Computer Arithmetic and Validity

1 - Foreword to the second edition [Seite 7]
2 - Preface [Seite 9]
3 - Introduction [Seite 23]
4 - I Theory of computer arithmetic [Seite 33]
4.1 - 1 First concepts [Seite 35]
4.1.1 - 1.1 Ordered sets [Seite 35]
4.1.2 - 1.2 Complete lattices and complete subnets [Seite 40]
4.1.3 - 1.3 Screens and roundings [Seite 46]
4.1.4 - 1.4 Arithmetic operations and roundings [Seite 57]
4.2 - 2 Ringoids and vectoids [Seite 65]
4.2.1 - 2.1 Ringoids [Seite 65]
4.2.2 - 2.2 Vectoids [Seite 76]
4.3 - 3 Definition of computer arithmetic [Seite 84]
4.3.1 - 3.1 Introduction [Seite 84]
4.3.2 - 3.2 Preliminaries [Seite 87]
4.3.3 - 3.3 The traditional definition of computer arithmetic [Seite 91]
4.3.4 - 3.4 Definition of computer arithmetic by semimorphisms [Seite 92]
4.3.5 - 3.5 A remark about roundings [Seite 100]
4.3.6 - 3.6 Uniqueness of the minus operator [Seite 101]
4.3.7 - 3.7 Rounding near zero [Seite 103]
4.4 - 4 Interval arithmetic [Seite 109]
4.4.1 - 4.1 Interval sets and arithmetic [Seite 110]
4.4.2 - 4.2 Interval arithmetic over a linearly ordered set [Seite 119]
4.4.3 - 4.3 Interval matrices [Seite 123]
4.4.4 - 4.4 Interval vectors [Seite 129]
4.4.5 - 4.5 Interval arithmetic on a screen [Seite 132]
4.4.6 - 4.6 Interval matrices and interval vectors on a screen [Seite 140]
4.4.7 - 4.7 Complex interval arithmetic [Seite 148]
4.4.8 - 4.8 Complex interval matrices and interval vectors [Seite 154]
4.4.9 - 4.9 Extended interval arithmetic [Seite 159]
4.4.10 - 4.10 Exception-free arithmetic for extended intervals [Seite 163]
4.4.11 - 4.11 Extended interval arithmetic on the computer [Seite 168]
4.4.12 - 4.12 Exception-free arithmetic for closed real intervals on the computer [Seite 171]
4.4.13 - 4.13 Comparison relations and lattice operations [Seite 174]
4.4.14 - 4.14 Algorithmic implementation of interval multiplication and division [Seite 175]
5 - II Implementation of arithmetic on computers [Seite 177]
5.1 - 5 Floating-point arithmetic [Seite 179]
5.1.1 - 5.1 Definition and properties of the real numbers [Seite 179]
5.1.2 - 5.2 Floating-point numbers and roundings [Seite 185]
5.1.3 - 5.3 Floating-point operations [Seite 194]
5.1.4 - 5.4 Subnormal floating-point numbers [Seite 202]
5.1.5 - 5.5 On the IEEE floating-point arithmetic standard [Seite 203]
5.2 - 6 Implementation of floating-point arithmetic on a computer [Seite 213]
5.2.1 - 6.1 A brief review of the realization of integer arithmetic [Seite 214]
5.2.2 - 6.2 Introductory remarks about the level 1 operations [Seite 223]
5.2.3 - 6.3 Addition and subtraction [Seite 228]
5.2.4 - 6.4 Normalization [Seite 232]
5.2.5 - 6.5 Multiplication [Seite 234]
5.2.6 - 6.6 Division [Seite 234]
5.2.7 - 6.7 Rounding [Seite 236]
5.2.8 - 6.8 A universal rounding unit [Seite 238]
5.2.9 - 6.9 Overflow and underflow treatment [Seite 239]
5.2.10 - 6.10 Algorithms using the short accumulator [Seite 242]
5.2.11 - 6.11 The level 2 operations [Seite 248]
5.3 - 7 Hardware support for interval arithmetic [Seite 258]
5.3.1 - 7.1 Introduction [Seite 258]
5.3.2 - 7.2 Arithmetic interval operations [Seite 259]
5.3.2.1 - 7.2.1 Algebraic operations [Seite 260]
5.3.2.2 - 7.2.2 Comments on the algebraic operations [Seite 262]
5.3.3 - 7.3 Circuitry for the arithmetic interval operations [Seite 263]
5.3.4 - 7.4 Comparisons and lattice operations [Seite 264]
5.3.4.1 - 7.4.1 Comments on comparisons and lattice operations [Seite 265]
5.3.4.2 - 7.4.2 Hardware support for comparisons and lattice operations [Seite 265]
