Front Cover; A New Approach to Scientific Computation; Copyright Page; Dedication; Table of Contents; Contributors; Preface; Acknowledgments; CHAPTER 1. A 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; REFERENCES; CHAPTER 2. COMPUTER DEMONSTRATION PACKAGES FOR STANDARD PROBLEMS OF NUMERICAL MATHEMATICS

Language extension PASCAL-SCComputing 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 equations; CHAPTER 3. SOLVING 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 PROBLEM9. REAL AND COMPLEX ZEROS OF POLYNOMIALS; 10. LINEAR, QUADRATIC AND CONVEX PROGRAMMING; 11. ARITHMETIC EXPRESSIONS; CONCLUSIONS; REFERENCES; CHAPTER 4. EVALUATION OF ARITHMETIC EXPRESSION SWITH MAXIMUM ACCURACY; INTRODUCTION; 1. EVALUATION OF POLYNOMIALS; 2. EVALUATION OF ARBITRARY ARITHMETIC EXPRESSIONS; 3. NUMERICAL RESULTS; REFERENCES; CHAPTER 5. SOLVING 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 ITERATIONS6. APPLICATIONS TO DIFFERENTIAL AND INTEGRAL EQUATIONS; 7. SOME EXAMPLES; REFERENCES; CHAPTER 6. ULTRA-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; REFERENCES; CHAPTER 7. A FORTRAN EXTENSION FOR SCIENTIFIC COMPUTATION; 1. MOTIVATION; 2. NOTATION OF THE LANGUAGE EXTENSION; 3. SYNTAX AND SEMANTICS OF THE EXTENSION; REFERENCES; CHAPTER 8. AN INTRODUCTION TO MATRIX PASCAL:A PASCAL EXTENSION FOR SCIENTIFIC COMPUTATION

A. DATA TYPESB. EXPRESSIONS; C. PROCEDURES, FUNCTIONS, OPERATORS; D. UNIVERSAL OPERATOR CONCEPT; E. EXPRESSIONS WITH MAXIMUM ACCURACY; F. STANDARD FUNCTIONS; REFERENCES; CHAPTER 9. REALIZATION 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; REFERENCES; CHAPTER 10. FEATURES OF A HARDWARE IMPLEMENTATION OF AN OPTIMAL ARITHMETIC; 1. INTRODUCTION; 2. IMPLEMENTATION OF SCALAR PRODUCTS