Exploiting Symmetry in Numerical Solving 3;4

It has recently been seen that the cost of numerically solving linear systems of equations in which the coeecient matrix is equi-variant with respect to a permutation group ? can be signiicantly reduced via the method of symmetry reduction. For systems in which the coeecient matrix is not equivariant, the numerical solving can be accelerated by using the ?-equivariant part of the coeecient matrix as a preconditioner. In the present paper the equivariant part of a general linear transformation is studied for general nite groups ?. For nding the ?-invariant zero points of ?-equivariant nonlinear maps, the Jacobian map can be replaced by its equivariant part in Newton-type methods. This increases the numerical eeciency of a Newton step signiicantly. For this modiication, it is shown that the familiar local convergence behavior to solutions is generally preserved. We give two examples. It is shown how these techniques can be used in numerical continuation and bifurcation. 1. Introduction Many problems in science and mathematics exhibit symmetry phenomena which may be exploited to analyze them, and also to eeect a signiicant cost reduction in their numerical treatment. Usually the symmetry stems from the domain or body on which the problem is considered. The numerical treatment of problems such as partial differential equations and integral equations generally involves discretiza-tions which ought (as far as possible) to incorporate or respect such symmetries.