On the construction of a normal pseudo-solution for a system of linear equations with a rectangular matrix

A modified $QR$ factorization algorithm is proposed. It allows us to construct a normal pseudo-solution of a System of Linear Algebraic Equations (SLAE) for a rectangular or square degenerate matrix with the same efficiency as for SLAE with a square nonsingular matrix. As an application, the construction of an analytic spline on a degenerate mesh is studied, and a modified algorithm is proposed to provide the “best” spline solution.