Math 573 Lecture Notes

  • Lecture 1: Polynomial Interpolation (Weierstrass appoximation theorem, Lagrange and Newton forms of the interpolating polynomial.) Revised 9/11/2015
  • Lecture 2: Polynomial Interpolation (Polynomial interpolation error, divided differences for repeated points.) Revised 9/11/2015
  • Lecture 3: Polynomial and Piecewise Polynomial Approximation(Interpolation of moments, Runge example, piecewise polynomial approximation.)
  • Lecture 4: Piecewise Polynomial Approximation (C^0 and C^1 piecewise polynomial approximation and error estimates, construction of basis functions.)
  • Lecture 5: Cubic Spline Approximation (cubic spline approximation, cubic spline basis functions, error in cubic spline approximation.)
  • Lecture 6: Trigonometric Interpolation (Interpolation by trigonometric functions, the finite Fourier transform, and fast Fourier transform.)
  • Lecture 7: Piecewise polynomial approximation in two dimensions(construction of continuous piecewise polynomial spaces on a triangulation of a polygonal domain). Revised 10/10/2015. Revised 10/13/2015.
  • Lecture 8: Approximation of Derivatives (numerical differentiation formulas, roundoff error in numerical differentiation.)
  • Lecture 9: Approximation of Integrals (basic numerical integration rules, composite numerical integration rules.)
  • Lecture 10: Approximation of Integrals -- Continued (iterative approaches to the approximation of integrals, Richardson extrapolation and Romberg integration.)
  • Lecture 11: Gaussian Quadrature (orthogonal polynomials and applications to quadrature.)
  • Lecture 12: Gaussian Quadrature continued (construction of Gaussian quadrature formulas.) Revised 10/28/2015.
  • Lecture 13: Adaptive Quadrature (estimation of local error and adaptive algorithms for numerical integration.) Revised 10/30/2015.
  • Lecture 14: Singular Integrals (techniques for evaluating singular integrals.)
  • Lecture 15: Numerical solution of ordinary differential equations (Euler's method and general Taylor series methods.) Revised 11/2/2015.
  • Lecture 16: Numerical solution of ODEs -- Continued (Runge-Kutta methods.) Revised 11/2/2015, Revised 11/11/2015.
  • Lecture 17: Estimation of local error (estimation of local error and step-size control.)
  • Lecture 18: Linear multistep methods (derivation, order, consistency, local truncation error.) Revised 11/18/2015. Revised 11/20/2015. Revised 11/25/2015.
  • Lecture 19: Convergence of multistep methods (linear difference equations, consistency as a necessary condition for convergence.)
  • Lecture 20: Stability of linear multistep methods (a necessary condition for convergence, maximum order of a zero-stable method, example of numerical instability) Revised 11/25/2015.
  • Lecture 21: Strong, weak, absolute, and relative stability (definitions and examples) Revised 12/7/2015.
  • Lecture 22: Predictor-corrector methods and generalizations to first order systems (comparison of Adams-Bashforth explicit and Adams-Moulton implicit methods, regions of absolute stability for first order systems)
  • Lecture 23: Additional types of stability and stiff differential equations (A-stability, Dahlquist theorems, methods for stiff problems based on numerical differentiation formulas)
  • Lecture 24: Discontinous Galerkin methods for odes (discussion of the basic methods)