This book constitutes the refereed post proceedings of the 16th International Symposium, SCAN 2014, held in Würzburg, Germany, in September 2014. The 22 full papers presented were carefully reviewed and selected from 60 submissions. The main concerns of research addressed by SCAN conferences are validation, verification or reliable assertions of numerical computations. Interval arithmetic and other treatments of uncertainty are developed as appropriate tools.

The major emphasis of the Dagstuhl Seminar on “Numerical Validation in C- rent Hardware Architectures” lay on numerical validation in current hardware architecturesand softwareenvironments. The generalidea wasto bring together experts who are concerned with computer arithmetic in systems with actual processor architectures and scientists who develop, use, and need techniques from veri?ed computation in their applications. Topics of the seminar therefore included: – The ongoing revision of the IEEE 754/854 standard for ?oating-point ari- metic – Feasible ways to implement multiple precision (multiword) arithmetic and to compute the actual precision at run-time according to the needs of input data – The achievement of a similar behavior of ?xed-point, ?oating-point and - terval arithmetic across language compliant implementations – The design of robust and e?cient numerical programsportable from diverse computers to those that adhere to the IEEE standard – The development and propagation of validated special-purpose software in di?erent application areas – Error analysis in several contexts – Certi?cation of numerical programs, veri?cation and validation assessment Computer arithmetic plays an important role at the hardware and software level, when microprocessors, embedded systems, or grids are designed. The re- ability of numerical softwarestrongly depends on the compliance with the cor- sponding ?oating-point norms. Standard CISC processors follow the 1985 IEEE norm 754, which is currently under revision, but the new highly performing CELL processor is not fully IEEE compliant.

The International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics SCAN is held biannually, the fourth conference took place in Wuppertal 1995. This volume contains contributions from outstanding research specialists based on their presentations at SCAN-95. It covers all aspects of scientific computing with validation, starting with the latest developments in the design of floating point units together with algorithms for floating point operations and elementary function evaluations with maximum accuracy. The book continues by treating scientific computing methods for many areas of applied mathematics such as numerical linear algebra, nonlinear equations, global optimization, ordinary and partial differential equations and dynamical systems. Some computer science aspects like complexity are also considered as are examples where validation methods have successfully be used in applications from the engineering sciences.

The SCAN conference, the International Symposium on Scientific Com puting, Computer Arithmetic and Validated Numerics, takes place bian nually under the joint auspices of GAMM (Gesellschaft fiir Angewandte Mathematik und Mechanik) and IMACS (International Association for Mathematics and Computers in Simulation). SCAN-98 attracted more than 100 participants from 21 countries all over the world. During the four days from September 22 to 25, nine highlighted, plenary lectures and over 70 contributed talks were given. These figures indicate a large participation, which was partly caused by the attraction of the organizing country, Hungary, but also the effec tive support system have contributed to the success. The conference was substantially supported by the Hungarian Research Fund OTKA, GAMM, the National Technology Development Board OMFB and by the J6zsef Attila University. Due to this funding, it was possible to subsidize the participation of over 20 scientists, mainly from Eastern European countries. It is important that the possibly first participation of 6 young researchers was made possible due to the obtained support. The number of East-European participants was relatively high. These results are especially valuable, since in contrast to the usual 2 years period, the present meeting was organized just one year after the last SCAN-xx conference.

Scan 2000, the GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics and Interval 2000, the International Conference on Interval Methods in Science and Engineering were jointly held in Karlsruhe, September 19-22, 2000. The joint conference continued the series of 7 previous Scan-symposia under the joint sponsorship of GAMM and IMACS. These conferences have traditionally covered the numerical and algorithmic aspects of scientific computing, with a strong emphasis on validation and verification of computed results as well as on arithmetic, programming, and algorithmic tools for this purpose. The conference further continued the series of 4 former Interval conferences focusing on interval methods and their application in science and engineering. The objectives are to propagate current applications and research as well as to promote a greater understanding and increased awareness of the subject matters. The symposium was held in Karlsruhe the European cradle of interval arithmetic and self-validating numerics and attracted 193 researchers from 33 countries. 12 invited and 153 contributed talks were given. But not only the quantity was overwhelming we were deeply impressed by the emerging maturity of our discipline. There were many talks discussing a wide variety of serious applications stretching all parts of mathematical modelling. New efficient, publicly available or even commercial tools were proposed or presented, and also foundations of the theory of intervals and reliable computations were considerably strengthened.

In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information. The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense. In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.

The articles in this book give a comprehensive overview on the whole field of validated numerics. The problems covered include simultaneous systems of linear and nonlinear equations, differential and integral equations and certain applications from technical sciences. Furthermore some papers which improve the tools are included. The book is a must for scientists working in numerical analysis, computer science and in technical fields.

Notes and Reports in Mathematics in Science and Engineering, Volume VII: Computer Arithmetic and Self-Validating Numerical Methods compiles papers presented at the first international conference on "Computer Arithmetic and Self-Validating Numerical Methods, held in Basel from October 2 to 6, 1989. This book begins by providing a tutorial introduction to computer arithmetic with operations of maximum accuracy, differentiation arithmetic and enclosure methods, and programming languages for self-validating numerical methods. The rest of the chapters discuss the determination of guaranteed bounds for eigenvalues by variational methods and guaranteed inclusion of solutions of differential equations. An appendix covering the IMACS-GAMM resolution on computer arithmetic is provided at the end of this publication. This volume is recommended for researchers and professionals working on computer arithmetic and self-validating numerical methods.

Floating-point arithmetic is ubiquitous in modern computing, as it is the tool of choice to approximate real numbers. Due to its limited range and precision, its use can become quite involved and potentially lead to numerous failures. One way to greatly increase confidence in floating-point software is by computer-assisted verification of its correctness proofs. This book provides a comprehensive view of how to formally specify and verify tricky floating-point algorithms with the Coq proof assistant. It describes the Flocq formalization of floating-point arithmetic and some methods to automate theorem proofs. It then presents the specification and verification of various algorithms, from error-free transformations to a numerical scheme for a partial differential equation. The examples cover not only mathematical algorithms but also C programs as well as issues related to compilation. - Describes the notions of specification and weakest precondition computation and their practical use - Shows how to tackle algorithms that extend beyond the realm of simple floating-point arithmetic - Includes real analysis and a case study about numerical analysis

To describe the true behavior of most real-world systems with sufficient accuracy, engineers have to overcome difficulties arising from their lack of knowledge about certain parts of a process or from the impossibility of characterizing it with absolute certainty. Depending on the application at hand, uncertainties in modeling and measurements can be represented in different ways. For example, bounded uncertainties can be described by intervals, affine forms or general polynomial enclosures such as Taylor models, whereas stochastic uncertainties can be characterized in the form of a distribution described, for example, by the mean value, the standard deviation and higher-order moments. The goal of this Special Volume on Modeling, Design, and Simulation of Systems with Uncertainties is to cover modern methods for dealing with the challenges presented by imprecise or unavailable information. All contributions tackle the topic from the point of view of control, state and parameter estimation, optimization and simulation. Thematically, this volume can be divided into two parts. In the first we present works highlighting the theoretic background and current research on algorithmic approaches in the field of uncertainty handling, together with their reliable software implementation. The second part is concerned with real-life application scenarios from various areas including but not limited to mechatronics, robotics, and biomedical engineering.