5.3.5 - 7.5 Alternative circuitry for interval operations and comparisons [Seite 266]
5.3.5.1 - 7.5.1 Hardware support for interval arithmetic on x86-processors [Seite 267]
5.3.5.2 - 7.5.2 Accurate evaluation of interval scalar products [Seite 269]
5.4 - 8 Scalar products and complete arithmetic [Seite 271]
5.4.1 - 8.1 Introduction and motivation [Seite 272]
5.4.2 - 8.2 Historical remarks [Seite 274]
5.4.3 - 8.3 The ubiquity of the scalar product in numerical analysis [Seite 279]
5.4.4 - 8.4 Implementation principles [Seite 282]
5.4.4.1 - 8.4.1 Long adder and long shift [Seite 284]
5.4.4.2 - 8.4.2 Short adder with local memory on the arithmetic unit [Seite 284]
5.4.4.3 - 8.4.3 Remarks [Seite 285]
5.4.4.4 - 8.4.4 Fast carry resolution [Seite 287]
5.4.5 - 8.5 Informal sketch for computing an exact dot product [Seite 289]
5.4.6 - 8.6 Scalar product computation units (SPUs) [Seite 289]
5.4.6.1 - 8.6.1 SPU for computers with a 32 bit data bus [Seite 291]
5.4.6.2 - 8.6.2 A coprocessor chip for the exact scalar product [Seite 294]
5.4.6.3 - 8.6.3 SPU for computers with a 64 bit data bus [Seite 297]
5.4.7 - 8.7 Comments [Seite 300]
5.4.7.1 - 8.7.1 Rounding [Seite 300]
5.4.7.2 - 8.7.2 How much local memory should be provided on an SPU? [Seite 301]
5.4.8 - 8.8 The data format complete and complete arithmetic [Seite 303]
5.4.8.1 - 8.8.1 Low level instructions for complete arithmetic [Seite 304]
5.4.8.2 - 8.8.2 Complete arithmetic in high level programming languages [Seite 305]
5.4.9 - 8.9 Top speed scalar product units [Seite 309]
5.4.9.1 - 8.9.1 SPU with long adder for 64 bit data word [Seite 309]
5.4.9.2 - 8.9.2 SPU with long adder for 32 bit data word [Seite 314]
5.4.9.3 - 8.9.3 An FPGA coprocessor for the exact scalar product [Seite 317]
5.4.9.4 - 8.9.4 SPU with short adder and complete register [Seite 317]
5.4.9.5 - 8.9.5 Carry-free accumulation of products in redundant arithmetic [Seite 323]
5.4.10 - 8.10 Hardware complete register window [Seite 324]
6 - III Principles of verified computing [Seite 327]
6.1 - 9 Sample applications [Seite 329]
6.1.1 - 9.1 Basic properties of interval mathematics [Seite 331]
6.1.1.1 - 9.1.1 Interval arithmetic, a powerful calculus to deal with inequalities [Seite 331]
6.1.1.2 - 9.1.2 Interval arithmetic as executable set operations [Seite 332]
6.1.1.3 - 9.1.3 Enclosing the range of function values [Seite 338]
6.1.1.4 - 9.1.4 Nonzero property of a function, global optimization [Seite 341]
6.1.2 - 9.2 Differentiation arithmetic, enclosures of derivatives [Seite 343]
6.1.3 - 9.3 The interval Newton method [Seite 351]
6.1.4 - 9.4 The extended interval Newton method [Seite 354]
6.1.5 - 9.5 Verified solution of systems of linear equations [Seite 355]
6.1.6 - 9.6 Accurate evaluation of arithmetic expressions [Seite 362]
6.1.6.1 - 9.6.1 Complete expressions [Seite 363]
6.1.6.2 - 9.6.2 Accurate evaluation of polynomials [Seite 364]
6.1.6.3 - 9.6.3 Arithmetic expressions [Seite 368]
6.1.7 - 9.7 Multiple precision arithmetics [Seite 369]
6.1.7.1 - 9.7.1 Multiple precision floating-point arithmetic [Seite 370]
6.1.7.2 - 9.7.2 Multiple precision interval arithmetic [Seite 373]
6.1.7.3 - 9.7.3 Applications [Seite 378]
6.1.7.4 - 9.7.4 Adding an exponent part as a scaling factor to complete arithmetic [Seite 380]
6.1.8 - 9.8 Remarks on Kaucher arithmetic [Seite 382]
6.1.8.1 - 9.8.1 The basic operations of Kaucher arithmetic [Seite 386]
7 - A Frequently used symbols [Seite 389]
8 - B On homomorphism [Seite 391]
9 - Bibliography [Seite 393]
10 - List of figures [Seite 443]
11 - List of tables [Seite 447]
12 - Index [Seite 449]