• K. Atkinson, Introduction to Numerical Analysis (except Chapter 1).
• G. Golub and C. Van Loan, Matrix Computations, Chapters 2-5, 7, 10.
• K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Chapters 2.2,  2.4,  2.6-2.9,  3.1,  3.2,  4.2,  5.1-5.5.

• J. Stoer and R. Bulirsch, Introduction to Numerical Analysis.

Interpolation Theory

• polynomial interpolation theory
• Newton divided differences
• finite differences and table-oriented interpolation formulae
• errors in data and forward differences
• further results on interpolation error
• Hermite interpolation
• piecewise polynomial interpolation, B-splines
• Fourier series, DFT and FFT
• trigonometric interpolation

Approximation of Functions

• the Weierstrass Theorem and Taylor's Theorem
• the minimax approximation problem
• the least squares approximation problem
• orthogonal polynomials
• minimax approximation
• near-minimax approximations

Rootfinding for Nonlinear Equations

• the bisection method
• Newton's method
• the secant method
• Muller's method
• a general theory for one-point iteration methods
• Aitken extrapolation for linearly convergent sequences
• the numerical evaluation of multiple roots
• Brent's rootfinding algorithm
• roots of polynomials
• systems of nonlinear equations
• Newton's method for nonlinear systems
• unconstrained optimization

Numerical Integration

• the trapezoidal rule and Simpson's rule
• Newton-Cotes integration formulae
• Gaussian quadrature
• asymptotic error formulae and their applications
• automatic numerical integration
• singular integrals
• numerical differentiation

Linear Algebra

• vector spaces, matrices, and linear systems
• eigenvalues and canonical forms for matrices
• vector and matrix norms, condition numbers
• convergence and perturbation theorems
• Sherman-Morrison formula

Numerical Solution of Systems of Linear Equations, Direct methods

• Gaussian elimination
• pivoting and scaling in Gaussian elimination
• variants of Gaussian elimination
• error analysis
• the residual correction method

Numerical Solution of Systems of Linear Equations, Iterative Methods

• Gauss-Jacobi, Gauss-Seidel
• error prediction and acceleration
• the numerical solution of Poisson's equation
• the conjugate gradient method

The Matrix Eigenvalue Problem

• eigenvalue location, error, and stability results
• the power method
• orthogonal transformations using Householder matrices
• the eigenvalues of a symmetric tridiagonal matrix
• the QR Method
• the calculation of eigenvectors and inverse iteration
• least squares solution of linear systems

Numerical Methods for Ordinary Differential Equations

• existence, uniqueness, and stability theory
• Euler's method
• multistep methods
• the midpoint method
• the trapezoidal method
• a low-order predictor-corrector algorithm
• derivation of higher order multistep methods
• convergence and stability theory for multistep methods
• stiff differential equations and the method of lines
• single-step and Runge-Kutta methods
• boundary value problems

Introduction to Linear Parabolic and Hyperbolic PDEs

• parabolic problems in one space dimension
• an explicit scheme, convergence, Fourier analysis
• an implicit scheme
• parabolic problems in two space dimensions
• an explicit scheme
• ADI
• hyperbolic problems in one space dimension
• the CFL condition
• finite differences
• stability, accuracy, and consistency
• the Lax Equivalence Theorem