1. Fast and Accurate Vector Operations.- 1.1 Introduction.- 1.1.1 Background.- 1.1.2 Historic Remarks.- 1.2 Implementation Principles.- 1.2.1 Solution A: Long Adder and Long Shift.- 1.2.2 Solution B: Short Adder with Local Memory on the Arithmetic Unit.- 1.2.3 Remarks.- 1.2.4 Fast Carry Resolution.- 1.3 High-Performance Scalar Product Units (SPU).- 1.3.1 SPU for Computers with a 32 Bit Data Bus.- 1.3.2 SPU for Computers with a 64 Bit Data Bus.- 1.4 Comments on the Scalar Product Units.- 1.4.1 Rounding.- 1.4.2 How much Local Memory should be Provided on a SPU?.- 1.4.3 A SPU Instruction Set.- 1.4.4 Interaction with High Level Programming Languages ..- 1.5.1 Scalar Product Units for Top-Performance Computers.- 1.5.2 Long Adder for 64 Bit Data Word (Solution A).- 1.5.3 Long Adder for 32 Bit Data Word (Solution A).- 1.5.4 Short Adder with Local Memory on the Arithmetic Unit for 64 Bit Data Word (Solution B).- 1.5.1 Short Adder with Local Memory on the Arithmetic Unit for 32 Bit Data Word (Solution B).- 1.6 Hardware Accumulation Window.- 1.7 Theoretical Foundation of Advanced Computer Arithmetic.- Bibliography and Related Literature.- 2. Rounding Near Zero.- 2.1 The one dimensional case.- 2.2 Rounding in product spaces.- Bibliography and Related Literature.- 3. Interval Arithmetic Revisited.- 3.1 Introduction and Historical Remarks.- 3.2 Interval Arithmetic, a Powerful Calculus to Deal with Inequalities.- 3.3 Interval Arithmetic as Executable Set Operations.- 3.4 Enclosing the Range of Function Values.- 3.5 The Interval Newton Method.- 3.6 Extended Interval Arithmetic.- 3.7 The Extended Interval Newton Method.- 3.8 Differentiation Arithmetic, Enclosures of Derivatives.- 3.9 Interval Arithmetic on the Computer.- 3.10 Hardware Support for Interval Arithmetic.- 3.10.1 Addition A + B and Subtraction A ? B.- 3.10.2 Multiplication A * B.- 3.10.3 Interval Scalar Product Computation.- 3.10.4 Division A / B.- 3.10.5 Instruction Set for Interval Arithmetic.- 3.10.6 Final Remarks.- Bibliography and Related Literature.