New condition numbers for matrices and linear systems (Q1116284)

The author states that the well-known condition number for matrices, \(\gamma (A)=\| A^{-1}\| \| A\|\), has two disadvantages, i.e., firstly, the dependence on a norm, secondly, that no information about the behaviour of the individual entries of \(A^{-1}\) is available. Hence, a new condition number, c(A), is proposed, where \(c(A)=\max \{(| A^{-1}| \cdot | A| \cdot | A^{- 1}|)_{ij}/| A^{-1}_{ij}|:A^{-1}_{ij}\neq 0\}.\) Further a related condition number c(A,b) for linear systems \(Ax=b\) is introduced.