Title | Perspectives on Enclosure Methods [electronic resource] / edited by Ulrich Kulisch, Rudolf Lohner, Axel Facius |
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Imprint | Vienna : Springer Vienna : Imprint: Springer, 2001 |
Connect to | http://dx.doi.org/10.1007/978-3-7091-6282-8 |
Descript | XII, 345 p. online resource |
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 โChaos conjecturesโ -- 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 Mooreโs 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 Mooreโ Interval Integration Method Really be? -- 1 Introduction -- 2 Mooreโs 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 Lohnerโs 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