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

摘要

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