A New Approach to Scientific Computation

ContributorsPrefaceAcknowledgmentsA New Arithmetic for Scientific Computation 1. Introduction 2. The Spaces of Numerical Computations 3. Traditional Definition of Computer Arithmetic: The Vertical Method 4. The New Definition of Computer Arithmetic: The Horizontal Method 5. Computer Arithmetic and Programming Languages 6. Realization and Applications ReferencesComputer Demonstration Packages for Standard Problems of Numerical Mathematics Language Extension PASCAL-SC Computing Inclusions-Old and New Precise Dot Product Linear Systems Inversion of a Matrix Eigenproblems Rounding Error and Cancellation Evaluation of a Polynomial Zero of a Polynomial Polynomial Package Arithmetic Expressions Systems of Non-linear Equations Differential EquationsSolving Algebraic Problems with High Accuracy Introduction 1. Computer Arithmetic 2. Linear Systems 3. Over- and Underdetermined Linear Systems 4. Linear Systems with Band Matrices 5. Sparse Linear Systems 6. Matrix Inversion 7. Non-linear Systems 8. The Algebraic Eigenvalue Problem 9. Real and Complex Zeros of Polynomials 10. Linear, Quadratic, and Convex Programming 11. Arithmetic Expressions Conclusions ReferencesEvaluation of Arithmetic Expressions with Maximum Accuracy Introduction 1. Evaluation of Polynomials 2. Evaluation of Arbitrary Arithmetic Expressions 3. Numerical Results ReferencesSolving Function Space Problems with Guaranteed Close Bounds 1. Introduction 2. Mathematical Preliminaries 3. Practical Use of the Fixed Point Theorems 4. Functional Arithmetic and Roundings 5. Algorithmic Execution of Iterations 6. Applications to Differential and Integral Equations 7. Some Examples ReferencesUltra-Arithmetic: The Digital Computer Set in Function Space 1. Introduction 2. A Review of Ultra-Arithmetic 3. Applications of Ultra-Arithmetic 4. The Arithmetic of Intervals of Polynomials ReferencesA FORTRAN Extension for Scientific Computation 1. Motivation 2. Notation of the Language Extension 3. Syntax and Semantics of the Extension ReferencesAn Introduction to MATRIX PASCAL: A PASCAL Extension for Scientific Computation A. Data Types B. Expressions C. Procedures, Functions, Operators D. Universal Operator Concept E. Expressions with Maximum Accuracy F. Standard Functions ReferencesRealization of an Optimal Computer Arithmetic 1. Introduction-Mathematical Foundations 2. Organization of the Arithmetic 3. Implementation of the Elementary Operations 4. Operations in the Higher Spaces 5. Realization on a Micro Computer ReferencesFeatures of a Hardware Implementation of an Optimal Arithmetic 1. Introduction 2. Implementation of Scalar Products 3. Algorithmic and Flowchart Description of a Hardware Unit 4. Parallelism in Scalar Products 5. Pipelining of the Arithmetic Operations Appendix ReferencesDifferentiation and Generation of Taylor Coefficients in PASCAL-SC 1. Automation of Evaluation and Differentiation of Functions 2. Derivative Data Types 3. Derivative Operators 4. Examples of Multiplication Operators in PASCAL-SC 5. Standard Derivative Functions 6. Applications of Derivative Types in Scientific Computation ReferencesMatrix Pascal A. Introduction B. Spaces, Operations, and Data Types C. Informal Description of the Language Extension D. Formal Description of MATRIX PASCAL E. Comments on the Implementation of the Arithmetic References