Floating-point numbers and floating-point arithmetic are widely used in numerical computations. A treatable problem size can quickly become large-scale due to the continual advancement of computational environments. If the number of floating-point operations increases, then problems caused by rounding errors become increasingly critical. In the worst case, an approximate solution obtained by a numerical computation can be inaccurate. Therefore, verified numerical computations are becoming increasingly important. This paper presents a survey of the basics related to verified numerical computations. We focus on floating-point arithmetic, interval arithmetic, rounding error analyses, and error-free transformations of floating-point operations.
Inverse eigenvalue problems arise in a variety of applications, and thus various Newton's methods, which quadratically converge, have been developed both in theory and practice. Among many studies over thirty years, two extremely significant developments are found. Firstly, smooth matrix decompositions have been successfully applied since the 1990s. Secondly, a matrix multiplication based method has been recently proposed. In this paper, such efficient modern solvers are classified in the context of classical Newton's methods according to their mathematical formulations, and then the corresponding convergence theorems and their relationship are surveyed.
This paper gives an overview of verification methods for finite dimensional conic linear programming problems. Besides the computation of verified tight enclosures for unique non-degenerate solutions to well-posed conic linear programming instances, we discuss a rigorous treatment of problems with multiple or degenerate solutions. It will be further shown how a priori knowledge about certain boundedness qualifications can be exploited to efficiently compute verified bounds for the optimal objective value. The corresponding approach is applicable even to ill-posed programming problems. Examples from linear and semidefinite programming are used to illustrate the respective approaches and give further explanations. Another topic is the treatment of programming problems whose parameters are subject to uncertainties. Rough but inclusive estimates for the variability of the corresponding programming problems are given, and also a best and worst case analysis is taken into account. At the end of this paper, special consideration is given to typical issues when applying interval arithmetic in the context of conic linear programming. Different ways are shown on how to resolve these issues.
We replace the cubic characteristics in the Duffing equation by two line segments connected at a point and investigate how an angle of that broken line conducts bifurcations to periodic orbits. Firstly we discuss differences in periodic orbits between the Duffing equation and a forced planar system including the broken line. In the latter system, a grazing bifurcation split the parameter space into the linear and nonlinear response domains. Also, we show that bifurcations of non-resonant periodic orbits appeared in the former system are suppressed in the latter system. Secondly, we obtain bifurcation diagrams by changing a slant parameter of the broken line. We also find the parameter set that a homoclinic bifurcation arises and the corresponding horseshoe map. It is clarified that a grazing bifurcation and tangent bifurcations form boundaries between linear and nonlinear responses. Finally, we explore the piecewise linear functions that show the minimum bending angles exhibiting bifurcation and chaos.