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.      

Antikythera

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.   

Bush Differential Analyzer

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.    

Eniac

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

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

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.    

Weather modelling

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

Chaos theory

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.    

Supercomputers

Most scientific simulation is done on supercomputers.    

Von Neumann Architecture

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

Seti at home

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.