Effective algorithms for spectral problems
This chapter examines few eigenvalues and corresponding eigenfunctions of eigenvalue problems involving symmetric elliptic operators. Special attention is paid to the algorithms that require computational work of the same type as for the corresponding boundary value problems provided the smoothness properties of eigenfunctions are of the same nature as those of the boundary problems solutions. The uses approximate orthogonalization with respect to those eigenvectors that have already been computed; simultaneous calculation of a group of the eigenvectors is also very important. P. Chebyshev, one of the founders of optimization in approximation theory, remarked more than a hundred years ago that the history of math ematics consists of two periods. The first is characterized by the fact that the mathematical problems were set by Gods, and in the second they were formulated by semigods. As a consequence of them, an international character of this theory should be stressed: many important steps have been made by many different people at different times.