r6.02.01 About decomposition methods like GAUSS

Summary:

This document shows a few aspects related to the decomposition method of GAUSS.

After a quick overview, we recall the main advantages and disadvantages associated with this direct method. Then, we detail the implementation of algorithm LDLT implemented in the*Code Aster*.

GAUSS (1777 - 1855) is at the origin of all direct methods for the numerical resolution of linear systems. Thank you for that here.

Table of Contents

• 1. Overview of GAUSS methods
  • 1.1. Presentation of the method.
  • 1.2. Notion of pivot
  • 1.3. Stability of the method
  • 1.4. Uniqueness of decomposition
  • 1.5. A variant: the factorization of CROUT
  • 1.6. The case of symmetric matrices
• 2. Disadvantages of type GAUSS methods
  • 2.1. The number of operations
  • 2.2. Filling the matrix
  • 2.3. The loss of precision during calculation
  • 2.4. Criteria for determining a zero pivot
• 3. Method LDLT by blocks implemented in Aster
  • 3.1. Implementation of factorization
  • 3.2. Implementation of the resolution
  • 3.3. Scaling
  • 3.4. Pivot tests
  • 3.5. Factorization of complex matrices
• 4. Bibliography
• 5. Description of document versions