Course Outline, Homework, and Exam Dates²

• Numerical Methods in Scientific Computing, Volume 1, Germund Dahlquist and Ake Bjorck, SIAM, 2008, ISBN: 978-0-898716-44-3.
• 1 Principles of Numerical Calculations
  • 1.1 Common Ideas and Concepts
    • 1.1.1 Fixed-Point Iteration, 1.1.2 Newton’s Method, 1.1.3 Linearization and Extrapolation, 1.1.4 Finite Difference Approximations
  • 1.5 Numerical Solution of Differential Equations
    • 1.5.1 Euler’s Method, 1.5.2 An Introductory Example, 1.5.3 Second Order Accurate Methods
• 4 Interpolation and Approximation
  • 4.1 The Interpolation Problem
    • 4.1.1 Introduction, 4.1.2 Bases for Polynomial Interpolation, 4.1.3 Conditioning of Polynomial Interpolation
  • 4.2 Interpolation Formulas and Algorithms
    • 4.2.1 Newton’s Interpolation Formula, 4.2.3 Barycentric Lagrange Interpolation, 4.2.4 Iterative Linear Interpolation, 4.2.5 Fast Algorithms for Vandermonde Systems, 4.2.6 The Runge Phenomenon
  • 4.3 Generalizations and Applications
    • 4.3.1 Hermite Interpolation, 4.3.4 Multivariate Interpolation
• Review Questions I: 1.1.5, 1.2.3, 1.3.4, 1.4.3, 1.5.2, 1.6.1, 2.1.2, 2.2.5, 2.3.3, 2.4.2, 2.4.6, 4.1.6, 4.2.4, 4.3.1
• Problems and Computer Exercises I: 1.1.3, 1.1.5, 1.5.3, 4.1.1, 4.2.3, 4.3.1, 4.3.3
  • 4.4 Piecewise Polynomial Interpolation
    • 4.4.1 Bernštein Polynomials and Bézier Curves, 4.4.2 Spline Functions
  • 4.5 Approximation and Function Spaces
    • 4.5.3 Inner Product Spaces and Orthogonal Systems, 4.5.4 Solution of the Approximation Problem, 4.5.5 Mathematical Properties of Orthogonal Polynomials
  • 4.6 Fourier Methods
    • 4.6.1 Basic Formulas and Theorems, 4.6.2 Discrete Fourier Analysis,
  • 4.7 The Fast Fourier Transform
    • 4.7.1 The Fast Fourier Algorithm, 4.7.2 Discrete Convolution by FFT
• Review Questions II: 3.1.1, 3.2.3, 3.3.4, 3.4.5, 4.4.1, 4.5.7, 4.6.1, 4.7.1
• Problems and Computer Exercises II: 4.4.2, 4.4.4, 4.5.6, 4.5.18, 4.6.10, 4.7.2
• 5 Numerical Integration
  • 5.1 Interpolatory Quadrature Rules
    • 5.1.1 Introduction, 5.1.2 Treating Singularities, 5.1.3 Some Classical Formulas, 5.1.4 Superconvergence of the Trapezoidal Rule
  • 5.2 Integration by Extrapolation
    • 5.2.1 The Euler–Maclaurin Formula, 5.2.2 Romberg’s Method, 5.2.3 Oscillating Integrands
  • 3.4 Acceleration of Convergence
    • 3.4.1 Introduction, 3.4.2 Comparison Series and Aitken Acceleration, 3.4.3 Euler’s Transformation, 3.4.5 Euler–Maclaurin’s Formula, 3.4.6 Repeated Richardson Extrapolation
  • 5.3 Quadrature Rules with Free Nodes
    • 5.3.1 Method of Undetermined Coefficients, 5.3.2 Gauss–Christoffel Quadrature Rules, 5.3.4 Matrices, Moments, and Gauss Quadrature, 5.3.5 Jacobi Matrices and Gauss Quadrature
  • Lecture 34. From Lanczos to Gauss Quadrature, Numerical Linear Algebra, Lloyd N . Trefethen and David Bau, SIAM, 1997, ISBN: 0898713619.
• Review Questions III: 5.1.3, 5.1.5, 5.2.2, 5.2.4, 3.4.5, 5.3.2, 5.3.4
• Problems and Computer Exercises III: 5.1.10, 5.1.16, 5.2.2, 5.2.7, 3.4.8, 5.3.3, 5.3.11
• EXAM I (March 24)
• A First Course in the Numerical Analysis of Differential Equations, 2nd Edition, Arieh Iserles, Cambridge University Press, 2008, ISBN 978-0-521734-90-5.
• Part I Ordinary differential equations
  • 1. Euler's method and beyond
    • 1.1 Ordinary differential equations and the Lipschitz condition
    • 1.2 Euler’s method
    • 1.3 The trapezoidal rule
    • 1.4 The theta method
  • 2. Multistep methods
    • 2.1 The Adams method
    • 2.2 Order and convergence of multistep methods
    • 2.3 Backward differentiation formulae
  • 3. Runge–Kutta methods
    • 3.2 Explicit Runge–Kutta schemes
    • 3.3 Implicit Runge–Kutta schemes
  • 4. Stiff equations
    • 4.1 What are stiff ODEs?
    • 4.2 The linear stability domain and A-stability
    • 4.3 A-stability of Runge–Kutta methods
    • 4.3 A-stability of Runge–Kutta methods
• PartII The Poisson equation
  • 8. Finite difference schemes
    • 8.1 Finite differences
    • 8.2 The five-point formula for $\nabla^2 u = f$
  • 9. The finite element method
    • 9.1 Two-point boundary value problems
    • 9.2 A synopsis of FEM theory
    • 9.3 The Poisson equation
  • 12. Classical iterative methods for sparse linear equations
    • 12.1 Linear one-step stationary schemes
    • 12.2 Classical iterative methods
• Problems and Computer Exercises IV: 1.2, 2.3, 3.4, 4.2
• EXAM II (May 5)