Numerical linear algebra and some problems in computational statistics

for any value of x in the interval (a, b). Chebyshev motivated his problem by investigation of limit theorems in probability theory, with some related work done even earlier by Heine. It was completely resolved by Markov; for more detailed comments and description of the related developments see Shohat and Tamarkin (1943), Akhiezer (1965) and Gautschi (1981). Here we will not describe the well known role of moments in approximation and determination of the characteristic and distribution functions of the random variable in probability theory. We point out, however, that an analogous approach is used in solving systems of linear algebraic equations using the matching moments model reduction described below. Chebyshev noticed a possible mechanical interpretation of his problem, but he did not investigate it. The mechanical interpretation was investigated by Stieltjes, who gave a complete solution to the following problem. Given a sequence of numbers ξk, k = 0, 1, . . . , a non-decreasing distribution function ω(λ) is sought such that the Riemann-Stieltjes integrals satisfy