Overview

Numerical Toolbox for Verified Computing presents an exten- sive set of sophisticated tools to solve basic numerical problems with a verification of the results using the featu- res of the scientific computer language PASCAL-XSC. The overriding concern of this book is reliability - the automa- tic verification of the result a computer returns for a gi- ven problem. This book is the first to offer a general dis- cussion on arithmetic and computational reliability, analy- tical mathematics and verification techniques, algorithms, and (most importantly) actual implementations in the form of working computer routines. In each chapter, examples, exer- cises, and numerical results demonstrate the application of the routines presented. The book introduces many computatio- nal verification techniques. It is not assumed that the rea- der has any prior formal knowledge of numerical verification or any familiarity with interval analysis. The necessary concepts are introduced. Some of the subjects that the book covers in detail are not usually found in standard numerical analysis texts. This book is intended primarily as a refe- rence text for anyone wishing to apply, modify, or develop routines to obtain mathematically certain and reliable re- sults. It can also be used as a textbook for an advanced course in scientific computation with automatic result veri- fication.

Full Product Details

Author:   Rolf Hammer ,  Matthias Hocks ,  Ulrich W. Kulisch ,  Dietmar Ratz
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Volume:   v. 21
Weight:   0.670kg
ISBN:  

9783540571186

ISBN 10:   3540571183
Pages:   354
Publication Date:   20 December 1993
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available.

