ON APPLICATION OF THE LANCZOS METHOD TO SOLUTION OF SOME PARTIAL-DIFFERENTIAL EQUATIONS

Let A be a square symmetric n x n matrix, phi be a vector from R(n), and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)phi arises very often in mathematical physics. We propose the following method to compute u. First, perform m steps of the Lanczos method with A and Define the spectral Lanczos decomposition method (SLDM) solution as u(m) = \\phi\\Qf(H)e1, where Q is the n x m matrix of the m Lanczos vectors and H is the m x m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for \\u - u(m)\\ that are stable in the presence of computer round-off errors when using the simple Lanczos method. We concentrate on computation of exp(- tA)phi, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.