The QR-Decomposition of a Matrix

Abstract

In theory, the linear equilibrium problem is simple to solve, requiring only the linear system of equations A ^⊤ A x = A ^⊤ b to solve. In practical applications, the matrix A usually has a lot of rows, so that solving with pencil and paper is no longer possible. But also the (naive) solving of the normal equation with a calculator is not recommended: The calculation of A ^⊤ A and subsequent solving of the LGS A ^⊤ A x = A ^⊤ b is unstable and thus leads to inaccurate results. In the numerical solution of the linear compensation problem, the Q R-decomposition of the matrix A is helpful. With the Q R decomposition, the linear equilibrium problem can be solved in a numerically stable way.