I need a little help clarifying what the difference is between analytic and numerical solutions as discussed in Weisberg’s “Forty Years…” article. He discusses this in talking about the limitations of brute-force models insofar as they may be incapable of either analytic solutions, numerical solutions, or both.

Numerical solutions seem to be something like being able to use the model, given some input, to achieve a numerical output. As such, it seems that the numerical solubility of a model is related to the complexity of the model and the current available computing power.

Analytic solutions seem to be some sort of “explicit description of how the parts of the model depend on one another and the magnitude of these dependencies.” (630) Is Weisberg here saying that analytic solutions are some sort of generalization of numerical data?

Regarding analytic solubility, Weisberg seems to say that there can be some models which admit of no analytical solutions regardless of simplicity. “Take a simple physical system that admits of no analytical solution, such as a three mass system with gravitation attraction between the masses… this type of system will admit of no analytic solution in closed form.” (631) Why is it the case that this model, or any other model, can never (he seems to imply) have an analytic solution? Is it some conceptual limitation characteristic of humans?

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Not all mathematical models have closed form solutions; i.e. one cannot solve the equations even if they “appear” to be a simple system. It is not a human limitation, except in the sense that human-designed mathematics cannot compute a unique solution. Some systems of this type can be “solved” using a numerical methods system that searches heuristically for an approximate solution by testing numerical outcomes within a pre-specified error around the unobtainable closed form solution. This is important in financial economics, complex simulation models, and “generative social science”.

An analytical solution finds the result in terms of the model parameters and the other endogenous and exogenous variables in the way we all learned to solve two equations in two unknown variables in algebra. Numerical solutions have calculated results but are not explicit in the unknowns and parameters.

Googling the two terms helps a lot.

Some random links:

MyPhysicsLab – Numerical vs. Analytic Solutions

http://www.myphysicslab.com/numerical_vs_analytic.html

The Importance of Finding Analytic Solutions to Problems

http://cnx.org/content/m12654/latest/