Background

Most of the material here is covered in the Gnu Octave Beginner's Guide, in particular

• GOBG Chapter 2
• GOBG Chapter 4
• GOBG Chapter 5

For some parts, we will refer to the GNU Octave Manual

We'll refer to the online text by Robert Beezer for linear algebra background. We can happily ignore the stuff about complex numbers.

Running Octave

• There is a GUI accessible from Activities -> GNU Octave, or by running from the command line

    >> octave --gui
  

• There is also a REPL accessible from the command line by running

    >> octave
  

• To sanity check your octave setup, run the following plots

    >> surf(peaks)
    >> countourf(peaks)
  

Fibonacci

Let's dive in to programming in Octave with a straight-forward port of our Python Fibonacci function. Save the following in ~/fcshome/cs2613/labs/L21/fib.m. It turns out to be important that the function name is the same as the file name.

function ret = fib(n)
  a = 0;
  b = 1;
  for i=0:n



  endfor
endfunction

%!assert (fib(0) == 0);
%!assert (fib(1) == 1);
%!assert (fib(2) == 1);
%!assert (fib(3) == 2);
%!assert (fib(4) == 3);
%!assert (fib(5) == 5);

• We can write this more consisely using the deal function, but we leave that for later.

• Note the %! assert. These are unit tests that can be run with

    >> test fib
  

• The syntax for %!assert is a bit fussy, in particular the parentheses are needed arround the logical test.

Matrix and vector review.

One of the main features of Octave we will discuss is vectorization. To understand it, we need some background material on Linear Algebra. If you've taken a linear algebra course recently, this should be easy, otherwise you will probably need to review

• Vector Operations, particularly

  • Addition
  • Subtraction
  • Scalar multiplication

• Inner product

• Matrix Operations

  • Addition
  • Subtraction
  • Scalar multiplication

• Matrix Vector Product

• Matrix Multiplication

Let's go through some simple examples on the board.

Fibonaccci as matrix multiplication

The following is a well known identity about the Fibonacci numbers F(i).

[ 1, 1;
  1, 0 ]^n = [ F(n+1), F(n);
               F(n),   F(n-1) ]

Since matrix exponentiation is built-in to octave, this is particularly to impliment in octave

Save the following as ~/fcshome/cs2613/labs/L21/fibmat.m, fill in the two matrix operations needed to complete the algorithm

function ret = fibmat(n)
  A = [1,1; 1,0];


endfunction

%!assert (fibmat(0) == 0);
%!assert (fibmat(1) == 1);
%!assert (fibmat(2) == 1);
%!assert (fibmat(3) == 2);
%!assert (fibmat(4) == 3);
%!assert (fibmat(5) == 5);
%!assert (fibmat(6) == 8);
%!assert (fibmat(25) == 75025);

Performance comparison

We can expect the second Fibonacci implimentation to be faster for two distinct reasons

• It's possible to compute matrix powers rather quickly (O(log n) compared O(n)), and since the fast algorithm is also simple, we can hope that octave impliments it. Since the source to octave is available, we could actually check this.

• Octave is interpreted, so loops are generally slower than matrix operations (which can be done in a single call to an optimized library). This general strategy is called vectorization, and applies in a variety of languages, usually for numerical computations. In particular most PC hardware supports some kind of hardware vector facility.

Of course, the first rule of performance tuning is to carefully test any proposed improvement. The following code gives an extensible way to run simple timing tests, in a manner analogous to the Python timeit method, whose name it borrows.

# Based on an example from the Julia microbenchmark suite.

function timeit(func, argument, reps)
    times = zeros(reps, 1);

    for i=1:reps
      tic(); func(argument); times(i) = toc();
    end

    times = sort(times);
    fprintf ('%s\tmedian=%.3fms mean=%.3fms total=%.3fms\n',func2str(func), median(times)*1000,
             mean(times)*1000, sum(times)*1000);
endfunction

What are the new features of octave used in this sample code?

• tic, toc, from GOBG8, GO 36.1
• Function Handles
• what else?

We can either use timeit from the octave command line, or build a little utility function like

function bench
  timeit(@fib, 42, 100000)
  timeit(@fibmat, 42, 100000)
endfunction

Measuring CPU time

Our timeit function measures Wall clock time. This means in particular that it is susceptible to interference from other activities on the same computer.

• Use the octave builtin cputime to write a function ctimeit that measures the cpu time taken by a function.

    >> ctimeit( @fibmat, 42, 100000)
    fibmat  mean = 0.018ms total = 1.812s
  

• Note that the cputime is not as precise as toc, you will need to change the way time is measured. There's a reason median is not reported here.

• What inaccuracy/overhead is built in to this method?

Using the profiler

Total time, as provided by timeit and ctimeit is useful for comparing two complete functions, it doesn't tell you where the time is being used within a given function. Octave supports profiling to help locate hotspots within your code.

• Use the profiler to measure a single call to fib(1000) and fibmat(1000). Don't forget to clear the profiling information between experiments. What do you observe about the precision of reported times? What other (more useful) information can you get from the profile?

• Repeat the experiment with a larger number of repeats of each function call. Compare the results with timeit and ctimeit for the same number of repetitions. There is at least one mysterious (large) difference. What do think might explain it?