Sum of Numbers
##############

..  image:: images/Carl_Friedrich_Gauss.jpg
    :align: center
    :width: 300

Surely, you have heard this man's name: Carl Friedrich Gauss. He grew up to be
a famous mathematician. In fact, his influence on the world of mathematics was
quite amazing. 

..  note::

    Get out of your box sometime, put your keyboard aside, and read some of the
    papers written by these folks. My favorite book on this is "God Created the
    Integers" :cite:`Hawking:2007` edited by Stephen Hawking. It is a tough
    read, but amazing to see how these thinkers came up with discoveries we use
    even today! This book also has papers by George Boole, and Alan Turing.

Busy Work!
**********

The story goes that Gauss was irritating his math teacher, way back in 1700, and
that teacher assigned a problem for Gauss to solve that would keep him busy.  

:: 
    
    Find the sum of all the numbers from 1 to 100

Gauss, being no dummy, took this problem on and immediately found a pattern.
Patterns help us generalize a problem and come up with a simple way to find the
answer.

In this case, Gauss found a pattern this way.

    * Split the list into two parts

    * Reverse the second list

    * Add up the two lists

    * Sum the two lists

Here is an example.: Sum up all the numbers from 1 to 100

::

      1   2   3   4 ...  47  48  49  50
    100  99  98  97 ...  54  53  52  51
    -----------------------------------
    101 101 101 101 ... 101 101 101 101

There are 50 terms in the final result, so the answer is 50*101 = 5050.
Furthermore, he came up with this general formula for the result:

..  math::
    
    sum(n) = n(n+1) / 2

Suppose we want the formula for the sum of numbers between any two numbers. We
can use Gauss's formula to find the answer:

..  math::

    sum(n1,n2) = sum(n2) - sum(n1)

We will use this formula in our testing later.

I Hate Math
***********

That is why we use computers, to do our busy work. 

..  note::

    Actually, I like math, and I wish I had studied it more in school. It is
    never too late to learn more. Go read that book!

Lets build a program to do Gauss's busy work for him. This should be easy:

..  literalinclude::    code/gauss1.cpp

Simple enough. We are going to use this code as a basis for exploring how
memory is really managed in a modern computer.

First, let's make this code more complex. We will load each number into an
array, then sum up that array:

..  literalinclude::    code/gauss2.cpp

To make it even more complex, let's rework this so it uses a two dimensional
array!

..  note::

    I said this was busy work!

..  literalinclude::    code/gauss3.cpp

And here is the final version. It has one subtle change:

..  literalinclude::    code/gauss4.cpp

Just for completeness, here is a ``Makefile`` you can use to build all of these
programs.

..  literalinclude::    code/Makefile
    :lines: 4-18

..  note::

    Look closely at this file, it uses some of Make's power to set the program
    name from the source file name. (This only works in a directory full of
    single-file programs. It is good for testing things!)

Memory Access Patterns
**********************

All of these programs generate the same result, but they get there using
different techniques for accessing memory!

    * Version 1: Simple random access of a few variables

    * Version 2: Simple sequential access into an array

    * Version 3: Still simple sequential array access (why)

    * Version 4: Not so simple random array access

In our next lecture, we will build a program that mimics how computers really
work. We will time our code and see what happens when we use these different
patterns to solve this simple problem.