A posteriori error estimates for linear equations (Q1185487)

Given a linear system \(A\hat x=b\) and an approximate solution \(x\) with corresponding residual \(r=b-Ax\), this short paper shows that \(\| x- \hat x\|_ p\) is proportional to \(\| r\|^ 2_ 2/\| A^ Tr\|_ q\), where \(p\in[1,\infty]\) and \(q=p/(p-1)\). Since the constant of proportionality is bounded below by 1, and since the presence of the \(A^ Tr\) term in the bound will cause it to change even when \(\| r\|\) stays constant, this error expression provides different information from the traditional error bound, which is based on the ratio \(\| r\|_ p/\| b\|_ p\). The author studies some of the properties of the constant of proportionality, and extends his theory to linear mappings in infinite-dimensional spaces.