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Linear Systems [M12]

Linear systems form an integral part of numerical methods, since continuous problems have to be linearised to be handled by a computer. At the end of the course students should be able to apply numerical techniques to solve linear systems with an understanding of their advantages and disadvantages.

The contents are:

Linear systems

  • Triangular systems and back substitution
  • Gaussian elimination with pivotal strategies and its equivalence to a LU factorisation
  • Choleski factorisation
  • QR factorization
  • Gram-Schmidt algorithm
  • Givens rotations
  • Householder reflections
  • Over-determined systems and linear least squares and singular value decomposition
  • Iterative methods and splitting
  • Jacobi and Gauss-Seidel methods
  • Spectral radius and relaxation
  • Steepest Descent Method
  • Conjugate gradient
  • Krylow subspaces and pre-conditioning
  • Eigenvalues and eigenvectors
  • Power Method
  • Inverse Iteration