Table of Contents

Chapter 1 Linear Algebra; Linear Systems of Equations

• 1.1 Introduction
• 1.2 Linear Systems of Algebraic Equations
• 1.3 Rank of a Matrix
• 1.4 Role of Rank in the Solution of Linear Systems
• 1.5 Determinant of a Matrix
• 1.6 Inverse of a Matrix

Chapter 2 Matrix Eigenvalue Problem

• 2.1 Introduction
• 2.2 Eigenvalue Properties of Special Matrices
• 2.3 Similarity Transformation and Matrix Diagonalization
• 2.4 Modal Decomposition
• 2.5 Simultaneous Diagonalization
• 2.6 Systems of Linear, First-Order Differential Equations
• 2.7 State-Variable Equations
• 2.8 Phase Plane Method for Linear Systems
• 2.9 Phase Plane Analysis of Nonlinear Systems

Chapter 3 Fundamentals of Numerical Methods

• 3.1 Introduction
• 3.2 Iterative Methods; Rate of Convergence and Stability
• 3.3 Bracketing and Fixed-Point Methods
• 3.4 Polynomial Approximation and Interpolation
• 3.5 Cubic Spline Interpolation
• 3.6 Fourier Approximation and Interpolation
• 3.7 Numerical Differentiation and Integration

Chapter 4 Numerical Methods for Ordinary Differential Equations

• 4.1 Introduction; Numerical Methods for First-Order Ordinary Differential Equations
• 4.2 Multi-step Methods
• 4.3 Numerical Solutions of Systems of First-Order ODE's

Chapter 5 Numerical Methods for Partial Differential Equations

• 5.1 Introduction
• 5.2 More on the Numerical Solution of Elliptic Equations
• 5.3 Numerical Solution of Parabolic and Hyperbolic PDE's

Chapter 6 Numerical Methods in Linear Algebra

• 6.1 Introduction; Direct Methods for Solving Linear Systems
• 6.2 Factorization Methods for Solving Linear Systems
• 6.3 Error Analysis
• 6.4 Iterative Methods for Solving Linear Algebraic Systems
• 6.5 Solution of Overdetermined Linear Systems; Least-Squares Method
• 6.6 Localization of Eigenvalues
• 6.7 Approximation of the Dominant Eigenvalue; Power Method
• 6.8 Deflation Methods; Inverse Power Method
• 6.9 Householder and Lanczos Tridiagonalization Methods; QR-Factorization

Appendix 1 References

Appendix 2 Useful Formulas

Appendix 3 Answers to Odd-Numbered Problems

Appendix 4 Tables

• Table 1 Gamma Function
• Table 2A Bessel Functions of the First Kind
• Table 2B Bessel Functions of the Second Kind
• Table 3 Laplace Transform Pairs

Index