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Numerical Methods

Book Description

Numerical Methods is a mathematical tool used by engineers and mathematicians to do scientific calculations. It is used to find solutions to applied problems where ordinary analytical methods fail. This book is intended to serve for the needs of courses in Numerical Methods at the Bachelors' and Masters' levels at various universities.

Table of Contents

  1. Cover
  2. Title page
  3. Contents
  4. Dedication
  5. Preface
  6. Chapter 1. Preliminaries
    1. 1.1 Approximate Numbers and Significant Figures
    2. 1.2 Classical Theorems Used In Numerical Methods
    3. 1.3 Types of Errors
    4. 1.4 General Formula for Errors
    5. 1.5 Order of Approximation
    6. Exercises
  7. Chapter 2. Non-Linear Equations
    1. 2.1 Classification of Methods
    2. 2.2 Approximate Values of the Roots
    3. 2.3 Bisection Method (Bolzano Method)
    4. 2.4 Regula–Falsi Method
    5. 2.5 Convergence of Regula–Falsi Method
    6. 2.6 Newton–Raphson Method
    7. 2.7 Square Root of a Number Using Newton–Raphson Method
    8. 2.8 Order of Convergence of Newton–Raphson Method
    9. 2.9 Fixed Point Iteration
    10. 2.10 Convergence of Iteration Method
    11. 2.11 Square Root of a Number Using Iteration Method
    12. 2.12 Sufficient Condition for the Convergence of Newton–Raphson Method
    13. 2.13 Newton’s Method for Finding Multiple Roots
    14. 2.14 Newton–Raphson Method for Simultaneous Equations
    15. 2.15 Graeffe’s Root Squaring Method
    16. 2.16 Muller’s Method
    17. 2.17 Bairstow Iterative Method
    18. Exercises
  8. Chapter 3. Linear Systems of Equations
    1. 3.1 Direct Methods
    2. 3.2 Iterative Methods for Linear Systems
    3. 3.3 The Method of Relaxation
    4. 3.4 Ill-Conditioned System of Equations
    5. Exercises
  9. Chapter 4. Eigenvalues and Eigenvectors
    1. 4.1 Eigenvalues and Eigenvectors
    2. 4.2 The Power Method
    3. 4.3 Jacobi’s Method
    4. 4.4 Given’s Method
    5. 4.5 Householder’s Method
    6. 4.6 Eigenvalues of a Symmetric Tri-diagonal Matrix
    7. 4.7 Bounds on Eigenvalues (Gerschgorin Circles)
    8. Exercises
  10. Chapter 5. Finite Differences and Interpolation
    1. 5.1 Finite Differences
    2. 5.2 Factorial Notation
    3. 5.3 Some More Examples of Finite Differences
    4. 5.4 Error Propagation
    5. 5.5 Numerical Unstability
    6. 5.6 Interpolation
    7. 5.7 Use of Interpolation Formulae
    8. 5.8 Interpolation with Unequal-Spaced Points
    9. 5.9 Newton’s Fundamental (Divided Difference) Formula
    10. 5.10 Error Formulae
    11. 5.11 Lagrange’s Interpolation Formula
    12. 5.12 Error in Lagrange’s Interpolation Formula
    13. 5.13 Hermite Interpolation Formula
    14. 5.14 Throwback Technique
    15. 5.15 Inverse Interpolation
    16. 5.16 Chebyshev Polynomials
    17. 5.17 Approximation of a Function with a Chebyshev Series
    18. 5.18 Interpolation by Spline Functions
    19. 5.19 Existence of Cubic Spline
    20. Exercises
  11. Chapter 6. Curve Fitting
    1. 6.1 Least Square Line Approximation
    2. 6.2 The Power Fit y = axm
    3. 6.3 Least Square Parabola (Parabola of Best Fit)
    4. Exercises
  12. Chapter 7. Numerical Differentiation
    1. 7.1 Centered Formula of Order O(h2)
    2. 7.2 Centered Formula of Order O(h4)
    3. 7.3 Error Analysis
    4. 7.4 Richardson’s Extrapolation
    5. 7.5 Central Difference Formula of Order O(h4) for f″(x)
    6. 7.6 General Method for Deriving Differentiation Formulae
    7. 7.7 Differentiation of a Function Tabulated in Unequal Intervals
    8. 7.8 Differentiation of Lagrange’s Polynomial
    9. 7.9 Differentiation of Newton Polynomial
    10. Exercises
  13. Chapter 8. Numerical Quadrature
    1. 8.1 General Quadrature Formula
    2. 8.2 Cote’s Formulae
    3. 8.3 Error Term in Quadrature Formula
    4. 8.4 Richardson Extrapolation (or Deferred Approach to the Limit)
    5. 8.5 Simpson’s Formula with End Correction
    6. 8.6 Romberg’s Method
    7. 8.7 Euler–Maclaurin Formula
    8. 8.8 Double Integrals
    9. Exercises
  14. Chapter 9. Difference Equations
    1. 9.1 Definitions and Examples
    2. 9.2 Homogeneous Difference Equation with Constant Coefficients
    3. 9.3 Particular Solution of a Difference Equation
    4. Exercises
  15. Chapter 10. Ordinary Differential Equations
    1. 10.1 Initial Value Problems and Boundary Value Problems
    2. 10.2 Classification of Methods of Solution
    3. 10.3 Single-Step Methods
    4. 10.4 Multistep Methods
    5. 10.5 Stability of Methods
    6. 10.6 Second Order Differential Equation
    7. 10.7 Solution of Boundary Value Problems by Finite Difference Method
    8. 10.8 Use of the Formula to Solve Boundary Value Problems
    9. 10.9 Eigenvalue Problems
    10. Exercises
  16. Chapter 11. Partial Differential Equations
    1. 11.1 Formation of Difference Equation
    2. 11.2 Geometric Representation of Partial Difference Quotients
    3. 11.3 Standard Five Point Formula and Diagonal Five Point Formula
    4. 11.4 Point Jacobi’s Method
    5. 11.5 Gauss–Seidel Method
    6. 11.6 Solution of Elliptic Equation by Relaxation Method
    7. 11.7 Poisson’s Equation
    8. 11.8 Eigenvalue Problems
    9. 11.9 Parabolic Equations
    10. 11.10 Iterative Method to Solve Parabolic Equations
    11. 11.11 Hyperbolic Equations
    12. Exercises
  17. Chapter 12. Elements of C Language
    1. 12.1 Programming Language
    2. 12.2 C Language
    3. 12.3 C Tokens
    4. 12.4 Library Functions
    5. 12.5 Input Operation
    6. 12.6 Output Operation
    7. 12.7 Control (Selection) Statements
    8. 12.8 Structure of a C Program
    9. 12.9 Programs of Certain Numerical Methods in C Language
  18. Appendix
    1. Model Paper 1
    2. Model Paper 2
    3. Model Paper 3
    4. Model Paper 4
    5. Model Paper 5
  19. Bibliography
  20. Copyright