A Modified Version of ABS Algorithms

Solving Ax=b where A=(a_1,…,a_m)~T∈R~m,n,x∈R~n,b∈R~m,by the ABS al-gorithm,we have the gelleral solution for the first i equations being of the form x=x-i 1 H_i 1~Tq,q∈R~n.Construct Z_i 1 such that Rang(Z_i 1)=Rang(H_i 1~T)=Null(A_i)and Z_i 1 is of full rank in column.Thus,x=x_(i 1) Z_i 1,q∈R~n-i.The modifiedalgorithms are based upon the idea that Z_i 1 is of full rank at each step.