A computable error bound for matrix functionals

Many problems in applied mathematics require the evaluation of matrix functionals of the form F(A):=uTf(A)u, where A is a large symmetric matrix and u is a vector. Golub and collaborators have described how approximations of such functionals can be computed inexpensively by using the Lanczos algorithm. The present note shows that error bounds for these approximations can be computed essentially for free when bounds for derivatives of f on an interval containing the spectrum of A are available.