Lecture 3 of 'Scientific Computing' (wi4201)

• The following subjects are discussed:
  
  
  • Finite difference method for 1D Laplace problem
  • Properties of the matrix
  • Eigenvalues and eigenvectors of the matrix
  • Positive definite matrix
  • Comparison of eigenvalues of the continuous operator and the discretized operator
  • Finite difference method for 2D Laplace problem
  • Boundary conditions
  • Lexicographic and red black ordering
  • Eigenvalues and eigenvectors of the 2D matrix
  • Proofs of properties of matrix A
  • M-matrix
    
    
• Material is described in pages 7, and 28 - 40 of the lecture notes. Recommended exercises: 2.12.11, 2.12.12, 3.12.11 - 3.12.17
  
  
• Typo's
  
  
  • In Definition 2.9.1 the following text should be added:
    The matrix A is a Z-matrix if a_i,j \lte 0 for all i,j such that i \no j.
    Furthermore in the definition of M-matrix it should be added that A is a Z-matrix.
    
  • Exercise 2.12.11 replace the final sentence of part 1 with:
    Show that if a symmetric matrix A is positive definite then all its leading principal minors are positive.
  • Exercise 2.12.16 The question is: Determine the sparsity structure of the discretization matrix for the five-point stencil if the grid points are ordered in alternating lines