Solving systems of linear equations
Basic numerical methods - Numerical algebra

Let A be a square matrix with n rows and n columns whose elements a_ij are known. Let b be a vector whose n components b_i are known. We search for the vector x, of components x₁ , x₂ , ..., x_n , which verifies the system of linear equations :

The solution to this problem is well known, and can be found in all linear algebra courses: x_i is equal to a ratio of determinants, with the determinant of the matrix A in the denominator and the same determinant in the numerator, in which the i^i-th column has been replaced by the second member vector b. The rules for calculating a determinant are also classic. However, we often forget to mention that such a calculation requires n · n! multiplications, i.e. of the order of n²...