Numerical Methods: Approximation
Numerical methods serve to approximate solutions to mathematical equations and solve mathematical problems using computational power after algorithms have been developed. Methods exist for finding roots of polynomial equations for example or factoring matrices. Generally a convergence process if stable leads to the right result much like Taylor polynomials approximating the values of various functions where the Taylor polynomials act as a series progressing towards infinite.
Relevant concepts from calculus are the mean value theorem and the intermediate value theorem as well as Rolle’s theorem which all serve to answer how a bounded well-behaved function reflects its various properties in terms of itself. This is all analysis.
What we want to understand most though is when things are intractable analytically we turn to numerical methods for approximations, much like when problems become intractable in let’s say, the business and economics sphere we turn to stories where as we process the stories we get closer to the solution of the business problems. The fundamental business problem was lack of liquidity not lack of assets, whereas we want to remember lack of liquidity is enough to bring you to bankruptcy if you have no hard assets, and we managed to solve this problem by approximating solutions with stories repeatedly until we saw that liquidity can only be created by trading as that is what trading is for, the generation of market liquidity, while asset heavy divisions like writing and art, do not generate windfalls usually that can solve financial crises. What is the difference between writing and art? I would say writing is an exact science while art is mostly an inexact science as practiced nowadays but if we bring art to writing’s level we would see that writing produces assets in terms of knowledge assets, while art produces aesthetics, and these are quite different. Knowledge assets make people more productive and trade more while aesthetics generally just make people happier. In any case, the danger was always that a financial liquidity crisis would spread but it has been contained by trading before a single dollar was spent and that was because of cost reductions through trading, and these cost reductions changed the base organization so that it is now more similar to a corporation while before it was a sole proprietorship. In other words, trading is not just about creating windfalls but about organizing your friends where a writer’s friends are organized mostly as audience to the writer, and same for an artist, while a trader has his friends as an information network. I share all this to show you that sometimes analytical solutions fail for long periods of time, and approximations have to be used, driven by sheer horsepower of the mind or computational power, and one such analytical solution which failed was revamping problems of the Second World War as a writer, but fortunately the trader stepped in with approximate methodology which solved the problem: the trader cut healthcare costs by fixing some health problems. Such is the power of a trader which is why they call him or her a master of the universe if done well. He isn’t a doctor but he can cut healthcare costs by fixing health problems through using what the doctor taught him. In other words a trader is basically an exceptional toolmaker and he can make his tools to mimick all professions that he understands.