Matrix computations (3rd ed.)
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Multigrid
Numerical Methods
| Hi-index | 0.00 |
In a previous paper we presented two variants of Kovarik's approximate orthogonalization algorithm for arbitrary symmetric matrices, one with and one without explicit matrix inversion. Here we propose another inverse-free version that has the advantage of a smaller bound on the convergence factor, while the computational costs per iteration are even less than in the initial inverse-free variant. We then investigate the application of the new algorithm for the numerical solution of linear least-squares problems with a symmetric matrix. The basic idea is to modify the right-hand side of the equation during the transformation of the matrix. We prove that the sequence of vectors generated in this way converges to the minimal norm solution of the problem. Numerical tests with the collocation discretization of a first-kind integral equation demonstrate a mesh-independent behaviour and stability with respect to numerical errors introduced by the use of numerical quadrature.