Efficient Solution of A, x ^(k) = b ^(k) Using A ⁻¹

In this work, we consider the problem of solving ${Ax^{(k)}=b^{(k)}}$ , ${k=1,\ldots, K}$ , where b ^(k+1) =  f(x ^(k)). We show that when A is a full ${n \times n}$ matrix and ${K\geqslant cn}$ , where ${c\ll1}$ depends on the specific software and hardware setup, it is faster to solve ${Ax^{(k)}=b^{(k)}}$ for ${{k = 1,\ldots, K}}$ by explicitly evaluating the inverse matrix A ⁻¹ rather than through the LU decomposition of A. We also show that the forward error is comparable in both methods, regardless of the condition number of A.