\contentsline {chapter}{\numberline {1}Mathematical Preliminaries}{1}
\contentsline {section}{\numberline {1.1}Review of Calculus}{1}
\contentsline {subsection}{\numberline {1.1.1}Limit, Continuity, Derivative, and Integral}{1}
\contentsline {subsection}{\numberline {1.1.2}Taylor's Theorem}{3}
\contentsline {section}{\numberline {1.2}Review of Linear Algebra and Matrix Analysis}{6}
\contentsline {subsection}{\numberline {1.2.1}Matrices and Vectors}{6}
\contentsline {subsection}{\numberline {1.2.2}Vector Space, Range Space, and Null Space}{9}
\contentsline {subsection}{\numberline {1.2.3}Orthogonality}{10}
\contentsline {section}{\numberline {1.3}Norms}{11}
\contentsline {subsection}{\numberline {1.3.1}Vector Norm Definition and Properties}{11}
\contentsline {subsection}{\numberline {1.3.2}Matrix Norm Definition and Properties}{12}
\contentsline {section}{\numberline {1.4}SVD: The Singular Value Decomposition}{16}
\contentsline {chapter}{\numberline {2}Computer Arithmetic}{19}
\contentsline {section}{\numberline {2.1}Floating-Point Number and Roundoff Error}{19}
\contentsline {section}{\numberline {2.2}Loss of Significance, Stability, and Conditioning}{24}
\contentsline {subsection}{\numberline {2.2.1}Loss of Significance}{24}
\contentsline {subsection}{\numberline {2.2.2}Numerical Stability}{30}
\contentsline {subsection}{\numberline {2.2.3}Conditioning}{31}
\contentsline {section}{\numberline {2.3}Floating-Point Error Analysis}{32}
\contentsline {section}{\numberline {2.4}Stability and Conditioning}{34}
\contentsline {subsection}{\numberline {2.4.1}Numerical Stability}{34}
\contentsline {subsection}{\numberline {2.4.2}Conditioning}{35}
\contentsline {chapter}{\numberline {3}Direct Methods for Solving Systems of Linear Equations}{37}
\contentsline {section}{\numberline {3.1}Triangular Systems}{38}
\contentsline {subsection}{\numberline {3.1.1}Diagonal System}{38}
\contentsline {subsection}{\numberline {3.1.2}Forward Substitution}{38}
\contentsline {subsection}{\numberline {3.1.3}Back Substitution}{40}
\contentsline {section}{\numberline {3.2}Gaussian Elimination and LU Factorization}{41}
\contentsline {subsection}{\numberline {3.2.1}Gaussian Elimination}{41}
\contentsline {subsection}{\numberline {3.2.2}Gaussian Transformation and LU Factorization}{44}
\contentsline {subsection}{\numberline {3.2.3}Existence and Uniqueness of LU Factorization}{49}
\contentsline {section}{\numberline {3.3}Pivoting}{50}
\contentsline {subsection}{\numberline {3.3.1}The Need for Pivoting}{51}
\contentsline {subsection}{\numberline {3.3.2}Partial Pivoting and Complete Pivoting}{52}
\contentsline {subsection}{\numberline {3.3.3}Scaled Row Pivoting}{54}
\contentsline {section}{\numberline {3.4}Some Special Linear Systems}{58}
\contentsline {subsection}{\numberline {3.4.1}Symmetric Positive Definite System and Cholesky Factorization}{58}
\contentsline {subsection}{\numberline {3.4.2}Diagonally Dominant Systems}{61}
\contentsline {subsection}{\numberline {3.4.3}Tridiagonal System}{63}
\contentsline {subsection}{\numberline {3.4.4}General Banded Systems}{64}
\contentsline {section}{\numberline {3.5}Perturbation Analysis}{64}
\contentsline {chapter}{\numberline {4}Iterative Methods for Solving Systems of Linear Equations}{67}
\contentsline {section}{\numberline {4.1}Classic Iterative Methods}{67}
\contentsline {subsection}{\numberline {4.1.1}Basic Concept}{67}
\contentsline {subsection}{\numberline {4.1.2}Richard's Method}{68}
\contentsline {subsection}{\numberline {4.1.3}Jacobi Method}{69}
\contentsline {subsection}{\numberline {4.1.4}Gauss-Seidel Method}{70}
\contentsline {subsection}{\numberline {4.1.5}Successive Over Relaxation (SOR) Method}{70}
\contentsline {subsection}{\numberline {4.1.6}Symmetric Successive Over Relaxation (SSOR) Method}{71}
\contentsline {section}{\numberline {4.2}Convergence Analysis}{71}
\contentsline {chapter}{\numberline {5}Solutions of Non-linear Equations}{77}
\contentsline {section}{\numberline {5.1}Preliminaries}{77}
\contentsline {section}{\numberline {5.2}Bisection Method}{79}
\contentsline {section}{\numberline {5.3}Newton's Method}{81}
\contentsline {subsection}{\numberline {5.3.1}Derivation of Newton's Method}{81}
\contentsline {subsection}{\numberline {5.3.2}Convergence Analysis}{82}
\contentsline {subsection}{\numberline {5.3.3}Examples and Pitfalls}{85}
