Ebook: Scientific Computation with Automatic Result Verification

Scientific Computation with Result Verification has been a persevering research topic at the Institute for Applied Mathematics of Karlsruhe University for many years. A good number of meetings have been devoted to this area. The latest of these meetings was held from 30 September to 2 October, 1987, in Karlsruhe; it was co-sponsored by the GAMM Committee on "Computer Arithmetic and Scientific Computation". - - This volume combines edited versions of selected papers presented at this confer­ ence, including a few which were presented at a similar meeting one year earlier. The selection was made on the basis of relevance to the topic chosen for this volume. All papers are original contributions. In an appendix, we have supplied a short account of the Fortran-SC language which permits the programming of algorithms with result verification in a natural manner. The editors hope that the publication of this material as a Supplementum of Computing will further stimulate the interest of the scientific community in this important tool for Scientific Computation. In particular, we would like to make application scientists aware of its potential. The papers in the second chapter of this volume should convince them that automatic result verification may help them to design more reliable software for their particular tasks. We wish to thank all contributors for adapting their manuscripts to the goals of this volume. We are also grateful to the Publisher, Springer-Verlag of Vienna, for an efficient and quick production.

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This Computing Supplementum collects a number of original contributions which all aim to compute rigorous and reliable error bounds for the solution of numerical problems. An introductory article by the editors about the meaning and diverse methods of automatic result verification is followed by 16 original contributions. The first chapter deals with automatic result verification for standard mathematical problems, such as enclosing the solution of ordinary boundary value problems, linear programming problems, linear systems of equations and eigenvalue problems. The second chapter deals with applications of result verification methods to problems of the technical sciences. The contributions consider critical bending vibrations stability tests for periodic differential equations, geometric algorithms in the plane, and the periodic solution of the oregonator, a mathematical model in chemical kinetics. The contributions of the third chapter are concerned with extending and developing the tools required in scientific computation with automatic result verification: evaluation of arithmetic expressions of polynomials in several variables and of standard functions for real and complex point and interval arguments with dynamic accuracy. As an appendix, a short account of the Fortran-SC language was added which permits the programming of algorithms with result verification in a natural manner.

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This Computing Supplementum collects a number of original contributions which all aim to compute rigorous and reliable error bounds for the solution of numerical problems. An introductory article by the editors about the meaning and diverse methods of automatic result verification is followed by 16 original contributions. The first chapter deals with automatic result verification for standard mathematical problems, such as enclosing the solution of ordinary boundary value problems, linear programming problems, linear systems of equations and eigenvalue problems. The second chapter deals with applications of result verification methods to problems of the technical sciences. The contributions consider critical bending vibrations stability tests for periodic differential equations, geometric algorithms in the plane, and the periodic solution of the oregonator, a mathematical model in chemical kinetics. The contributions of the third chapter are concerned with extending and developing the tools required in scientific computation with automatic result verification: evaluation of arithmetic expressions of polynomials in several variables and of standard functions for real and complex point and interval arguments with dynamic accuracy. As an appendix, a short account of the Fortran-SC language was added which permits the programming of algorithms with result verification in a natural manner.
Content:
Front Matter....Pages I-VIII
Automatic Result Verification....Pages 1-6
Front Matter....Pages 7-7
A Method for Producing Verified Results for Two-Point Boundary Value Problems....Pages 9-22
A Kind of Difference Methods for Enclosing Solutions of Ordinary Linear Boundary Value Problems....Pages 23-31
A Self-Validating Method for Solving Linear Programming Problems with Interval Input Data....Pages 33-45
Enclosing the Solutions of Systems of Linear Equations by Interval Iterative Processes....Pages 47-58
Errorbounds for Quadratic Systems of Nonlinear Equations Using the Precise Scalar Product....Pages 59-67
Inclusion of Eigenvalues of General Eigenvalue Problems for Matrices....Pages 69-78
Verified Inclusion for Eigenvalues of Certain Difference and Differential Equations....Pages 79-87
Front Matter....Pages 89-89
VIB — Verified Inclusions of Critical Bending Vibrations....Pages 91-98
Stability Test for Periodic Differential Equations on Digital Computers with Applications....Pages 99-110
The Periodic Solutions of the Oregonator and Verification of Results....Pages 111-121
On Arithmetical Problems of Geometric Algorithms in the Plane....Pages 123-136
Front Matter....Pages 137-137
Precise Evaluation of Polynomials in Several Variables....Pages 139-148
Evaluation of Arithmetic Expressions with Guaranteed High Accuracy....Pages 149-157
Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy....Pages 159-184
Inverse Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy....Pages 185-211
Inclusion Algorithms with Functions as Data....Pages 213-224
Front Matter....Pages 225-225
FORTRAN-SC A Study of a FORTRAN Extension for Engineering/Scientific Computation with Access to ACRITH....Pages 227-244

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This Computing Supplementum collects a number of original contributions which all aim to compute rigorous and reliable error bounds for the solution of numerical problems. An introductory article by the editors about the meaning and diverse methods of automatic result verification is followed by 16 original contributions. The first chapter deals with automatic result verification for standard mathematical problems, such as enclosing the solution of ordinary boundary value problems, linear programming problems, linear systems of equations and eigenvalue problems. The second chapter deals with applications of result verification methods to problems of the technical sciences. The contributions consider critical bending vibrations stability tests for periodic differential equations, geometric algorithms in the plane, and the periodic solution of the oregonator, a mathematical model in chemical kinetics. The contributions of the third chapter are concerned with extending and developing the tools required in scientific computation with automatic result verification: evaluation of arithmetic expressions of polynomials in several variables and of standard functions for real and complex point and interval arguments with dynamic accuracy. As an appendix, a short account of the Fortran-SC language was added which permits the programming of algorithms with result verification in a natural manner.
Content:
Front Matter....Pages I-VIII
Automatic Result Verification....Pages 1-6
Front Matter....Pages 7-7
A Method for Producing Verified Results for Two-Point Boundary Value Problems....Pages 9-22
A Kind of Difference Methods for Enclosing Solutions of Ordinary Linear Boundary Value Problems....Pages 23-31
A Self-Validating Method for Solving Linear Programming Problems with Interval Input Data....Pages 33-45
Enclosing the Solutions of Systems of Linear Equations by Interval Iterative Processes....Pages 47-58
Errorbounds for Quadratic Systems of Nonlinear Equations Using the Precise Scalar Product....Pages 59-67
Inclusion of Eigenvalues of General Eigenvalue Problems for Matrices....Pages 69-78
Verified Inclusion for Eigenvalues of Certain Difference and Differential Equations....Pages 79-87
Front Matter....Pages 89-89
VIB — Verified Inclusions of Critical Bending Vibrations....Pages 91-98
Stability Test for Periodic Differential Equations on Digital Computers with Applications....Pages 99-110
The Periodic Solutions of the Oregonator and Verification of Results....Pages 111-121
On Arithmetical Problems of Geometric Algorithms in the Plane....Pages 123-136
Front Matter....Pages 137-137
Precise Evaluation of Polynomials in Several Variables....Pages 139-148
Evaluation of Arithmetic Expressions with Guaranteed High Accuracy....Pages 149-157
Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy....Pages 159-184
Inverse Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy....Pages 185-211
Inclusion Algorithms with Functions as Data....Pages 213-224
Front Matter....Pages 225-225
FORTRAN-SC A Study of a FORTRAN Extension for Engineering/Scientific Computation with Access to ACRITH....Pages 227-244
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