Advances in Numerical Analysis Emphasizing with Interval Data

✍ Scribed by Tofigh Allahviranloo, Witold Pedrycz, Armin Esfandiari

• Publisher: CRC Press
• Year: 2022
• Tongue: English
• Leaves: 219
• Edition: 1
• Category: Library

✦ Synopsis

Numerical analysis forms a cornerstone of numeric computing and optimization, in particular recently, interval numerical computations play an important role in these topics. The interest of researchers in computations involving uncertain data, namely interval data opens new avenues in coping with real-world problems and deliver innovative and efficient solutions. This book provides the basic theoretical foundations of numerical methods, discusses key technique classes, explains improvements and improvements, and provides insights into recent developments and challenges.

The theoretical parts of numerical methods, including the concept of interval approximation theory, are introduced and explained in detail. In general, the key features of the book include an up-to-date and focused treatise on error analysis in calculations, in particular the comprehensive and systematic treatment of error propagation mechanisms, considerations on the quality of data involved in numerical calculations, and a thorough discussion of interval approximation theory.

Moreover, this book focuses on approximation theory and its development from the perspective of linear algebra, and new and regular representations of numerical integration and their solutions are enhanced by error analysis as well. The book is unique in the sense that its content and organization will cater to several audiences, in particular graduate students, researchers, and practitioners.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Authors
1 About the Book
2 Error Analysis
2.1 Introduction
2.2 Error Analysis
2.2.1 Errors in an Algorithm
2.2.1.1 Problem
2.2.1.2 Problem
2.2.1.3 Definition-Absolute Error
2.2.1.4 Example
2.2.1.5 Definition – Relative Error
2.2.1.6 Problem
2.2.1.7 Theorem
2.2.1.8 Remark
2.2.1.9 Example
2.2.1.10 Different Types of Error Sources
2.2.2 Round of Error and Floating Points Arithmetic
2.2.2.1 Note
2.2.2.2 Definition
2.2.2.3 Definition
2.2.2.4 Note
2.2.2.5 Remark
2.2.2.6 Problem
2.2.2.7 Problem
2.2.2.8 Problem
2.2.2.9 Problem
2.2.2.10 Problem
2.2.2.11 Problem
2.2.2.12 Problem
2.3 Interval Arithmetic
2.4 Interval Error
2.5 Interval Floating Point Calculus
2.6 Problem
2.7 Algorithm Error Propagation
2.7.1 Problem
2.7.2 Scientific Representation of Numbers
2.7.3 Definition
2.7.4 Example
2.8 Exercises
3 Interpolation
3.1 Introduction
3.2 Lagrange Interpolation
3.2.1 Problem
3.2.2 Problem
3.2.3 Problem
3.2.4 Problem
3.2.5 Problem
3.3 Iterative Interpolation
3.3.1 Problem
3.4 Interpolation by Newton’s Divided Differences
3.4.1 Problem
3.4.2 Problem
3.4.3 Problem
3.4.4 Problem
3.4.5 Problem
3.4.6 Point
3.4.7 Problem
3.4.8 Problem
3.4.9 Point
3.4.10 Problem
3.4.11 Problem
3.4.12 Problem
3.4.13 Problem
3.4.14 Problem
3.5 Exercise
4 Advanced Interpolation
4.1 Hermit Interpolation
4.1.1 Problem
4.1.2 Problem
4.1.3 Problem
4.1.4 Problem
4.2 Fractional Interpolation
4.2.1 Problem
4.2.2 Problem
4.2.3 Problem
4.3 Inverse Newton’s Divided Difference Interpolation
4.3.1 Problem
4.3.2 Problem
4.3.3 Problem
4.4 Trigonometric Interpolation
4.4.1 Problem
4.4.2 Problem
4.4.3 Problem
4.4.4 Problem
4.4.5 Problem
4.5 Spline Interpolation
4.5.1 Spline Space
4.5.2 Definition- Spline Polynomial Function
4.5.3 Example
4.5.4 Definition
4.5.5 Approximation
4.5.6 Example
4.5.7 Example
4.5.8 Definition The Best Approximation
4.5.9 Existence of the Best Approximation
4.5.10 Minimum Sequence
4.5.11 Lemma
4.5.12 Theorem
