r6.03.01 Resolution of non-regular systems by a method of decomposition into singular values

Summary:

This document is devoted to the resolution of systems of non-regular linear equations. The matrices taken into account may be square, non-invertible, or rectangular.

After recalling the theoretical framework of solutions in the sense of least squares, we focus the paper on the method by decomposition into singular values, which provides, on the one hand, a tool for diagnosing the degree of regularity of the system, and, on the other hand, a family of resolution algorithms that are both more general and more stable than those deriving from the normal equation approach.

Finally, we detail the algorithm implemented in the*Code_Aster* which resolves the equivalent functionality of the Nag library (F04 JDF for version 12 and F04 JDE for version 15) used for modeling the metallurgical behavior of steels [R4.04.01].

• 1. Introduction
• 2. Solution of a rectangular linear system
  • 2.1. Least squares formalism
  • 2.2. Existence of optimums
  • 2.3. Uniqueness of the optimum and ranking of the system
  • 2.4. Solution in the sense of least squares
• 3. Singular values
  • 3.1. Breakdown into singular values
  • 3.2. Rank, image and core
  • 3.3. Pseudo-inverse and least squares solutions
• 4. Solving a rectangular linear system
  • 4.1. Method of normal equations
  • 4.2. Singular value decomposition method
  • 4.3. Comparison of the method of normal equations to the method of decomposition into singular values
• 5. Algorithm SVD for solving an equi or substressed linear system
  • 5.1. Problem reduction and algorithm principle
  • 5.2. Reduction to the upper triangular shape
  • 5.3. Reduction to the upper bi-diagonal shape
  • 5.4. Decomposition SVD of a higher bidiagonal
• 6. Bibliography
• 7. Description of document versions