Fast Algorithms Of The Minimal Norm Least Square Solution And Left Or Right Inverse For Special Rectangular Matrices

Solving linear equations arise in a surprising number in the computing problems of engineering, but sometimes they are unsolvable. In this paper fast algorithms are presented which compute the minimal norm least square solutions for linear equations with special rectangular matrices coefficients, such as Vandermonde matrices, Toeplitz matrices, Loewner matrices etc. And then, this paper presents an algorithm of computing the left inverse or right inverse for these special rectangle matrices. In thenormal algorithms for solving these problems, we need O(m2n) multiplications ordivisions. The algorithms in this paper only need O(mn) + O(n2) multiplications ordivisions. The paper is built as follows.In Chap 1, we introduce the applied context of the minimal norm least square solutions for Vandermonde matrix first. The fast algorithm of the minimal norm least square solutions for Vandermonde matrix with mxn order and its transpose arepresented later through constructing VTV (or WT ) and seeking their inverses.In Chap 2, through researching the fast triangular factorization of (ATA)-1and(AAT)-1, another fast algorithm of the minimal norm least square solutions forlinear equations which coefficients are mxn Loewner matrix, Vandermonde matrix, Toeplitz matrix are given.In Chap 3, we give the definitions of left inverse and right inverse; present the fast algorithms of left inverse and right inverse for Toeplitz matrix, Hankel matrix, Loewner matrix and Vandermonde matrix with mxn order.In Chap 4, some numerical examples are presented to check the correctness of the algorithms.