In minisymposium: Advances in Krylov Subspace MethodsMon 10:30–11:00, Auditorium Jean Monnet
Matrix functions are of increasing importance in many applications of scientific computing and engineering. Of particular interest is the problem of computing $f(A)b$ for a large and sparse matrix $A$, scalar function $f$, and vector $b$. Given the nature of $A$, Krylov methods are an essential tool for developing methods to compute $f(A)b$ efficiently. \\ We present a new iterative method for computing $f(A)b$, when $f$ is a Stieltjes function and $A$ is Hermitian positive definite. This new method is derived from a relationship between the standard Lanczos relation and a Gauss–Radau quadrature rule; we henceforth call it the Radau-Lanczos method. We also present theoretical results regarding the method's convergence properties, as well as numerical results demonstrating the method's improvement over the standard Lanczos method for matrix functions. In particular, we show that the Radau-Lanczos method is effective in maintaining accuracy when one must use restarts, a common procedure implemented with Krylov methods to handle computer storage limitations.