\contentsline {subsection}{\numberline {5.3.4}System of Nonlinear Equations}{86}
\contentsline {section}{\numberline {5.4}Quasi-Newton's Method (Secant Method)}{87}
\contentsline {subsection}{\numberline {5.4.1}The Secant Method}{87}
\contentsline {subsection}{\numberline {5.4.2}Error Analysis of Secant Method}{89}
\contentsline {section}{\numberline {5.5}Fixed Point and Functional Iteration}{93}
\contentsline {subsection}{\numberline {5.5.1}Functional Iteration}{94}
\contentsline {subsection}{\numberline {5.5.2}Convergence Analysis}{96}
\contentsline {chapter}{\numberline {6}Interpolation}{99}
\contentsline {section}{\numberline {6.1}Polynomial Interpolation}{99}
\contentsline {subsection}{\numberline {6.1.1}Existence And Uniqueness}{99}
\contentsline {subsection}{\numberline {6.1.2}Naive Approach for Polynomial Interpolation}{100}
\contentsline {subsection}{\numberline {6.1.3}Lagrange Form and Neville's Method}{101}
\contentsline {subsection}{\numberline {6.1.4}Newton's Form of Polynomial Interpolation}{103}
\contentsline {subsection}{\numberline {6.1.5}Divided Differences Scheme}{105}
\contentsline {subsection}{\numberline {6.1.6}Error Analysis for Polynomial Interpolation}{111}
\contentsline {section}{\numberline {6.2}Hermite Interpolation}{112}
\contentsline {subsection}{\numberline {6.2.1}Existence and Uniqueness}{113}
\contentsline {subsection}{\numberline {6.2.2}Lagrange Form for Hermite Interpolation}{113}
\contentsline {subsection}{\numberline {6.2.3}Divided Difference Method for Hermite Interpolation}{114}
\contentsline {subsection}{\numberline {6.2.4}Error Analysis for Hermite Interpolation}{115}
\contentsline {section}{\numberline {6.3}Spline Interpolation}{116}
\contentsline {subsection}{\numberline {6.3.1}Cubic Spline}{117}
\contentsline {chapter}{\numberline {7}Numerical Differentiation and Integration}{121}
\contentsline {section}{\numberline {7.1}Numerical Differentiation}{121}
\contentsline {subsection}{\numberline {7.1.1}Finite Difference Method}{121}
\contentsline {subsection}{\numberline {7.1.2}Polynomial Interpolation Method}{122}
\contentsline {subsection}{\numberline {7.1.3}Richardson Extrapolation Method}{123}
\contentsline {section}{\numberline {7.2}Numerical Integration}{125}
\contentsline {subsection}{\numberline {7.2.1}Elements of Numerical Integration}{125}
\contentsline {subsection}{\numberline {7.2.2}Newton-Cotes Formulas}{127}
\contentsline {paragraph}{Trapezoidal Rule:}{128}
\contentsline {paragraph}{Simpson's Rule:}{129}
\contentsline {paragraph}{Midpoint Rule:}{131}
\contentsline {subsection}{\numberline {7.2.3}Composite Newton-Cotes Forumlas}{132}
\contentsline {paragraph}{Composite Simpson's Rule:}{134}
\contentsline {paragraph}{Composite Midpoint Rule:}{134}
\contentsline {paragraph}{Composite Trapezoidal Rule}{135}
\contentsline {section}{\numberline {7.3}Gaussian Quadrature}{135}
\contentsline {subsection}{\numberline {7.3.1}Orthogonal Polynomials and Quadrature Rule}{136}
\contentsline {subsection}{\numberline {7.3.2}Gaussian Quadrature Rule}{141}
\contentsline {paragraph}{Gaussian Quadrature Rule:}{142}
\contentsline {subsection}{\numberline {7.3.3}Error Analysis}{144}
\contentsline {section}{\numberline {7.4}Adaptive Quadrature}{145}
\contentsline {section}{\numberline {7.5}Romberg Integration}{145}
\contentsline {chapter}{\numberline {8}Numerical Solutions of Ordinary Differential Equations}{147}
\contentsline {section}{\numberline {8.1}Existence and Uniqueness of Solutions}{147}
\contentsline {section}{\numberline {8.2}Euler's Method}{148}
\contentsline {section}{\numberline {8.3}Runge-Kutta Methods}{148}
\contentsline {section}{\numberline {8.4}Systems and Higher-Order Ordinary Differential Equations}{150}
\contentsline {chapter}{\numberline {9}Boundary-Value Problems for Ordinary Differential Equations}{153}
\contentsline {section}{\numberline {9.1}Mathematical Theories}{153}
\contentsline {section}{\numberline {9.2}Finite Difference Method For Linear Problems}{158}
\contentsline {subsection}{\numberline {9.2.1}The Finite Difference Formulation}{158}
\contentsline {subsection}{\numberline {9.2.2}Convergence Analysis}{160}
\contentsline {subsection}{\numberline {9.2.3}Higher Order Approximations}{161}
\contentsline {section}{\numberline {9.3}Finite Difference Method For Nonlinear Problems}{161}
\contentsline {section}{\numberline {9.4}Shooting Methods}{161}