Introductory Programming
Fall 2006

For today, you should have:

1) Read Chapter 2


Today:

1) update continued

2) matrices


For next time you should:

1) modify update so it stores successive values of a and b
   in vectors named A and B, and then plots them

2) prepare for Evaluation 2, which will cover everything up
   to what we did today.


Update
------

Last time we wrote a script that looks like this:

anew = a - 0.03 * a + 0.05 * b
bnew = b + 0.03 * a - 0.05 * b
a = anew
b = bnew

Today we will gradually improve this code, following
these steps:

1) use round() to make sure we are moving integral cars!
   
   suggestion: define variables froma and fromb

2) add a loop, so the script runs a number of iterations

3) add plotting

   some commands we might want:

   clf                  % clear the current figure
   hold on              % allow mulitple plots on the same figure
   plot(x, y, 'ro')     % plot a red o at x, y
   plot(x, y, 'bx')     % plot a blue cross at x, y





Matrices
--------

In linear algebra, 

1) a single number all by itself is a scalar.

2) a sequence of numbers is a vector

   arranged horizontally, it's a row vector; vertical is a column vector

3) a 2-D grid of numbers is a matrix

4) an n dimensional grid of numbers is a tensor, for n>2


In most programming languages,

1) scalars are represented by integers and floating point numbers

2) vectors, matrices and tensors are represented by 1-D, 2-D and
   n-D arrays


In MATLAB, Everything Is a Matrix!

1) a matrix that happens to be 1x1 is treated as a scalar

2) a matrix that is 1 x n is a row vector

3) a matrix that is n x 1 is a column vector

4) a matrix that is n x m is a matrix


Creating matrices
-----------------

Try the following code:

a = 1
b = [1, 2]
c = [3, 4]
d = [1, 2; 3, 4]    % notice the semi-colon!
e = [b; c]
whos

The whos command is similar to who, but it gives the size
and type of each value in your workspace.

The [] operator creates a new matrix.

Inside square brackets:

* the comma operator separates elements in a row

* the semi-colon operator separates rows

What do you think [b, c] does?

To be more concise, you can leave out the commas.

As usual, let's think of everything we can do wrong with matrices!


length and size
---------------

What are the length() and size() of a, b, c, d and e?


The colon operator
------------------

The colon operator, which we have seen in the context of a
for loop, also creates a matrix:

x = 1:10
whos

What happens if we try:

y = 10:1
whos

What about:

z = []
whos


Matrix operations
-----------------

Most operators and functions work on matrices

w = 0:0.1:2
x = w * pi      % multiplication by a scalar does what you expect
y = sin(x)      % most math functions do what you expect


While we are at it, if you have two matrices that are the
same length, you can plot one as a function of the other.

plot(x,y)

Most operations between matrices and scalars do what you expect.

Addition and subtraction between matrices does what you expect:

a = 1:3
b = 1:3
c = a+b
d = a-b

It performs the operations "elementwise"

Multiplication DOES NOT do what you expect:

e = a*b

??? Error using ==> mtimes
Inner matrix dimensions must agree.

The .* operator performs elementwise multiplication.

e = a .* b


Same with exponentiation

f = a^2
??? Error using ==> mpower
Matrix must be square.

f = a.^2      % elementwise exponentiation

You can divide a matrix by a scalar.

For now, you should not divide by a matrix.


Indices
-------

(we'll start out talking about vectors, then generalize to
matrices.)

The elements of a vector are numbered from 1..n, where n
is the length of the vector.

You can read, or access, an element with the () operator

x = 1:10
y = x .^ 2
z = y(10)

The expression in paretheses is called an index, because
it indicates which element you want.

It must be an integer between 1 and n, inclusive.

y(0)
y(11)

What do you think will happen for the following:

y(1.0)
y(1.000000001)
y(1.000000000000000000001)
y(i)
y('1')

Keep that last one in mind!