Perspectives on Enclosure Methods

Enclosure methods and their applications have been developed to a high standard during the last decades. These methods guarantee the validity of the computed results. This means they are of the same standard as the rest of mathematics. The book deals with a wide variety of aspects of enclosure methods. All contributions follow the common goal to push the limits of enclosure methods forward. Topics that are treated include basic questions of arithmetic, proving conjectures, bounds for Krylow type linear system solvers, bounds for eigenvalues, the wrapping effect, algorithmic differencing, differential equations, finite element methods, application in robotics, and nonsmooth global optimization.

Proving Conjectures by Use of Interval Arithmetic.- 1 Computer Assisted Proofs in Analysis.- 2 The Kepler Conjecture.- 3 The Double Bubble Conjecture.- 4 The Dirac-Schwinger Conjecture.- 5 ‘Chaos conjectures’.- References.- Advanced Arithmetic for the Digital Computer Interval Arithmetic Revisited.- 1 Introduction and Historical Remarks.- 2 Interval Arithmetic, a Powerful Calculus to Deal with Inequalities.- 3 Interval Arithmetic as Executable Set Operations.- 4 Enclosing the Range of Function Values.- 5 The Interval Newton Method.- 6 Extended Interval Arithmetic.- 7 The Extended Interval Newton Method.- 8 Differentiation Arithmetic, Enclosures of Derivatives.- 9 Interval Arithmetic on the Computer.- 10 Hardware Support for Interval Arithmetic.- References.- Highly Accurate Verified Error Bounds for Krylov Type Linear System Solvers.- 1 Introduction.- 2 Iterative Solvers and Finite Precision.- 3 Krylov Subspace Methods.- 4 Improved Arithmetic.- 5 Verified Error Bounds.- 6 Computational Results.- References.- Elements of Scientific Computing.- 1 Hardware Requirements.- 2 Software Requirements.- 3 Modelling Requirements.- 4 Conclusion.- References.- Biography.- The Mainstreaming of Interval Arithmetic.- 1 Introduction.- 2 Moore’s Law and Precision.- 3 Interval Physics.- 4 Summary.- References.- Bounds for Eigenvalues with the Use of Finite Elements.- 1 Introduction.- 2 Setting for the Problem.- 3 Calculation of Bounds.- 4 Verified Computation.- 5 Application: the Membrane Problem.- 6 Numerical Examples.- References.- Algorithmic Differencing.- 1 Algorithmic Representation of Functions.- 2 Transformation of Algorithms.- 3 Finite Precision Calculations.- 4 First Order Difference Operators.- 5 Differences of Inverse Functions.- 6 Higher Order Divided Differences.- References.- A Comparison of Techniques for Evaluating Centered Forms.- 1 Introduction.- 2 Methods for Computing Slope Vectors.- 3 A Numerical Example.- 4 Summary and Recommendations.- References.- On the Limit of the Total Step Method in Interval Analysis.- 1 Introduction.- 2 Notations.- 3 Results.- References.- How Fast can Moore’ Interval Integration Method Really be?.- 1 Introduction.- 2 Moore’s Algorithm.- 3 Estimation of the Integration Error.- 4 Conclusions.- References.- Numerical Verification and Validation of Kinematics and Dynamical Models for Flexible Robots in Complex Environments.- 1 Introduction.- 2 Error Propagation Control and Reliable Numerical Algorithms in MOBILE.- 3 Verified Calculation of the Solution of Discrete-Time Algebraic Riccati Equation.- 4 Accurate Distance Calculation Algorithms.- 5 Accurate Robot Reliability Estimation.- 6 Further Work.- 7 Acknowledgement.- References.- On the Ubiquity of the Wrapping Effect in the Computation of Error Bounds.- 1 Introduction or What is the Wrapping Effect?.- 2 Where does the Wrapping Effect appear?.- 3 How can we Reduce the Wrapping Effect?.- 4 Conclusion.- References.- A New Perspective on the Wrapping Effect in Interval Methods for Initial Value Problems for Ordinary Differential Equations.- 1 Introduction.- 2 Preliminaries.- 3 How the Wrapping Effect Arises in Interval Methods for IVPs for ODEs: A Traditional Explanation.- 4 The Wrapping Effect as a Source of Instability in Interval Methods for IVPs for ODEs.- 5 The Parallelepiped and Lohner’s QR-Factorization Methods.- 6 Why the Parallelepiped Method Often Fails.- 7 When Does the Parallelepiped Method Work Well.- 8 How the QR Method Improves Stability.- 9 Conclusions.- A Lemmas.- References.- A Guaranteed Bound of the Optimal Constant in the Error Estimates for Linear Triangular Elements.- 1 Introduction.- 2 Strategy.- 3 The Method to Calculate a Rigorous Solution.- 4 Checking the Condition.- 5 Some Computational Techniques for Efficient Enclosure Methods.- 6 Numerical Results.- References.- Nonsmooth Global Optimization.- 1 Introduction.- 2 Preliminaries.- 3 A Pruning Technique for Global Optimizationx.- 4 Multidimensional Pruning Techniques for Global Optimization.- References.