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The topics
of this exam are covered in undergraduate courses such as MATE 4061-4062
and in the graduate course MATE 6672.
Topics:
·
Floating
point numbers: representation of real numbers in finite precision, arithmetic in
finite precision, error bounds.
·
Numerical
methods for solving linear systems: Gaussian elimination and its
variants, LU factorization, Basic iterative methods: Richardson, Jacobi,
Gauss-Seidel and SOR.
·
Numerical
methods for solving non-linear systems: Fixed point iteration, Newton method,
bisection, Regula-Falsi, Aitken’s
method.
·
Basic
theory of polynomial approximation: Weierstrass
approximation theorem, Berstein polynomials,
Lagrange interpolation polynomials, Legendre, Chebyshev, Hermite and Laguerre polynomials, least squares approximation.
·
Numerical
integration: interpolatory and Gaussian quadratures, integration rules for singular and
multiple integrals.
·
Numerical
differentiation; approximation formulas of derivatives, error and order
of approximation, introduction to the finite differences method for
solving two-point boundary problems or partial differential equations,
Von Neumann stability analysis.
·
Numerical
methods for solving ordinary differential equations: Euler-Cauchy method,
Runge-Kutta method, predictor-corrector, one
step and multi-step methods, consistency, convergence and stability,
shooting method for boundary value problems.
Bibliography:
·
An introduction to numerical Analysis, E. Süli and D. Mayers,
Cambridge University Press, 2003.
·
Scientific Computation, Deuflard,
Springer Verlag, 2000.
·
A first course in numerical analysis, A. Iserles, Cambridge University Press, 1999.
·
Numerical analysis: an introduction, W. Gautschi, Birkhäuser,
1997.
·
Numerical methods for Ordinary differential Equations, Hairer and Wanner, Springer Verlag
1999.
·
Finite difference Schemes and Partial Differential Equations, J.C. Strikwerda, 2nd Ed, SIAM
2004.
·
Numerical Analysis,
K. Atkinson and W. Han.
Springer Verlag, 2000.
·
Analysis of Numerical Methods, E.
Isaacson and H. B. Keller, Dover
Pub. Inc., 1994.
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