Zur Kondition des linearen Ausgleichsproblems mit linearen Gleichungen als Nebenbedingungen (Q760166)

The author studies the condition for the linear least squares problem (LS) with linear equality constraints (LSE): Compute \(x\in R^ n\) with \(A_ 1x=b_ 1\), \(\min \| A_ 2x-b_ 2\|,\quad A=(A_ 1,A_ 2)^ T,\) \(rank(A)=n\). He characterizes the solution by pseudo-inverses and related projections and shows some perturbation results. Then he demonstrates, that the condition number \(\| A\| \cdot \| X\|\) (with X left inverse of A) gives too pessimistic error estimations. He proposes instead three different condition numbers, which give better results. It is noted, that component-wise estimations are possible in the same manner, if component-wise bounds for the data errors are available. Then the question of optimal scaling and minimal condition arises, but this question is not pursued further by the author. If the problems LS and LSE have identical solutions, the condition of LS may be better than that of LSE or vice versa. The different cases are demonstrated with an interesting example.