This paper gives an iterative procedure for approximately finding the unique solution \(x_ 1,x_ 2,\dots\), \(x_ n\) of equations in the form of \(a_{i1} x_ 1+ a_{i2} x_ 2+\cdots+ a_{in} x_ n+ b=0\) \((i= 1,2,\dots, n)\) as follows:
Step 1: Multiplying with corresponding constants so that \(\sum^ n_{k= 1} a^ 2_{ik}\simeq 1\) \((i= 1,2,\dots, n)\).
Step 2: Choosing a 1st approximation \(x^ 0_ 1,x^ 0_ 2,\dots, x^ 0_ n\) arbitrarily as the zero group for the following solution groups: \(x^{(p,1)}_ 1, x^{(p,1)}_ 2,\dots, x^{(p,1)}_ n,\dots, x^{(p,n)}_ 1, x^{(p,n)}_ 2,\dots, x^{(p,n)}_ n\) \((p= 0,1,2,\dots)\), each of which consists of \(n\) approximate solutions.
Step 3: Executing the following iteration \(x^{(p+1, 1)}_ i= x^{(p,n)}_ i- a_{1i}\cdot L^{(p,n)}_ 1\), \(x^{(p+ 1,2)}_ i= x^{(p+1, 1)}_ i- a_{2i}\cdot L^{(p+ 1,1)}_ 2,\dots, x^{(p+1, n)}_ i= x^{(p+ 1,n-1)}_ i- a_{ni}\cdot L^{(p+1, n-1)}_ n\) \((i= 1,2,\dots, n)\) until the sequence of answers \(x^{(r,s)}_ 1, x^{(r,s)}_ 2,\dots, x^{(r,s)}_ n\) \((s= 1,2,\dots, n;\;r= 1,2,\dots)\) converges towards \(x_ 1,x_ 2,\dots, x_ n\), where \(L^{(r,s)}_ i= \sum^ n_{k=1} a_{ik} x^{(r,s)}_ k+ b_ i\) is the left-hand side of the \(i\)th equation, and the unknowns \(x^{(r,s)}_ 1, x^{(r,s)}_ 2,\dots, x^{(r,s)}_ n\) are the \(s\)th solution of the \(r\)th group.
The convergence of the iteration is proved.