Summary: These are the notes for two lectures I gave at the Belfast Functional Analysis Day 1999. The purpose of these notes is to give an idea of how \(C^*\)-algebra techniques can be successfully employed in order to solve some concrete problems of Numerical Analysis. I focus my attention on several questions concerning the asymptotic behavior of large Toeplitz matrices. This limitation ignores the potential and the triumphs of \(C^*\)-algebra methods in connection with large classes of other operators and plenty of different approximation methods, but it allows me to demonstrate the essence of the \(C^*\)-algebra approach and to illustrate it with nevertheless nontrivial examples.