitters. From the time differences of the incoming signals the boat obtains differences of distances to the transmitters. This leads to two equations that represent the distances such as;
x 2 x2
f x x1 1( , 2)  1861 2  3002 21862 1
f2(x x1, 2)  2279500)2 2  500(x1 2300)27922 1 (x
Implement a Newton solver for this system and define the Jacobian matrix.
Using (400,400) as a starting point and run the solution for 25 iterations. This is enough to get to convergence?. Repeat for all starting values from (401, 401) to (600,600) in increments of 1 (402,402) etc . Which values does the method converge to for which starting values and how many different solutions are there?
4. Use the simple gradient descent program provided in the lecture slides to solve Problem
8.21 in Holmes.

5. (a) Use the provided simple gradient descent program supplied to solve the Rosenbock Problem. Use and accuracy requirement of 1.0e-8 and find how many iterations are needed to get the solution when starting from (0.5, 0.5) (0.1, 0.1) and (-0.5 0.5).
5. (b) Implement the Armijo algorithm on page 361-3 of Holmes. How much does this reduce the number of iterations needed?
5 (c) Implement the Adam method which is one of the most commonly used methods in machine learning and see if that helps. Notes on this method are on line as are many examples of its use with the Rosenbrock function.

What to turn in
For these assignments, we expect both SOURCE CODE and a written REPORT be uploaded as a zip or tarball file to Canvas.
• Source code for all programs that you write, thoroughly documented. o Include a README file describing how to compile and run your code.
• Your report should be in PDF format and should stand on its own.