Exact generalized inverses and solution to linear least squares problems using multiple modulus residue arithmetic

The floating point number system on the computer does not accurately model the real number system; consequently, the condition of data and error due to rounding must be considered. A multiple modulus number system, however, is capable of exactly modeling the rational number system. Therefore, with only the assumption that the problem has rational entries, an exact solution can be sought. Methods are developed which use multiple modulus residue arithmetic to compute exact reflexive and exact Moore-Penrose inverses for an arbitrary nxs matrix. The methods are then applied to find exact solutions to linear least squares problems. An important distinction between working over the field of rational numbers and working over a Galois field generated by the integers 0, 1, 2, ..., p-1, p-prime, is made. This enables us to develop the multiple modulus theory necessary to find exact generalized inverses. Exact generalized inverses have been previously computed using single modulus arithmetic; however, the single modulus solution is unsatisfactory from an applied point of view because the size of the modulus necessary to complete the problem is usually too large to fit in a standard computer fixed-point word. Algorithms are developed and implemented which compute these exact generalized inverses and exact solutions to the linear least squares problems