We consider the linear equation Ax = b where A is a sparse symmetric positive definite matrix arising from a finite element discretisation. We use the preconditioned conjugate gradient method to solve this equation, introducing an element-by-element preconditioner which is based on a Crout's decomposition of the element matrices and an element-by-element product of them. When the mesh is colour...

We consider the solution of ill-conditioned symmetric and positive definite large sparse linear systems of equations. These arise, for instance, when using some symmetrizable preconditioning technique for solving a general (possibly unsymmetric) ill-conditioned linear system, or in domain decomposition of a numerically difficult elliptic problem. We are also concerned with the consecutive solut...

The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the nonEuclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular,...

in the present paper, among other results, a decomposition formula is given for the w-bounded continuous negative definite functions of a topological *-semigroup s with a weight function w into a proper h*-algebra a in terms of w-bounded continuous positive definite a-valued functions on s. a generalization of a well-known result of k. harzallah is obtained. an earlier conjecture of the author ...

The task of finding the singular-value decomposition (SVD) of a finite-dimensional complex Iinear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self-equivalent flows. The work constitutes a generalization of that of Brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated iso...

The statistical analysis of covariance matrix data is considered, and in particular methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix, and in particular ...