Table of Contents

1 Introduction.- 1 Introduction.- 1.1 Advice for Quick Readers.- 1.2 Structure of the Book.- 1.3 Typography.- 1.4 Algorithmic Notation.- 1.5 Implementation.- 1.6 Computational Environment.- 1.7 Why Numerical Result Verification?.- 1.7.1 A Brief History of Computing.- 1.7.2 Arithmetic on Computers.- 1.7.3 Extensions of Ordinary Floating-Point Arithmetic.- 1.7.4 Scientific Computation with Automatic Result Verification...- 1.7.5 Program Verification versus Numerical Verification.- I Preliminaries.- 2 The Features of PASCAL-XSC.- 2.1 Predefined Data Types, Operators, and Functions.- 2.2 The Universal Operator Concept.- 2.3 Overloading of Procedures, Functions, and Operators.- 2.4 Module Concept.- 2.5 Dynamic Arrays and Subarrays.- 2.6 Data Conversion.- 2.7 Accurate Expressions (#-Expressions).- 2.8 The String Concept.- 2.9 Predefined Arithmetic Modules.- 2.10 Why PASCAL-XSC?.- 3 Mathematical Preliminaries.- 3.1 Real Interval Arithmetic.- 3.2 Complex Interval Arithmetic.- 3.3 Extended Interval Arithmetic.- 3.4 Interval Vectors and Matrices.- 3.5 Floating-Point Arithmetic.- 3.6 Floating-Point Interval Arithmetic.- 3.7 The Problem of Data Conversion.- 3.8 Principles of Numerical Verification.- II One-Dimensional Problems.- 4 Evaluation of Polynomials.- 4.1 Theoretical Background.- 4.1.1 Description of the Problem.- 4.1.2 Iterative Solution.- 4.2 Algorithmic Description.- 4.3 Implementation and Examples.- 4.3.1 PASCAL-XSC Program Code.- 4.3.1.1 Module rpoly.- 4.3.1.2 Module rpeval.- 4.3.2 Examples.- 4.3.3 Restrictions and Hints.- 4.4 Exercises.- 4.5 References and Further Reading.- 5 Automatic Differentiation.- 5.1 Theoretical Background.- 5.2 Algorithmic Description.- 5.3 Implementation and Examples.- 5.3.1 PASCAL-XSC Program Code.- 5.3.1.1 Module ddf_ari.- 5.3.2 Examples.- 5.3.3 Restrictions and Hints.- 5.4 Exercises.- 5.5 References and Further Reading.- 6 Nonlinear Equations in One Variable.- 6.1 Theoretical Background.- 6.2 Algorithmic Description.- 6.3 Implementation and Examples.- 6.3.1 PASCAL-XSC Program Code.- 6.3.1.1 Module xi_ari.- 6.3.1.2 Module nlfzero.- 6.3.2 Example.- 6.3.3 Restrictions and Hints.- 6.4 Exercises.- 6.5 References and Further Reading.- 7 Global Optimization.- 7.1 Theoretical Background.- 7.1.1 Midpoint Test.- 7.1.2 Monotonicity Test.- 7.1.3 Concavity Test.- 7.1.4 Interval Newton Step.- 7.1.5 Verification.- 7.2 Algorithmic Description.- 7.3 Implementation and Examples.- 7.3.1 PASCAL-XSC Program Code.- 7.3.1.1 Module 1st1_ari.- 7.3.1.2 Module gopl.- 7.3.2 Examples.- 7.3.3 Restrictions and Hints.- 7.4 Exercises.- 7.5 References and Further Reading.- 8 Evaluation of Arithmetic Expressions.- 8.1 Theoretical Background.- 8.1.1 A Nonlinear Approach.- 8.2 Algorithmic Description.- 8.3 Implementation and Examples.- 8.3.1 PASCAL-XSC Program Code.- 8.3.1.1 Module expreval.- 8.3.2 Examples.- 8.3.3 Restrictions, Hints, and Improvements.- 8.4 Exercises.- 8.5 References and Further Reading.- 9 Zeros of Complex Polynomials.- 9.1 Theoretical Background.- 9.1.1 Description of the Problem.- 9.1.2 Iterative Approach.- 9.2 Algorithmic Description.- 9.3 Implementation and Examples.- 9.3.1 PASCAL-XSC Program Code.- 9.3.1.1 Module cpoly.- 9.3.1.2 Module cipoly.- 9.3.1.3 Module cpzero.- 9.3.2 Example.- 9.3.3 Restrictions and Hints.- 9.4 Exercises.- 9.5 References and Further Reading.- III Multi-Dimensional Problems.- 10 Linear Systems of Equations.- 10.1 Theoretical Background.- 10.1.1 A Newton-like Method.- 10.1.2 The Residual Iteration Scheme.- 10.1.3 How to Compute the Approximate Inverse.- 10.2 Algorithmic Description.- 10.3 Implementation and Examples.- 10.3.1 PASCAL-XSC Program Code.- 10.3.1.1 Module matinv.- 10.3.1.2 Module linsys.- 10.3.2 Example.- 10.3.3 Restrictions and Improvements.- 10.4 Exercises.- 10.5 References and Further Reading.- 11 Linear Optimization.- 11.1 Theoretical Background.- 11.1.1 Description of the Problem.- 11.1.2 Verification.- 11.2 Algorithmic Description.- 11.3 Implementation and Examples.- 11.3.1 PASCAL-XSC Program Code.- 11.3.1.1 Module lop_ari.- 11.3.1.2 Module rev_simp.- 11.3.1.3 Module lop.- 11.3.2 Examples.- 11.3.3 Restrictions and Hints.- 11.4 Exercises.- 11.5 References and Further Reading.- 12 Automatic Differentiation for Gradients, Jacobians, and Hessians.- 12.1 Theoretical Background.- 12.2 Algorithmic Description.- 12.3 Implementation and Examples.- 12.3.1 PASCAL-XSC Program Code.- 12.3.1.1 Module hess_axi.- 12.3.1.2 Module grad_ari.- 12.3.2 Examples.- 12.3.3 Restrictions and Hints.- 12.4 Exercises.- 12.5 References and Further Reading.- 13 Nonlinear Systems of Equations.- 13.1 Theoretical Background.- 13.1.1 Gauss-Seidel Iteration.- 13.2 Algorithmic Description.- 13.3 Implementation and Examples.- 13.3.1 PASCAL-XSC Program Code.- 13.3.1.1 Module nlss.- 13.3.2 Example.- 13.3.3 Restrictions, Hints, and Improvements.- 13.4 Exercises.- 13.5 References and Further Reading.- 14 Global Optimization.- 14.1 Theoretical Background.- 14.1.1 Midpoint Test.- 14.1.2 Monotonicity Test.- 14.1.3 Concavity Test.- 14.1.4 Interval Newton Step.- 14.1.5 Verification.- 14.2 Algorithmic Description.- 14.3 Implementation and Examples.- 14.3.1 PASCAL-XSC Program Code.- 14.3.1.1 Module 1st_ari.- 14.3.1.2 Module gop.- 14.3.2 Examples.- 14.3.3 Restrictions and Hints.- 14.4 Exercises.- 14.5 References and Further Reading.- A Utility Modules.- A.l Module b_util.- A.2 Module r_util.- A.3 Module i_util.- A.4 Module mvi_util.- Index of Special Symbols.

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