642:573 Numerical Analysis: Course Syllabus

Material on most of the topics listed below can be found in A. Quarteroni, R. Sacco, and F. Saleri: Numerical Mathematics, referred to as [QSS] or Kendall Atkinson, An Introduction to Numerical Analysis, referred to as [A].

0: Overview of the Course

1: Approximation by Polynomials, Piecewise Polynomials, and Trigonometric Functions

1.1  Weierstrass approximation theorem, Lagrange and Newton forms of the
     interpolating polynomial. [A: 4.1, 3.1, 3.2], [QSS: 8.1, 8.2]
1.2  Polynomial interpolation error, divided differences for repeated points,
     [A: 3.2, 3.6], [QSS: 8.5]
1.3  Interpolation of moments, Runge example. [A: 3.5], [QSS: 8.1]
1.4  Piecewise polynomial approximation: C^0 and C^1 piecewise polynomial
     approximation and error estimates; construction of basis functions.
     [A: 3.7], [QSS: 8.3]
1.5  Piecewise Polynomial Approximation: cubic spline approximation, basis
     functions, and error estimates [A:3.7], [QSS: 8.7]
1.6  Trigonometric interpolation; fast Fourier transform [A:3.8], [QSS: 10.9]
1.7  Piecewise polynomial approximation in higher dimensions [QSS: 8.6]

2: Numerical Differentiation and Integration

2.1  Numerical differentiation [A: 5.7], [QSS: 10.10]
2.2  Basic and composite rules for numerical integration [A: 5.1, 5.2],
     [QSS: 9.1, 9.2, 9.3, 9.4]
2.3  Extrapolation and Romberg integration [A: 5.4], [QSS: 9.6]
2.4  Orthogonal polynomials and Gaussian quadrature [A: 4.4, 5.3],
     [QSS: 10.1, 10.2, 10.3, 10.4, 10.5, 10.6]
2.5  Orthogonal polynomials and Gaussian quadrature (continued)
2.6  Adaptive quadrature [A: 5.5], [QSS: 9.7]
2.7  Singular integrals [A: 5.6], [QSS: 9.8]

3: Numerical Methods for Ordinary Differential Equations

3.1  The initial value problem for ordinary differential equations 
     [A: 6.1], [QSS: 11.1]
3.2  Euler and general Taylor series methods [A: 6.2], [QSS: 11.1, 11.2, 11.3]
3.3  Runge-Kutta methods [A: 6.10], [QSS: 11.8]
3.4  Estimation of local error and adaptive methods [A: 6.10], [QSS: 11.8.2]
3.5  Linear multistep methods (derivation, order, consistency, local 
     truncation error) [A: 6.3, 6.4, 6.5], [QSS: 11.5, 11.6]
3.6  Convergence of linear multistep methods [A: 6.8], [QSS: 11.4, 11.6]
3.7  Stability of linear multistep methods [A: 6.8], [QSS: 11.6]
3.8  Stability of linear multistep methods (continued)
3.9  Predictor-corrector methods [A: 6.6, 6.7], [QSS: 11.7]
3.10 First order systems of odes [QSS: 11.9]
3.11 Stiff systems of odes [6.9], [QSS: 11.10]
3.12 The discontinuous Galerkin method