I worked for a company where some people had done some computations (for plants and lines) in their heads or with notebooks or whatever for decades. Noone else knew how to do what they did, so the company decided to write a program to do it to ensure the future.
These guys told the programmers what they did, and then did not like the program. Indeed, they often decided what they wanted to see, and then put the numbers into the program to get the result they wanted, even though the numbers they put in were bogus and were only there to contrive the results they wanted to see.
So, the answer to favorite method of computation is simply, what i'm used to.
This sounds like the much(over) cited Expert System example of Campbell Soup Company mind melding with the really old guy that they flew all over the US fixing their soup cookers. They were terrified that he would retire or fall out.
<warstory>Took a trip to Alabama years ago to see a Unisys factory control system. Roll steel came in one end of the plant and steel buildings went out the other. Their major production problem was that the heat from the continous welding of the web on the I-beams tended to warp the beam. They had their own old guy who would walk the beam with a torch and a bucket of water. This "artist" would heat and cool spots on the beam and pull the warping out! He was training he son to take over from him.</warstory>
There's never enough time to do it right, but always enough time to do it over...
Wait, what? No ANN option? This is my favourite model which I layer over my natural neural network to get the job done. After computing, I write the stuff down in perl. The rest is done by The Machine.
I do apologize for missing that one -- and rule 110, which QM mentions below and which came to my mind just today when I thought of Conway Life (and then of this poll) again.
Others are missing, too, notably boolean logic circuits, various kinds of automata, Boltzmann machines, Conway's FRACTRAN (yes, that's the same Conway), and many more. Even diophantine equations can be coaxed to yield a model of computation.
Mea culpa, fellow monks! Accept my apologies, after all I'm just a silly pony.