##
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)