Local convergence for the curve tracing of the homotopy method.
Max-min interval systems of linear equations with bounded solution
Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by and , where . The notation represents an interval system of linear equations, where , are given interval matrix and interval vector, respectively, and a solution is from a given interval vector . We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.
Measures of fuzziness and operations with fuzzy sets.
We discuss the effects that the usual set theoretic and arithmetic operations with fuzzy sets and fuzzy numbers have with respect to the energies and entropies of the fuzzy sets connected and of the resulting fuzzy sets, and we also compare the entropies and energies of the results of several of those operations.
Measuring and maintaining consistency: a hybrid FTF algorithm
Due to the versatility as well as its ease of implementation, the Fast Transversal Filters algorithm is attractive for many adaptive filtering applications. However, it is not widely used because of its undesirable tendency to diverge when operating in finite precision arithmetic. To compensate, modifications to the algorithm have been introduced that are either occasional (performed when a predefined condition(s) is violated) or structured as part of the normal update iteration. However, in neither...
Method of curvature in circular complex arithmetic.
Méthode numérique de détection de la singularité d'une matrice.
Mixed precision GMRES-based iterative refinement with recycling
With the emergence of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes for solving linear systems have recently been developed. However, in certain settings, GMRES may require too many iterations per refinement step, making it potentially more expensive than the alternative of recomputing the LU factors in a higher precision. In this work, we incorporate the idea of Krylov subspace recycling, a well-known technique for reusing information across sequential invocations,...
Monotone interval eigenproblem in max–min algebra
The interval eigenproblem in max-min algebra is studied. A classification of interval eigenvectors is introduced and six types of interval eigenvectors are described. Characterization of all six types is given for the case of strictly increasing eigenvectors and Hasse diagram of relations between the types is presented.
Multiple-Precision Correctly rounded Newton-Cotes quadrature
Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics. We study the Newton-Cotes quadrature scheme in the context of multiple-precision arithmetic and give enough details on the algorithms and the error bounds to enable software...
Můžeme věřit numerickým výpočtům?
Nikdy neztotožnujme realitu s jejím matematickým či numerickým modelem. (Věnováno Emilu Vitáskovi k jeho 80. narozeninám.)
New sufficient convergence conditions for the secant method
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
Nonsingularity and -matrices.
New proofs of two previously published theorems relating nonsingularity of interval matrices to -matrices are given.
Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding
For a real square matrix and an integer , let denote the matrix formed from by rounding off all its coefficients to decimal places. The main problem handled in this paper is the following: assuming that has some property, under what additional condition(s) can we be sure that the original matrix possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number...
Note sur la suppression des étiquettes en programmation
Numerical Stability for Solving Nonlinear Equations.
Numerical Stability of Decent Methods for Solving Linear Equations.
Numerical Stability of the Chebyshev Method for the Solution of Large Linear Systems.
Numerické metody a počítače
Object oriented design philosophy for scientific computing
This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...
Object oriented design philosophy for scientific computing
This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...