642:573 Numerical Analysis: Course Syllabus

Material on most of the topics listed below can be found in:

K. Atkinson, An Introduction to Numerical Analysis,,Wiley, 1989 (second edition), referred to as [A].

D. Kincaid and W. Cheney: Mathematics of Scientific Computing, AMS, 2002 (third edition), referred to as [KC].

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Springer, 2004 (second edition), referred to as [QRS]

J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, Springer, 2002, (third edition) referred to as [SB].

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], [KC: 6.1, 6.2],
[QSS: 8.1, 8.2], [SB: 2.1.1, 2.1.3]
1.2  Polynomial interpolation error, divided differences for repeated points,
[A: 3.2, 3.6], [KC: 6.2, 6.3], [QSS: 8.5], [SB: 2.1.4, 2.1.5]
1.3  Interpolation of moments, Runge example. [A: 3.5], [KC: 6.1],
[QSS: 8.1], [SB: 2.1.4]
1.4  Piecewise polynomial approximation: C^0 and C^1 piecewise polynomial
approximation and error estimates; construction of basis functions.
[A: 3.7], [KC: 6.4], [QSS: 8.3]
1.5  Piecewise Polynomial Approximation: cubic spline approximation, basis
functions, and error estimates [A:3.7], [KC: 6.5, 6.6], [QSS: 8.7],
[SB: 2.4.1, 2.4.2, 2.4.3, 2.4.4, 2.4.5]
1.6  Trigonometric interpolation; fast Fourier transform [A:3.8],
[KC: 6.12, 6.13], [QSS: 10.9], [SB: 2.3.1, 2.3.2]
1.7  Piecewise polynomial approximation in higher dimensions [KC: 6.10],
[QSS: 8.6]
```

2: Numerical Differentiation and Integration

```2.1  Numerical differentiation [A: 5.7], [KC: 7.1], [QSS: 10.10]
2.2  Basic and composite rules for numerical integration [A: 5.1, 5.2],
[KC: 7.2], [QSS: 9.1, 9.2, 9.3, 9.4], [SB: 3.1, 3.2]
2.3  Extrapolation and Romberg integration [A: 5.4], [KC: 7.4], [QSS: 9.6],
[SB: 3.4, 3.5]
2.4  Orthogonal polynomials and Gaussian quadrature [A: 4.4, 5.3],
[KC: 6.8, 7.3], [QSS: 10.1, 10.2, 10.3, 10.4, 10.5, 10.6], [SB: 3.6]
2.5  Orthogonal polynomials and Gaussian quadrature (continued)
2.7  Singular integrals [A: 5.6], [QSS: 9.8], [SB: 3.7]
```

3: Numerical Methods for Ordinary Differential Equations

```3.1  The initial value problem for ordinary differential equations
[A: 6.1], [KC: 8.1], [QSS: 11.1], [SB: 7.0,7.1]
3.2  Euler and general Taylor series methods [A: 6.2], [KC: 8.2],
[QSS: 11.1, 11.2, 11.3], [SB: 7.2.1, 7.2.2, 7.2.3, 7.2.4]
3.3  Runge-Kutta methods [A: 6.10], [KC: 8.3], [QSS: 11.8],
[SB: 7.2.1, 7.2.2, 7.2.3, 7.2.4]
3.4  Estimation of local error and adaptive methods [A: 6.10], [QSS: 11.8.2],
[SB: 7.2.5]
3.5  Linear multistep methods (derivation, order, consistency, local
truncation error) [A: 6.3, 6.4, 6.5], [KC: 8.4], [QSS: 11.5, 11.6],
[SB: 7.2.6, 7.2.7]
3.6  Convergence of linear multistep methods [A: 6.8], [KC: 8.5],
[QSS: 11.4, 11.6], [SB: 7.2.7, 7.2.8, 7.2.9, 7.2.10, 7.2.11]
3.7  Stability of linear multistep methods [A: 6.8], [KC: 8.5], [QSS: 11.6],
[SB: 7.2.9, 7.2.11, 7.2.12, 7.2.16]
3.8  Stability of linear multistep methods (continued)
3.9  Predictor-corrector methods [A: 6.6, 6.7], [QSS: 11.7], [SB: 7.2.6]
3.10 First order systems of odes [KC: 8.6], [QSS: 11.9]
3.11 Stiff systems of odes [A: 6.9], [QSS: 11.10], [SB: 7.2.16]
3.12 The discontinuous Galerkin method
```