Numerical Methods (88-376)
Syllabus
The list of topics below is intended as a rough guide only. The numbers
in square brackets indicate the number of lectures expected to be devoted to
each main topic; note the sum of these numbers is 11, though there will
hopefully be 13 lectures in the semester.
- Error Analysis [1].
Relative and absolute error, vector and matrix norms,
propagation of error in calculations,
error in function calculations, rounding error.
- Solution of Nonlinear Equations [2].
1 dimensional root finding: interval bisection,
Newton's method,
general iterative schemes
(fixed points,
rate of convergence, possibilities of
periodic orbits and chaos).
The multidimensional Newton method.
- Numerical Linear Algebra [2].
Implementation and conditioning of the Ax=b problem.
Implementation and conditioning of the eigenvalue problem.
- Interpolation [2].
Polynomial interpolation, by Newton's and Lagrange's methods.
Cubic splines.
- Approximation Theory [2].
General notions of approximation theory
Least squares for approximating a set of points by a curve.
Orthogonal polynomials.
Least squares for approximating a curve by a curve.
- Numerical Quadrature [1].
Trapezium rule,
Simpson's rule,
Romberg extrapolation,
Gaussiam quadrature.
- Integration of Ordinary Differential Equations [1].
Writing general ODEs as first order systems.
Euler method,
Runge Kutta method,
order of convergence,
choice of step size.
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