cs249 lecture notes, Fall 2001
Week 7, Monday

Quiz 5

Schedule exam... October 22 or 25?

Vocabulary:

array:   a sequence of values (not necessarily numbers)

element: one of the values in an array

index:   an integer used to indicate one of the values in
         an array



Quiz 5 Solution
---------------

Part 1:

Most idiomatic version

function result = sum (a)
    total = 0;
    for i = 1:length(a)
        total = total + a(i);
    end
    result = total;

1) length is a built-in function

2) a(i) uses i as an index to select an element of a

3) the accumulator pattern:  initialize before the loop,
   modify inside the loop, use the result outside the loop


Alternate version

function result = sum (a)
    total = 0;
    for x = a
        total = total + x;
    end
    result = total;

The latter version emphasizes the semantics of the
for loop

    for variable = array

Assigns each element of the array to the variable, in turn.


Part 2:

function result = selectOdd (a)
    index = 1:2:length(a)
    result = a(index)


Two things here

1) an array constructor with an increment

2) using an array as an index

   Strictly, the elements of the index array should be integers.


Array concatenation
-------------------

Sections 5.2 and 5.3 lead right into Section 5.10

1)  array concatenation looks like [head tail]

2) the empty array is []

3) you can accumulate an array by concatenating onto the
   beginning or the end


Willis equation
---------------

1) Using linspace and for loops to enumerate points in
   the complex plane.

2) Checking parity with the modulus operator.

3) How to use clf, hold and axis

4) a tiny bit about object handles


function result = mandel2 (n)
    clf; hold on;
    f = gcf;
    set (f, 'DefaultLineMarkerSize', 2);
    willis (n)

function willis (n)
    coords = linspace (-3, 3, n);
    
    for real = coords
        for imag = coords
            z = real + i * imag;
            a = iterate (z);
            if (a == -1)
                continue
            elseif (mod(a,2) == 0)
                plotPoint (z, 'rs', 'red');
            else
                plotPoints (z, 'bs', 'blue');
            end
        end
    end
    axis ([-3 3 -3 3]);

    
function plotPoint (z, format, color)
    h = plot (z, format);
    set (h, 'MarkerSize', 2);
    set (h, 'MarkerFaceColor', color);


function result = iterate (z0)
    z = z0;
    lastz = z0;
    
    for j = 1:100
        z2 = z*z;

        f = z*z2 - 2*z - 5;
        fp = 3 * z2 - 2;

        z = z - f / fp;
                
        diff = z - lastz;
        if (abs (diff) < 1e-8)
            result = j;
            return
        end
        lastz = z;
    end
    result = -1;



Array operations
----------------

+ and - perform elementwise addition and subtraction

We used that for vector arithmetic.

array * scalar or scalar * array is elementwise

array * array is dot product, which only works if the
              sizes of the array are compatible

array / scalar is elementwise

scalar / array  and array / array are something else

array ^. scalar is elementwise exponentiation


Plotting
--------

The second figure in Section 5.7 is wrong.  Sorry!


Euler's method
--------------

Most general form

dy/dt = f (t, y)


For many physical problems, the rate is a function of
"position" or "time", but not both.

Constant acceleration

dy/dt = gt            "position" is position

Cooling

dy/dt = -r (y - Ts)   "position" is temperature

Draw quivers for these problems.


Thursday: let's do acceleration of gravity with wind
resistance and find terminal velocity.