1. Introduction#

Given a real matrix \(A\) of order \(m\times n\) and a vector \(b\) element of \({ℝ}^{m}\), we consider the problem of determining a vector \(x\) element of \({ℝ}^{n}\) that verifies the following linear system:

\(\text{Ax}=b\) eq 1-1

It is well known ([bib3] p. 9) that this system admits one, and only one solution, for every \(b\) element of \({ℝ}^{m}\) under the necessary and sufficient conditions that it be equi-constrained \((m=n)\) and that its matrix \(A\) be regular. Also, the investigation of the cases under stress \((m\le n)\) and under stress \((m\ge n)\) will confront us with one of the following three situations:

  1. The linear system [éq 1-1] has one solution and only one solution,

  2. The linear system [éq 1-1] does not allow a solution,

  3. The linear system [éq 1-1] has an infinity of solutions.

In practice, situation 2) is generally encountered in the case of an over-stressed system, whereas singular and under-stressed equi-constrained systems generally lead to situation 3).

To pretend to solve a linear system like [éq 1-1], we must first define what we will call solution. This is the purpose of paragraph 2, which is based mainly on the concept of least squares and on differentiable optimization to define, regardless of the type of system, a solution that is always unique.

Paragraph 3 is devoted to the decomposition into singular values of matrices (abbreviated SVD: Singular Value Decomposition), which not only constitutes a tool for diagnosing which of the three previous situations corresponds to the linear system studied, but also provides a method for determining the solution defined in paragraph 2.

The method using decomposition SVD is presented in paragraph 4 and is compared there to the normal equations method.

paragraph 5 algebraically details the application of the SVD method to the resolution of an equi or sub-constrained linear system as implemented in the*Code_Aster*.

In the following paragraphs, we will use the following notations:

  • \(\parallel x\parallel\) and \((\text{x,y})\) for, respectively, the Euclidean norm of the vector \(x\) and the associated dot product of the vectors \(x\) and \(y\) elements of \({ℝ}^{m}\) or \({ℝ}^{n}\), respectively,

  • \({M}^{T}\) for the transpose of the matrix \(M\),

  • \(\text{Ker}M\) and \(\text{Im}M\) for, respectively, the kernel and the image of (the linear application associated with) the matrix \(M\),

  • \({X}^{\perp }\) for the orthogonal of the subspace \(X\) of \({ℝ}^{m}\) or \({ℝ}^{n}\).