Advances in Numerical Analysis Emphasizing Interval Data

Advances in Numerical Analysis Emphasizing Interval Data

by Tofigh Allahviranloo, Witold Pedrycz, Armin Esfandiari
Released February 2022
Publisher(s): CRC Press
ISBN: 9781000540314

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Book description

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

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Table of Contents
  6. Preface
  7. Authors
  8. 1 About the Book
  9. 2 Error Analysis
    1. 2.1 Introduction
    2. 2.2 Error Analysis
      1. 2.2.1 Errors in an Algorithm
        1. 2.2.1.1 Problem
        2. 2.2.1.2 Problem
        3. 2.2.1.3 Definition-Absolute Error
        4. 2.2.1.4 Example
        5. 2.2.1.5 Definition – Relative Error
        6. 2.2.1.6 Problem
        7. 2.2.1.7 Theorem
        8. 2.2.1.8 Remark
        9. 2.2.1.9 Example
        10. 2.2.1.10 Different Types of Error Sources
      2. 2.2.2 Round of Error and Floating Points Arithmetic
        1. 2.2.2.1 Note
        2. 2.2.2.2 Definition
        3. 2.2.2.3 Definition
        4. 2.2.2.4 Note
        5. 2.2.2.5 Remark
        6. 2.2.2.6 Problem
        7. 2.2.2.7 Problem
        8. 2.2.2.8 Problem
        9. 2.2.2.9 Problem
        10. 2.2.2.10 Problem
        11. 2.2.2.11 Problem
        12. 2.2.2.12 Problem
    3. 2.3 Interval Arithmetic
    4. 2.4 Interval Error
    5. 2.5 Interval Floating Point Calculus
    6. 2.6 Problem
    7. 2.7 Algorithm Error Propagation
      1. 2.7.1 Problem
      2. 2.7.2 Scientific Representation of Numbers
      3. 2.7.3 Definition
      4. 2.7.4 Example
    8. 2.8 Exercises
  10. 3 Interpolation
    1. 3.1 Introduction
    2. 3.2 Lagrange Interpolation
      1. 3.2.1 Problem
      2. 3.2.2 Problem
      3. 3.2.3 Problem
      4. 3.2.4 Problem
      5. 3.2.5 Problem
    3. 3.3 Iterative Interpolation
      1. 3.3.1 Problem
    4. 3.4 Interpolation by Newton’s Divided Differences
      1. 3.4.1 Problem
      2. 3.4.2 Problem
      3. 3.4.3 Problem
      4. 3.4.4 Problem
      5. 3.4.5 Problem
      6. 3.4.6 Point
      7. 3.4.7 Problem
      8. 3.4.8 Problem
      9. 3.4.9 Point
      10. 3.4.10 Problem
      11. 3.4.11 Problem
      12. 3.4.12 Problem
      13. 3.4.13 Problem
      14. 3.4.14 Problem
    5. 3.5 Exercise
  11. 4 Advanced Interpolation
    1. 4.1 Hermit Interpolation
      1. 4.1.1 Problem
      2. 4.1.2 Problem
      3. 4.1.3 Problem
      4. 4.1.4 Problem
    2. 4.2 Fractional Interpolation
      1. 4.2.1 Problem
      2. 4.2.2 Problem
      3. 4.2.3 Problem
    3. 4.3 Inverse Newton’s Divided Difference Interpolation
      1. 4.3.1 Problem
      2. 4.3.2 Problem
      3. 4.3.3 Problem
    4. 4.4 Trigonometric Interpolation
      1. 4.4.1 Problem
      2. 4.4.2 Problem
      3. 4.4.3 Problem
      4. 4.4.4 Problem
      5. 4.4.5 Problem
    5. 4.5 Spline Interpolation
      1. 4.5.1 Spline Space
      2. 4.5.2 Definition-Spline Polynomial Function
      3. 4.5.3 Example
      4. 4.5.4 Definition
      5. 4.5.5 Approximation
      6. 4.5.6 Example
      7. 4.5.7 Example
      8. 4.5.8 Definition The Best Approximation
      9. 4.5.9 Existence of the Best Approximation
      10. 4.5.10 Minimum Sequence
      11. 4.5.11 Lemma
      12. 4.5.12 Theorem
      13. 4.5.13 Best Approximation Uniqueness
      14. 4.5.14 Definition Convex Set
      15. 4.5.15 Theorem Uniqueness
      16. 4.5.16 Theorem-Best Approximation Theory in the Normed Linear Space
      17. 4.5.17 Best Approximation in Spline Space
      18. 4.5.18 Definition
      19. 4.5.19 Example
      20. 4.5.20 Example
      21. 4.5.21 Theorem
      22. 4.5.22 Lemma
      23. 4.5.23 Haar Condition
      24. 4.5.24 Remark
      25. 4.5.25 Haar Space
      26. 4.5.26 Example
      27. 4.5.27 Remark
      28. 4.5.28 Types of Splines
      29. 4.5.29 Remark-Integral Relation
      30. 4.5.30 Remark
      31. 4.5.31 Remark
      32. 4.5.32 B-Spline
      33. 4.5.33 Existence of B-Spline
      34. 4.5.34 Definition
      35. 4.5.35 B-Spline Positivity
      36. 4.5.36 Theorem (Representation)
      37. 4.5.37 Other Properties of B-Splines
      38. 4.5.38 Problem
      39. 4.5.39 Problem
