Singular Value Decomposition, Hessian Errors, and Linear Algebra of Non-parametric Extraction of Partons from DIS

Abstract

By singular value decomposition (SVD) of a numerically singular Hessian matrix and a numerically singular system of linear equations for the experimental data (accumulated in the respective ${\chi ^2}$ function) and constraints, least square solutions and their propagated errors for the non-parametric extraction of Partons from $F_2$ are obtained. SVD and its physical application is phenomenologically described in the two cases. Among the subjects covered are: identification and properties of the boundary between the two subsets of ordered eigenvalues corresponding to range and null space, and the eigenvalue structure of the null space of the singular matrix, including a second boundary separating the smallest eigenvalues of essentially no information, in a particular case. The eigenvector-eigenvalue structure of "redundancy and smallness" of the errors of two pdf sets, in our simplified Hessian model, is described by a secondary manifestation of deeper null space, in the context of SVD.