A concise algorithm to solve over-/under-determined linear systems

Abstract

An O(mn2) direct algorithm to compute a solution of a system of m linear equations Ax=b with n variables is presented. It is concise and matrix inversion- free. It provides an in-built consistency check and also produces the rank of the matrix A. Further, if necessary, it can prune the redundant rows of A and convert A into a full row rank matrix thus preserving the complete information of the system. In addition, the algo rithm produces the unique projection operator that projects the real (n)-dimensional space orthogonally onto the null space of A and that provides a means of computing a relative error bound for the solution vector as well as a nonnegative solution.

The minimum-norm least-squares solution of a linear system and symmetric rank-one updates

“…By employing the rank-one update formula for generalized inverse, the MNLS solutions to all systems of linear equations in the sequence can be successively computed. Like the algorithms in [9,11], there is no need to compute the generalized inverses of all the intermediate matrices in the procedure.…”

Solving linear program as linear system in polynomial time

“…Or, one may choose to bypass this linearly dependent equation and go to the next one. The following algorithm which is a modified version of the algorithms in Lord et al [10,11] and which is better from comprehension point of view, finds a particular solution x. It obtains as a by-product the orthogonal projection operator [12][13][14][15][16], without explicitly computing the minimum-norm least-squares inverse A + = A − mt , where I is the identity (unit) matrix of order n. Also, in-built in the algorithm is the computation of the rank r of A.…”

“…It may be observed that there exists so far no mathematically non-iterative technique to solve an LP. Only in the case of a special optimization problem where we are required to obtain the minimumnorm least-squares solution or a minimum-norm solution or a least-squares solution, instead of optimizing a linear function c t x, we have a direct (non-iterative) algorithm [6,7,9,18,10,12,14,15]. Solving an LP deterministically mathematically noniteratively in polynomial time is yet an open problem.…”

A direct heuristic algorithm for linear programming

“…In step S.3, we may compute the orthogonal projection operator P I À H I À A A directly (without computing A or A A using the concise algorithm for linear systems [18,32,19] However, instead, we compute the n  n matrix H A A in Omn 2 operations, c H I À Hc in On 2 operations, the scalar s k in On operations, and the n  1 solution vector x d À c H s k in On operations.…”