Course Outline
Advanced Numerical Analysis
 
 
1.Error Analysis
Trefethen's Definition of Numerical Analysis, (Trefethen and Bau, pp. 321-327)
Representation of Numbers
IEEE Floating Point Arithmetic
Floating Point Error Analysis, Conditioning
Numerical stability, Inherent error.


2.Interpolation and Approximation
Polynomial interpolation: Lagrange, Nevil, Newton and Hermite methods
Trigonometric interpolation( FFT algorithm)
Ratinal interpolation: inverse and reciprocal differences
Spline and Bspline interpolation


3.Numerical Differentiation and Integration
Newton-Cotes methods
Peano's error representation
The Euler-Maclaurin summation formula
Extrapolation
Gaussian methods
Evaluation of singular integrals
Adaptive methods


4.Numerical solution of Nonlinear equations
Roots of polynomials
Zeros and minimum points (iterative methods)
High order methods
Optimization


MAIN TEXT: Introduction to Numerical Analysis, Stoer and Bulirsch, Springer, 2002.


Further references:
1. Trefethen, Definition of Numerical Analysis , see homepage.
2. Ramon E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.
3. G.M.Phillips, Interpolation and Approximation by Polynomials, Springer, 2003.
4. M. L. Overton, Numerical Computing with IEEE Floating Point Arithmetic, SIAM, 2001.
5. Kahan's 1981 paper, "Why do we need a floating-point arithmetic standard?"
6. C. Moler, Numerical Computing with MATLAB}, SIAM, 2005. "Cleve Moler's floatgui.m"
7. N. J. Higham, Accuracy and Stability of Numerical Algorithms}, 2nd edition, SIAM, Philadelphia, 2002.
8. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, Second Edition, CAMBRIDGE UNIVERSITY PRESS, 2002.
9. P. Henrici, Elements of numerical analysis, Wiley, 1964.
10. Kincaid D., Cheney W. Numerical analysis, Brooks-Cole Publishing Company, 1991.
11. S. D. Conte, Carl de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third Edition, McGraw-Hill Book Company, 1980.
12. Alfio Quarteroni and Riccardo Sacco and Fausto Saleri, Numerical Mathematics, Springer-Verlag, 2000.
13. Steven T. Karris, Numerical Analysis Using MATLAB and Spreadsheets, Second Edition, Orchard Publications, 2004.
14. J. M. McNAMEE, Numerical Methods for Roots of Polynomials, Part I, Elsevier, 2007.
15. D . A. MACDONALD, A NOTE ON THE SUMMATION OF SLOWLY CONVERGENT ALTERNATING SERIES, BIT 36 :4 (1996), 766-774 .
16. O. CAPRANI, ROUNDOFF ERRORS IN FLOATING-POINT SUMMATION, BIT 15 (1975), 5-9.
17. Riesel H. A case of numerical divergence, springer BIT,Volume 1 No. 2 June, 1961.
18. R. L. Burden and J. D. Faires. Numerical Analysis, (Brooks/Cole, 1997).
19. K. Atkinson, An Introduction to Numerical Analysis, second edition, WILEY, 1989.
20. Lars Eldén, Linde Wittmeyer-Koch, Hans Bruun Nielsen: Introduction to Numerical Computation, Analysis and Matlab Illustrations, Studentlitteratur, 2004.
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