Preface -- Contents -- Foundations of Matrix Analysis -- 1.1 Vector Spaces -- 1.2 Matrices -- 1.3 Operations with Matrices -- 1.4 Trace and Determinant of a Matrix -- 1.5 Rank and Kernel of a Matrix -- 1.6 Special Matrices -- 1.7 Eigenvalues and Eigenvectors -- 1.8 Similarity Transformations -- 1.9 The Singular Value Decomposition (SVD) -- 1.10 Scalar Product and Norms in Vector Spaces -- 1.11 Matrix Norms -- 1.12 Positive Definite, Diagonally Dominant and M-matrices -- 1.13 Exercises -- Principles of Numerical Mathematics -- 2.1 Well-posedness and Condition Number of a Problem -- 2.2 Stability of Numerical Methods -- 2.3 A priori and a posteriori Analysis -- 2.4 Sources of Error in Computational Models -- 2.5 Machine Representation of Numbers -- 2.6 Exercises -- Direct Methods for the Solution of Linear Systems -- 3.1 Stability Analysis of Linear Systems -- 3.2 Solution of Triangular Systems -- 3.3 The Gaussian Elimination Method (GEM) and LU Factorization -- 3.4 Other Types of Factorization -- 3.5 Pivoting -- 3.6 Computing the Inverse of a Matrix -- 3.7 Banded Systems -- 3.8 Block Systems -- 3.9 Sparse Matrices -- 3.10 Accuracy of the Solution Achieved Using GEM -- 3.11 An Approximate Computation of K(A) -- 3.12 Improving the Accuracy of GEM -- 3.13 Undetermined Systems -- 3.14 Applications -- 3.15 Exercises -- Iterative Methods for Solving Linear Systems -- 4.1 On the Convergence of Iterative Methods -- 4.2 Linear Iterative Methods -- 4.3 Stationary and Nonstationary Iterative Methods -- 4.4 Methods Based on Krylov Subspace Iterations -- 4.5 The Lanczos Method for Unsymmetric Systems -- 4.6 Stopping Criteria -- 4.7 Applications -- 4.8 Exercises -- Approximation of Eigenvalues and Eigenvectors -- 5.1 Geometrical Location of the Eigenvalues -- 5.2 Stability and Conditioning Analysis -- 5.3 The Power Method -- 5.4 The QR Iteration.

5.5 The Basic QR Iteration -- 5.6 The QR Method for Matrices in Hessenberg Form -- 5.7 The QR Iteration with Shifting Techniques -- 5.8 Computing the Eigenvectors and the SVD of a Matrix -- 5.9 The Generalized Eigenvalue Problem -- 5.10 Methods for Eigenvalues of Symmetric matrices -- 5.11 The Lanczos Method -- 5.12 Applications -- 5.13 Exercises -- Rootfinding for Nonlinear Equations -- 6.1 Conditioning of a Nonlinear Equation -- 6.2 A Geometric Approach to Root nding -- 6.3 Fixed-point Iterations for Nonlinear Equations -- 6.4 Zeros of Algebraic Equations -- 6.5 Stopping Criteria -- 6.6 Post-processing Techniques for Iterative Methods -- 6.7 Applications -- 6.8 Exercises -- Nonlinear Systems and Numerical Optimization -- 7.1 Solution of Systems of Nonlinear Equations -- 7.2 Unconstrained Optimization -- 7.3 Constrained Optimization -- 7.4 Applications -- 7.5 Exercises -- Polynomial Interpolation -- 8.1 Polynomial Interpolation -- 8.2 Newton Form of the Interpolating Polynomial -- 8.3 Piecewise Lagrange Interpolation -- 8.4 Hermite-Birko Interpolation -- 8.5 Extension to the Two-Dimensional Case -- 8.6 Approximation by Splines -- 8.7 Splines in Parametric Form -- 8.8 Applications -- 8.9 Exercises -- Numerical Integration -- 9.1 Quadrature Formulae -- 9.2 Interpolatory Quadratures -- 9.3 Newton-Cotes Formulae -- 9.4 Composite Newton-Cotes Formulae -- 9.5 Hermite Quadrature Formulae -- 9.6 Richardson Extrapolation -- 9.7 Automatic Integration -- 9.8 Singular Integrals -- 9.9 Multidimensional Numerical Integration -- 9.10 Applications -- 9.11 Exercises -- Orthogonal Polynomials in Approximation Theory -- 10.1 Approximation of Functions by Generalized Fourier Series -- 10.2 Gaussian Integration and Interpolation -- 10.3 Chebyshev Integration and Interpolation -- 10.4 Legendre Integration and Interpolation.

10.5 Gaussian Integration over Unbounded Intervals -- 10.6 Programs for the Implementation of Gaussian Quadratures -- 10.7 Approximation of a Function in the Least-Squares Sense -- 10.8 The Polynomial of Best Approximation -- 10.9 Fourier Trigonometric Polynomials -- 10.10 Approximation of Function Derivatives -- 10.11 Transforms and Their Applications -- 10.12 The Wavelet Transform -- 10.13 Applications -- 10.14 Exercises -- Numerical Solution of Ordinary Differential Equations -- 11.1 The Cauchy Problem -- 11.2 One-Step Numerical Methods -- 11.3 Analysis of One-Step Methods -- 11.4 Difference Equations -- 11.5 Multistep Methods -- 11.6 Analysis of Multistep Methods -- 11.7 Predictor-Corrector Methods -- 11.8 Runge-Kutta (RK) Methods -- 11.9 Systems of ODEs -- 11.10 Stiff Problems -- 11.11 Applications -- 11.12 Exercises -- Two-Point Boundary Value Problems -- 12.1 A Model Problem -- 12.2 Finite Difference Approximation -- 12.3 The Spectral Collocation Method -- 12.4 The Galerkin Method -- 12.5 Advection-Diffusion Equations -- 12.6 A Quick Glance to the Two-Dimensional Case -- 12.7 Applications -- 12.8 Exercises -- Parabolic and Hyperbolic Initial Boundary Value Problems -- 13.1 The Heat Equation -- 13.2 Finite Difference Approximation of the Heat Equation -- 13.3 Finite Element Approximation of the Heat Equation -- 13.4 Space-Time Finite Element Methods for the Heat Equation -- 13.5 Hyperbolic Equations: A Scalar Transport Problem -- 13.6 Systems of Linear Hyperbolic Equations -- 13.7 The Finite Difference Method for Hyperbolic Equations -- 13.8 Analysis of Finite Difference Methods -- 13.9 Dissipation and Dispersion -- 13.10 Finite Element Approximation of Hyperbolic Equations -- 13.11 Applications -- 13.12 Exercises -- References -- Index of MATLAB Programs -- Index.