4.5.13 Best Approximation Uniqueness
4.5.14 Definition Convex Set
4.5.15 Theorem Uniqueness
4.5.16 Theorem-Best Approximation Theory in the Normed Linear Space
4.5.17 Best Approximation in Spline Space
4.5.18 Definition
4.5.19 Example
4.5.20 Example
4.5.21 Theorem
4.5.22 Lemma
4.5.23 Haar Condition
4.5.24 Remark
4.5.25 Haar Space
4.5.26 Example
4.5.27 Remark
4.5.28 Types of Splines
4.5.29 Remark-Integral Relation
4.5.30 Remark
4.5.31 Remark
4.5.32 B–Spline
4.5.33 Existence of B-Spline
4.5.34 Definition
4.5.35 B-Spline Positivity
4.5.36 Theorem (Representation)
4.5.37 Other Properties of B-Splines
4.5.38 Problem
4.5.39 Problem
4.5.40 Problem
4.5.41 Problem
4.5.42 Problem
4.5.43 Problem
4.5.44 Problem
4.5.45 Problem
4.5.46 Problem
4.6 Reciprocal Interpolation
4.6.1 Transforming Reciprocal Interpolation to Direct Interpolation
4.6.2 Example
4.7 Exercise
5 Interval Interpolation
5.1 Interval Interpolation
5.1.1 Theorem
5.1.2 Corollary
5.1.3 Theorem
5.1.4 Point
5.1.5 Theorem
5.1.6 Example
5.1.7 Example
5.1.8 Theorem-Interval Interpolating Polynomial Error
5.1.9 Interval Lagrange Interpolation
6 Interpolation from the Linear Algebra Point of View
6.1 Introduction
6.1.1 Remark
6.1.2 Remark
6.1.3 Remark
6.1.4 Corollary
6.2 Lagrange Interpolation
6.3 Taylor’s Interpolation
6.4 Abelian Interpolation
6.5 Lidestone’s Interpolation
6.6 Simple Hermite Interpolation
6.7 Complete Hermite Interpolation
6.8 Fourier Interpolation
6.8.1 Problem
6.8.2 Problem
6.8.3 Problem
6.8.4 Problem
6.8.5 Problem
6.8.6 Problem
6.8.7 Problem
7 Newton-Cotes Quadrature
7.1 Newton-Cotes Quadrature
7.1.1 Problem
7.1.2 Problem
7.1.3 Problem
7.2 The Peano’s Kernel Error Representation
7.2.1 Problem
7.3 Romberg’s Quadrature Rule
7.3.1 Problem
7.3.2 Problem
7.3.3 Problem
7.3.4 Problem
8 Interval Newton-Cotes Quadrature
8.1 Introduction
8.2 Some Definitions
8.2.1 Lemma
8.2.2 Definition-Distance between Two Intervals
8.2.3 Definition-Continuity of an Interval Function
8.2.4 Definition
8.3 Newtons-Cotes Method
8.3.1 Peano’s Error Representation
8.3.2 Theorem
8.4 Trapezoidal Integration Rule
8.5 Simpson Integration Rule
8.6 Example
8.7 Example
9 Gauss Integration
9.1 Gaussian Integration
9.1.1 Gauss Legendre
9.1.2 Problem
9.1.3 Problem
9.1.4 Problem
9.1.5 Problem
9.1.6 Gauss Laguerre
9.1.7 Gauss Hermite
9.2 Gauss-Kronrod Quadrature Rule
9.3 Gaussian Quadrature for Approximate of Interval Integrals
9.4 Gauss-Legendre Integration Rules for interval valued functions
9.4.1 One-Point Gauss-Legendre Integration Rule
9.4.2 Two-Point Gauss-Legendre Integration Rule
9.4.3 Three-Point Gauss-Legendre Integration Rule
9.5 Gauss-Chebyshev Integration Rules for interval valued functions
9.5.1 One-Point Gauss-Chebyshev Integration Rule
9.5.2 Two-Point Gauss-Chebyshev Integration Rule
9.6 Gauss-Laguerre Integration Rules for interval valued functions
9.6.1 One-Point Gauss-Laguerre Integration Rule
9.6.2 Two-Point Gauss-Laguerre Integration Rule
9.7 Gaussian Multiple Integrals Method
9.8 Gauss-Legendre Multiple Integrals Rules for interval valued functions
9.8.1 Composite One-Point Gauss-Legendre Integration Rule
9.8.2 Composite Two-Point Gauss-Legendre Integration Rule
9.8.3 Composite One- and Three-Point Gauss-Legendre Integration Rule
9.9 Gauss-Chebyshev Multiple Integrals Rules for interval valued functions
9.9.1 Composite One-Point Gauss-Chebyshev Integration Rule
9.9.2 Composite One-and Two-Point Gauss-Chebyshev Integration Rule
9.10 Composite Gauss-Legendre and Gauss-Chebyshev Integration Rule
9.10.1 Composite One-Point Gauss-Legendre and One- Point Gauss-Chebyshev Multiple Integral Rule
9.11 Adaptive Quadrature Rule
9.11.1 Introduction of Adaptive Quadrature Based on Simpson’s Method
Index