The Solution of Updating or Regionalizing a Matrix with both Positive and Negative Entries

Normally, when updating or regionalizing input-output matrices with negative entries, the negative numbers are first brought outside the matrix, then the matrix is updated or regionalized, then the negative numbers are added back to the result. This is theoretically, and sometimes also empirically, a rather unsatisfactory procedure. This paper proposes a theoretically sound alternative for the presently used ad hoc procedure. Based on the first-order conditions of a restated information loss problem, we generalize the RAS-procedure using reciprocals of the exponential transformations of the related Lagrange multipliers. The diagonal matrices that update or regionalize a given matrix optimally are the solutions of a fixed-point problem. To derive a numerical solution, the paper presents the GRAS-algorithm, which is illustrated in terms of a simple updating example.