| Popis: |
Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g., in quasi-Newton methods. Motivated by the latter, we study a nonclassic matrix condition number, the $\omega$-condition number. We do this in the context of optimal conditioning for: (i) our application to low rank updating of generalized Jacobians; (ii) iterative methods for linear systems: (iia) clustering of eigenvalues and (iib) convergence rates. For a positive definite matrix, the $\omega$-condition measure is the ratio of the arithmetic and geometric means of the eigenvalues. In particular, our applications concentrate on linear systems with low rank updates of ill-conditioned positive definite matrices. These systems arise in the context of nonsmooth Newton methods using generalized Jacobians. We are able to use optimality conditions and derive explicit formulae for $\omega$-optimal preconditioners and preconditioned updates. Connections to partial Cholesky sparse preconditioners are made. Evaluating or estimating the classical condition number $\kappa$ can be expensive. We show that the $\omega$-condition number can be evaluated explicitly following a Cholesky or LU factorization. Moreover, the simplicity of $\omega$ allows for the derivation of formulae for optimal preconditioning in various scenarios, i.e., this avoids the need for expensive algorithmic calculations. Our empirics show that $\omega$ estimates the actual condition of a linear system significantly better. Moreover, our empirical results show a significant decrease in the number of iterations required for a requested accuracy in the residual during an iterative method, i.e., these results confirm the efficacy of using the $\omega$-condition number compared to the classical condition number. |
