Before the lab

• Complete any needed linear algebra review

Background

• GOBG Chapter 2 (element-by-element operations, dot product)
• GOBG Chapter 4 (plotting)
• Octave Unit Test
• Benchmarking tools
• Octave anonymous functions

Working with grids

• Octave has as special notation for ranges that we can use to initialize arrays.

    r = [-1:0.5:1]
  

• The builtin function meshgrid can be used to take Cartesian Products of two arrays.

    [X,Y] = meshgrid(r,r)
  

• Despite what best practice is in languages like Java, Octave is quite fond of parallel arrays. In this example, the (r(i),r(j)) element of the cartesian product is [X(j,i),Y(j,i)]. Since we just care about hitting all combinations, we won't worry too much about flipping indices

• For each point in our grid, we can compute a height as follows

    Z = X.^2 - Y.^2
  

• We can then plot a surface using those heights with

    surf(X,Y,Z)
  

• Things are very blocky here because we don't have many points. This is so we can understand the intermediate matrices, and how surf works. We'll use more points to give a smoother appearence in what follows.

Defining and testing a function

In the rest of this lab we want to use Octave to explore the function

    δ(β, a, b) = βa·b - (β-1)b·b

Here β is a scalar (a number) and a and b are vectors of length 2. We curious what the zeros of this function are for fixed β and a. One approach is to try plotting that surface as a function of b. We'll start by defining and testing a function for beta

• Translate the mathematical function δ into an Octave function.

    ## compute βa·b - (β-1)b·b
    ## a, b are assumed to be column vectors, beta is a scalar
    function ret = delta(beta, a, b)

    endfunction

• Complete the tests according to given comments. Use rand to generate a and b. You will also have to look up the documentation for assert for how have a floating point error tolerance.

%% delta(0,a,b) = |b|²
%!test
%!
%!
%!

%% delta(1,a,b) = a.b
%!test
%!
%!
%!

%% delta(2,a,b) = 2a.b - |b|²
%!test
%!
%!
%!

Making 3D plots

• Write one or more for loops to initialize Z

a = [4;4];
beta = 7.5;

%% Generating vectors
range = [-4:0.1:8];

% Compute cartesian product (grid)
[X,Y] = meshgrid(range,range);







surf(X,Y,Z);

• The resulting plot should look something like the following.

• Try replacing surf with contourf to get a different view of the surface. You can also "zoom" by changing range, but change it back for the timing tests below. What can you guess about the zero set (where the surface intersects the z=0 plane).

Python Quiz

• Questions will be distributed on paper at or before 9:30.