23 research outputs found
Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization
Krylov subspace residual and restarting for certain second order differential equations
We propose algorithms for efficient time integration of large systems of
oscillatory second order ordinary differential equations (ODEs) whose solution
can be expressed in terms of trigonometric matrix functions. Our algorithms are
based on a residual notion for second order ODEs, which allows to extend the
``residual-time restarting'' Krylov subspace framework -- which was recently
introduced for exponential and -functions occurring in time
integration of first order ODEs -- to our setting. We then show that the
computational cost can be further reduced in many cases by using our restarting
in the Gautschi cosine scheme. We analyze residual convergence in terms of
Faber and Chebyshev series and supplement these theoretical results by
numerical experiments illustrating the efficiency of the proposed methods
Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation
Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations
Approximation of functions of large matrices with Kronecker structure
We consider the numerical approximation of f(A) b where b 08 RNand A is the sum of Kronecker products, that is A= M2 97 I+ I 97 M1 08 RN
7N. Here f is a regular function such that f(A) is well defined. We derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations, with special emphasis on the exponential and on completely monotonic functions, for which the new procedure becomes particularly advantageous. Our findings are illustrated by numerical experiments with typical functions used in applications