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A hybrid method for nonlinear least squares that uses quasi-Newton updates applied to an approximation of the Jacobian matrix

Lukšan, Ladislav, Vlček, Jan (2019)

Programs and Algorithms of Numerical Mathematics

In this contribution, we propose a new hybrid method for minimization of nonlinear least squares. This method is based on quasi-Newton updates, applied to an approximation A of the Jacobian matrix J , such that A T f = J T f . This property allows us to solve a linear least squares problem, minimizing A d + f instead of solving the normal equation A T A d + J T f = 0 , where d R n is the required direction vector. Computational experiments confirm the efficiency of the new method.

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

Alexandre Caboussat, Roland Glowinski, Danny C. Sorensen (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the...

A one parameter method for the matrix inverse square root

Slobodan Lakić (1997)

Applications of Mathematics

This paper is motivated by the paper [3], where an iterative method for the computation of a matrix inverse square root was considered. We suggest a generalization of the method in [3]. We give some sufficient conditions for the convergence of this method, and its numerical stabillity property is investigated. Numerical examples showing that sometimes our generalization converges faster than the methods in [3] are presented.

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