Experimental mathematics (Q5902756)

The authors make here an earnest attempt to highlight and propagate the value and importance of experimental mathematics which is an emerging area of mathematics. Due to the advent of widespread use of computers as well as the potential for tackling increasingly complex practical problems, the science of experimental mathematics is determined to become an interesting subfield of mathematics (and/or computer science) which could occupy a status similar to that of, say, operations research. To make the discussion more explicit and concrete the whole idea is illustrated with an example of a recent mathematical experience from linear algebra. Considering a (real) positive definite Toeplitz matrix \(R: R=\{r_{| s-t|}\); \(s,t=1,2,...,n\}\), constructing the vector \(V=R^{-1}e\), \(e=col(1,1,...)\) it is conjectured that V always has non- negative entries only. Several aspects of the problem are further discussed.