The Error Analysis Of The Krylov Subspace Methods For Computing The Bilinear Form Of Matrix Functions

The bilinear form of matrix functions is of wide interest in many applications.The most commonly used approach is to reduce it to a Riemann-Stieltjes integral,so that we can use different types of quadrature formulas to approximate it,and the most common type is Gauss quadrature rules.But this method is not equipped with a universal posteriori error expression that can be used for determining the accuracy of the algorithm or designing reliable stopping criterion.In this dissertation,an accurate posteriori error estimate will be established,which can serve a reliable stopping criterion.In this dissertation,Krylov subspace methods will be mostly used to compute the approximatations of the bilinear form of matrix function.The original problem is split into two parts: first,calculate the matrix function times a vector f(A)v,then calculate the inner product with vector u.For f(A)v,we already have a reliable posteriori error estimates based on the Krylov subspace algorithms for f(t)sufficiently smooth.We generalize it to the bilinear form of matrix functions.Both the theory and the numerical experiments are reported to indicate that the first term of the error expansion can be used as a reasonable posteriori error estimate.The existing algorithms and stopping criterion for calculating the bilinear form of matrix function deal with some special cases,such as the hyperbola function f(t)= 1/t.In this dissertation,the relative error estimates are derived,which can be used in f(t)=1/t and some other sufficiently smooth funcions.