### Week 2 - Modelling the natural world

Click on the title of each topic below (except Numeric Methods) to be
linked to an article about that topic. Each student is responsible for
reading every article.

The Antikythera mechanism is an ancient greek astronomical computer
found in a shipwreck at the bottom of the sea. It is a far more
advanced mechanism than we believed that the classical greeks were
capable of building.

Vanevar Bush designed and built the first tryly versatile analog
computer in the late 20's and early 30's at MIT, under the auspices of
the US Army Ballistics Research Lab. It was capable of solving
complicated calculus problems.

The same Ballistics Research Laboratory paid for the construction of
Eniac, the first digital computer designed to model the natural world,
in this case, the trajectory of cannon fire.

#### Numerical Methods

I have not found a reasonable reading on numerical methods at an
appropriate level. Numerical methods or numerical analysis is the
branch of mathematics that studies how certain hard kinds of problems
can be solved. A common approach is the iterative approach, where a
sequence of ever more accurate approximate answers, using the error in
the previous answer to find a more accurate next answer.

Numerical methods are important for computers because they provide a
good approach for computers to solve these problems.

Newton's method is an iterative mechanism for mechanically solving many
forms of equations. At its essence, you take a guess and a formula that
will help you bound the real answer, given that guess.

We will look at finding square roots, where Newton's method simplifies
to the much older Babylonian method. To find the square root of NUM,
you start with a reasonable guess, which I will call X0

If X0 is the initial guess, the next guess (X1) is given by

X1 = (X0 + NUM/X0) /2.

If you are mathematically inclined, you can realize that the real
square root has to be between X0 and NUM/X0. We will guess that it is
exactly in the middle, which is wrong, but close.

The next guess X2 is computed as

X2 = (X1+ NUM/X1) / 2

X3, X4 and so on are each calculated the same way, giving a number ever
closer to the real answer. You stop the calculation when you two
consecutive numbers that are sufficiently close together for your
purposes.

Finite element analysis is a mechanism to use numerical approximations
to model physical systems, such as heat transfer or stress. The
analysis breaks the system down to a set of relatively simple nodes.
The physics of any individual node is simple enough to solve for a
given time slice. The solution for each time slice for all the
surrounding nodes becomes the inputs for the next time slice for the
surrounded node.

Scientists use numerical modelling to understand the climate and
predict the weather.

Chaos explains that the natural world may not be easy to predict or
model. In some cases, miniscule pertubations in the initial state
causes fundamental shifts in the results of the model.

Most scientific simulation is done on supercomputers.

The Von Neumann architecture adds the final element of modern
computing, the stored program.

Using many, many computers is an alternative to using a single large
supercomputer. SETI@home was one of the early and the best known
example of this kind of approach.