      40. 4.5.40 Problem
      41. 4.5.41 Problem
      42. 4.5.42 Problem
      43. 4.5.43 Problem
      44. 4.5.44 Problem
      45. 4.5.45 Problem
      46. 4.5.46 Problem
    6. 4.6 Reciprocal Interpolation
      1. 4.6.1 Transforming Reciprocal Interpolation to Direct Interpolation
      2. 4.6.2 Example
    7. 4.7 Exercise
  12. 5 Interval Interpolation
    1. 5.1 Interval Interpolation
      1. 5.1.1 Theorem
      2. 5.1.2 Corollary
      3. 5.1.3 Theorem
      4. 5.1.4 Point
      5. 5.1.5 Theorem
      6. 5.1.6 Example
      7. 5.1.7 Example
      8. 5.1.8 Theorem-Interval Interpolating Polynomial Error
      9. 5.1.9 Interval Lagrange Interpolation
  13. 6 Interpolation from the Linear Algebra Point of View
    1. 6.1 Introduction
      1. 6.1.1 Remark
      2. 6.1.2 Remark
      3. 6.1.3 Remark
      4. 6.1.4 Corollary
    2. 6.2 Lagrange Interpolation
    3. 6.3 Taylor’s Interpolation
    4. 6.4 Abelian Interpolation
    5. 6.5 Lidestone’s Interpolation
    6. 6.6 Simple Hermite Interpolation
    7. 6.7 Complete Hermite Interpolation
    8. 6.8 Fourier Interpolation
      1. 6.8.1 Problem
      2. 6.8.2 Problem
      3. 6.8.3 Problem
      4. 6.8.4 Problem
      5. 6.8.5 Problem
      6. 6.8.6 Problem
      7. 6.8.7 Problem
  14. 7 Newton-Cotes Quadrature
    1. 7.1 Newton-Cotes Quadrature
      1. 7.1.1 Problem
      2. 7.1.2 Problem
      3. 7.1.3 Problem
    2. 7.2 The Peano’s Kernel Error Representation
      1. 7.2.1 Problem
    3. 7.3 Romberg’s Quadrature Rule
      1. 7.3.1 Problem
      2. 7.3.2 Problem
      3. 7.3.3 Problem
      4. 7.3.4 Problem
  15. 8 Interval Newton-Cotes Quadrature
    1. 8.1 Introduction
    2. 8.2 Some Definitions
      1. 8.2.1 Lemma
      2. 8.2.2 Definition-Distance between Two Intervals
      3. 8.2.3 Definition-Continuity of an Interval Function
      4. 8.2.4 Definition
    3. 8.3 Newtons-Cotes Method
      1. 8.3.1 Peano’s Error Representation
      2. 8.3.2 Theorem
    4. 8.4 Trapezoidal Integration Rule
    5. 8.5 Simpson Integration Rule
    6. 8.6 Example
    7. 8.7 Example
  16. 9 Gauss Integration
    1. 9.1 Gaussian Integration
      1. 9.1.1 Gauss Legendre
      2. 9.1.2 Problem
      3. 9.1.3 Problem
      4. 9.1.4 Problem
      5. 9.1.5 Problem
      6. 9.1.6 Gauss Laguerre
      7. 9.1.7 Gauss Hermite
    2. 9.2 Gauss-Kronrod Quadrature Rule
    3. 9.3 Gaussian Quadrature for Approximate of Interval Integrals
    4. 9.4 Gauss-Legendre Integration Rules for Interval Valued Functions
      1. 9.4.1 One-Point Gauss-Legendre Integration Rule
      2. 9.4.2 Two-Point Gauss-Legendre Integration Rule
      3. 9.4.3 Three-Point Gauss-Legendre Integration Rule
    5. 9.5 Gauss-Chebyshev Integration Rules for Interval Valued Functions
      1. 9.5.1 One-Point Gauss-Chebyshev Integration Rule
      2. 9.5.2 Two-Point Gauss-Chebyshev Integration Rule
    6. 9.6 Gauss-Laguerre Integration Rules for Interval Valued Functions
      1. 9.6.1 One-Point Gauss-Laguerre Integration Rule
      2. 9.6.2 Two-Point Gauss-Laguerre Integration Rule
    7. 9.7 Gaussian Multiple Integrals Method
    8. 9.8 Gauss-Legendre Multiple Integrals Rules for Interval Valued Functions
      1. 9.8.1 Composite One-Point Gauss-Legendre Integration Rule
      2. 9.8.2 Composite Two-Point Gauss-Legendre Integration Rule
      3. 9.8.3 Composite One- and Three-Point Gauss-Legendre Integration Rule
    9. 9.9 Gauss-Chebyshev Multiple Integrals Rules for Interval Valued Functions
      1. 9.9.1 Composite One-Point Gauss-Chebyshev Integration Rule
      2. 9.9.2 Composite One- and Two-Point Gauss-Chebyshev Integration Rule
    10. 9.10 Composite Gauss-Legendre and Gauss-Chebyshev Integration Rule
      1. 9.10.1 Composite One-Point Gauss-Legendre and One-Point Gauss-Chebyshev Multiple Integral Rule
    11. 9.11 Adaptive Quadrature Rule
      1. 9.11.1 Introduction of Adaptive Quadrature Based on Simpson’s Method
  17. Index

Product information

  • Title: Advances in Numerical Analysis Emphasizing Interval Data
  • Author(s): Tofigh Allahviranloo, Witold Pedrycz, Armin Esfandiari
  • Release date: February 2022
  • Publisher(s): CRC Press
  • ISBN: 9781000540314