Computational Error and Complexity in Science and Engineering

Computational Error and Complexity

1st Edition, Volume 201 - March 4, 2005
Authors: Vangipuram Lakshmikantham, Syamal Kumar Sen
Language: English
Hardback ISBN:
9 7 8 - 0 - 4 4 4 - 5 1 8 6 0 - 6
Paperback ISBN:
9 7 8 - 0 - 4 4 4 - 5 4 5 8 3 - 1
eBook ISBN:
9 7 8 - 0 - 0 8 - 0 4 5 9 5 1 - 6

The book “Computational Error and Complexity in Science and Engineering” pervades all the science and engineering disciplines where computation occurs. Scientific and engineering… Read more

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The book “Computational Error and Complexity in Science and Engineering” pervades all the science and engineering disciplines where computation occurs. Scientific and engineering computation happens to be the interface between the mathematical model/problem and the real world application. One needs to obtain good quality numerical values for any real-world implementation. Just mathematical quantities symbols are of no use to engineers/technologists. Computational complexity of the numerical method to solve the mathematical model, also computed along with the solution, on the other hand, will tell us how much computation/computational effort has been spent to achieve that quality of result. Anyone who wants the specified physical problem to be solved has every right to know the quality of the solution as well as the resources spent for the solution. The computed error as well as the complexity provide the scientific convincing answer to these questions. Specifically some of the disciplines in which the book will be readily useful are (i) Computational Mathematics, (ii) Applied Mathematics/Computational Engineering, Numerical and Computational Physics, Simulation and Modelling. Operations Research (both deterministic and stochastic), Computing Methodologies, Computer Applications, and Numerical Methods in Engineering.Key Features:- Describes precisely ready-to-use computational error and complexity- Includes simple easy-to-grasp examples wherever necessary.- Presents error and complexity in error-free, parallel, and probabilistic methods.- Discusses deterministic and probabilistic methods with error and complexity. - Points out the scope and limitation of mathematical error-bounds.- Provides a comprehensive up-to-date bibliography after each chapter.

Preface
Chapter 1: Introduction
- 1.1 Science versus engineering
- 1.2 Capability and limit of computation
- 1.3 What is computation in science and engineering
- 1.4 Tools for Computation
- 1.5 Algorithms and Complexity
- 1.6 Types of computation
- 1.7 Models of computation
- 1.8 Computer representable numbers: Scope and error
- 1.9 Problem-solving: Stages and error
- 1.10 Stages of problem-solving: Equivalence and hierarchical structure
Chapter 2: Error: Precisely What, Why, and How
- 2.1 Introduction
- 2.2 Error: Precisely what and how to compute
- 2.3 Error-free environment/quantity — How far is it possible
- 2.4 Error analysis
- 2.5 Limitation of interval arithmetic and significant digit arithmetic
- 2.6 Visualization of error
- 2.7 Mathematical error versus computable error
- 2.8 Confidence versus error
- 2.9 Error-bound is non-decreasing while actual error need not be
- 2.10 Stability and error
Chapter 3: Complexity: What, Why, and How
- 3.1 Introduction
- 3.2 Algorithm as Turing machine and algorithmic complexity
- 3.3 Pspace
- 3.4 Alternation
- 3.5 Logspace
- 3.6 Probabilistic complexity
- 3.7 Descriptive complexity
- 3.8 Boolean circuit complexity
- 3.9 Communication complexity
- 3.10 Quantum complexity
- 3.11 Parallel complexity
Chapter 4: Errors and Approximations in Digital Computers
- 4.1 Introduction
- 4.2 Number representation
- 4.3 Fixed- and floating-point representation and a arithmetic
- 4.4 Error in function with approximate arguments (direct problem)
- 4.5 Error in arguments with prescribed accuracy in function (inverse problem)
- 4.6 Significance of a function
- 4.7 Error in series approximation
- 4.8 Base 2 system: best in computer/communication
- 4.9 IEEE 754 floating-point format
Chapter 5: Error and Complexity in Numerical Methods
- 5.1 Introduction
- 5.2 Error in quantities and computations
- 5.3 Computational complexity
- 5.4 What computer can represent
- 5.5 Algorithms and related errors
- 5.6 Conclusions
Chapter 6: Error and Complexity in Error-Free, Parallel, and Probabilistic Computations
- 6.1 Introduction
- 6.2 Actual error-bound in exact computation: Exponential problem
- 6.3 Parallel computation: error and complexity
- 6.4 Error-bounds in probabilistic computation
- 6.5 Shrinking-rectangle randomized algorithm for complex zero: Error and complexity
Index

Life Sciences

Physical Sciences & Engineering

Social Sciences & Humanities

Health

Computational Error and Complexity in Science and Engineering