A Lie algebra view of matrix splittings
By: Michele Benzi, Milo Viviani
Potential Business Impact:
Makes computer math problems solve much faster.
In this paper we use some basic facts from the theory of (matrix) Lie groups and algebras to show that many of the classical matrix splittings used to construct stationary iterative methods and preconditioniers for Krylov subspace methods can be interpreted as linearizations of matrix factorizations. Moreover, we show that new matrix splittings are obtained when we specialize these splittings to some of the classical matrix groups and their Lie and Jordan algebras. As an example, we derive structured generalizations of the HSS (Hermitian and skew-Hermitian splitting) iteration, and provide sufficient conditions for their convergence.
Similar Papers
On alternating-conjugate splitting methods
Numerical Analysis
Makes computer math problems solve better over time.
Positive semidefinite/positive semidefinite splitting iteration methods for solving nonsingular non-Hermitian positive semidefinite systems
Numerical Analysis
Solves hard math problems faster for computers.
Integral Forms in Matrix Lie Groups
Robotics
Simplifies math for robots and computer graphics.