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Fundamentals, Non-linear Systems and Interpolation [M12]

The aims of this course are to provide introductions to floating-point arithmetic and numerical techniques. The principles of good numerical methods will be illustrated by examples, but it will be shown that the design of a numerical algorithm is not necessarily straightforward, even for simple problems - that the solution has to fit the problem. At the end of the course students should be able to apply numerical techniques with an understanding of their underlying principles.

The contents are:

Fundamentals

  • Floating point arithmetic
  • Overflow and underflow
  • Floating point arithmetic
  • Absolute, relative error and machine epsilon
  • Forward and backward error analysis
  • Loss of significance
  • Robustness
  • Convergence, error testing and order of convergence
  • Condition
  • Computational complexity

Non-linear Systems

  • Bisection, rule of false position, secant method
  • Newton-Raphson method
  • Broyden's method
  • Householder methods
  • Muller's method and inverse quadratic interpolation
  • Fixed-point iteration theory
  • Mixed methods

Interpolation

  • Polynomial interpolation
  • Lagrange formula
  • Divided Differences and Newton's formula
  • Polynomial Best Approximations
  • Orthogonal (Chebychev) polynomials and their recurrence relations
  • Least-squares Polynomial Fitting
  • Peano Kernel Theorem
  • Splines and B-splines