ANALYTICAL MATRIX INVERSE CALCULATIONS, APPLYING PREDICTIVE COORDINATION T. Stoilov and K. Stoilova Institute of Computer and Communication Systems - Bulgarian Academy of Sciences Acad.G.Bonchev str. Bl.2, 1113 Sofia, tel. (359 2) 979 27 74; e -mail: todor@hsi.iccs.bas.bg Abstract : The research derived analytical relations between the components of a square matrix and it s inverse. The relations have been worked out, applying a hierarchical approach and non-iterative concept of coordination. The analytical relations have been assessed by computational performance. It has been proved that for large -scale matrices, these relations are computationaly efficient. Copyright © 2004 IFAC Key words: hierarchical systems theory, coordination, matrix inverse calculations 1. INTRODUCTION Some numerical algorithms involve solving a succession of linear systems Ax=b, each of which slightly differs from its predecessor in matrix A * . Instead of solving each time the equations from scratch, one can often update a matrix factorizations of A to find the new inverse A *-1 having A -1 . Thus the calculation of matrix A -1 , which is the inverse of a square matrix A , is a key stone in the implementation of real time control, optimization and management algorithms and on- line decision policies. Practically, three types of factorizations are under way by numerical calculations: LU factorizations (Fausett, 1999), QR decomposition and SVD-singular value decomposition (Flannery, 1997). The peculiarities of the LU, QR and SVD factorization techniques define the computational efficiency in finding A -1 . It is worth to find methods for evaluating an inverse matrices A * , which slightly differ from the initial one A. The new A * can skip one or several columns/rows of A, or to have only few new components a ij . Thus the utilization of the inverse A -1 or several of its components in finding the new inverse A *-1 can speed a lot the computational efforts. Attempts in finding results for matrix inversion are given in (Strassen, 1969). For a given matrix A with a structure (a) where a ij could not be only scalar components, but also appropriate matrices, the inverse matrix (b) can be calculated. 22 21 12 11 a a a a A = (a) 22 21 12 11 1 α α α α α= = - A (b) If it is noted 1 5 6 22 4 5 3 21 4 12 1 3 1 21 2 1 11 1 - - = - = = = = = R R a R R R a R a R R R a R a R (1) the components ij α of the inverse matrix A -1 =a are found according to the equalities: 7 1 11 2 6 21 6 22 21 3 7 6 3 12 R R R R R R R R R - = = - = = = α α α α α (2) In these relations the “inverse operator” occurs only twice on matrices, which have lower dimensions in comparison with the initial A. Thus it is worth to perform the inversion of A not by factorization, but applying relations (1) and (2), especially for the case of large scale N of A. This research presents an appropriate result in finding the inverse matrix of an initial one A with dimension NxN (N is a big value). Here A is a symmetric, A=A T . Analytical relations are derived between the components a ij of A and ij α of a=A -1 . These relations have been worked out, applying a hierarchical approach and non-iterative concept of coordination (Stoilov and Stoilova, 1999). The analytical relations have been assessed by computational performance. It has been proved IFAC DECOM-TT 2004 Automatic Systems for Building the Infrastructure in Developing Countries October 3 - 5, 2004 Bansko, Bulgaria This research is supported in part by the Information and Communication Technology Development Agency, Ministry of Transportation, Republic Bulgaria, contract o ИД 14/1.7.2004. 1 1