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A comparison of some efficient numerical methods for a nonlinear elliptic problem

Balázs Kovács (2012)

Open Mathematics

The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them...

A fixed point method to compute solvents of matrix polynomials

Fernando Marcos, Edgar Pereira (2010)

Mathematica Bohemica

Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.

A Newton-Kantorovich-SOR type theorem

Béla Finta (2005)

Open Mathematics

In this paper we propose a new method for solving nonlinear systems of equations in finite dimensional spaces, combining the Newton-Raphson's method with the SOR idea. For the proof we adapt Kantorovich's demonstration given for the Newton-Raphson's method. As applications we reobtain the classical Newton-Raphson's method and the author's Newton-Kantorovich-Seidel type result.

A note on direct methods for approximations of sparse Hessian matrices

Miroslav Tůma (1988)

Aplikace matematiky

Necessity of computing large sparse Hessian matrices gave birth to many methods for their effective approximation by differences of gradients. We adopt the so-called direct methods for this problem that we faced when developing programs for nonlinear optimization. A new approach used in the frame of symmetric sequential coloring is described. Numerical results illustrate the differences between this method and the popular Powell-Toint method.

A parallel projection method for linear algebraic systems

Fridrich Sloboda (1978)

Aplikace matematiky

A direct projection method for solving systems of linear algebraic equations is described. The algorithm is equivalent to the algorithm for minimization of the corresponding quadratic function and can be generalized for the minimization of a strictly convex function.

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