Verified numerical computations play important roles for rigorously solving systems of equations arising in various scientific problems. Following the previous special section on verified numerical computations published in Vol.2 No.1 (2011), this special section is devoted to self-validating methods and computer-assisted proofs, which have attracted attention of researchers in a wide spectrum of science and engineering. The following variety of topics is presented: error-free transformation for matrix multiplication, verified solutions of linear systems, verification methods and their related topics for ordinary and partial differential equations, a topological computation approach to the interior crisis bifurcation, and verified computation for the linking number of links. The guest editors believe that this special section invites many readers to the frontier of this emerging research field. They would like to express their sincere thanks to all authors for their contributions. They also thank the reviewers and the secretaries of this special section, especially Takashi Hisakado (Kyoto University), Kenta Kobayashi (Hitotsubashi University) and Katsuhisa Ozaki (Shibaura Institute of Technology), and the editorial staff of NOLTA journal for their supports on publishing this special section.
This paper is concerned with accurate numerical algorithms for matrix multiplication. Recently, an error-free transformation from a product of two floating-point matrices into an unevaluated sum of floating-point matrices has been developed by the authors. Combining this technique and accurate summation algorithms, new algorithms for accurate matrix multiplication could be investigated. In this paper, it is mentioned that the previous work is not the unique way to achieve an error-free transformation and the constraint of the error-free transformation is clarified. For the application, a new algorithm is developed reducing the number of matrix products compared to the previous algorithm.
This paper is concerned with verification methods for numerical solutions of linear systems. Many methods for the verification require switches of rounding modes defined by the IEEE 754 standard. However, the switches cannot be supported in several computational environments. In such cases, Ogita-Rump-Oishi's method can work on such environments. Recently, Rump developed new error estimates of floating-point summation and dot product. The aim of this paper is to improve Ogita-Rump-Oishi's error estimates by using the error estimates by Rump. In addition, the computational cost of our method is comparable to that of Ogita-Rump-Oishi's method.
This paper describes a numerical verification of solutions for infinite dimensional functional equations based on residual form and sequential iteration. Comparing with other verification procedures as typified by Newton-type iterations, the proposed algorithm can be done at low computational cost, although it needs that the formulated compact map is retractive in some neighborhood of the fixed-point to be verified. Several computer-assisted proofs for differential equations, including nonlinear partial differential equations will be shown.
In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.
We show a criterion for verifying the locally unique existence of stationary solutions of a class of partial differential equations and the local dynamics around them with computer assistance. Our method is designed so that, in near future, it can be extended to further analysis of dynamics, like connecting orbits between equilibria. Several sample examples for verifying hyperbolic equilibria of a concrete PDE are also shown.
We show a strange property of an approximate solution for the initial value problem of ODEs. This study is originally motivated by the numerical verification methods of solutions for parabolic initial-boundary value problems. We prove that an a priori norm estimate of the derivative of approximation for simple initial value problems could not be stable, even though the concerned numerical scheme is a natural finite element approximation. Since the corresponding estimates for the exact solution is bounded by a given function, it implies that the discretization could no longer maintain the same property at all. Therefore, this result should be an interesting example which shows an essential gap between the infinite and finite dimensional problems.
We study the interior crisis bifurcation from the viewpoint of the graph-based topological computation developed in . We give a new formulation of the interior crisis bifurcation in terms of a change of the attractor-repeller decompositions of the dynamics, and prove that the attractor before the crisis disappears by creating a chain connecting orbit to the repeller at the moment of the interior crisis. As an illustration, we discuss the interior crisis bifurcation in the Ikeda map.
We propose a rigorous numerical algorithm for computing the linking number of links defined by spatially distributed data points. The key idea is to use an analytic expression for the solid angle of a tetrahedron for quick evaluation of the degree of the Gauss map. An implementation of the algorithm with INTLAB, a Matlab toolbox for reliable computing, is also provided.
A spiking neuron model described by an asynchronous cellular automaton is introduced. Our model can be implemented as an asynchronous sequential logic circuit and its control parameter is adjustable after implementation in an FPGA. It is shown that our model can reproduce twenty types of dynamic response behaviors observed in biological and other model neurons. It is also shown that our model can reproduce the features of four groups into which biological and other model neurons are classified. In addition, underlying bifurcations of the four groups are analyzed, and the results yield basic guides to the synthesis of our model.