On Preconditioning

Abstract

As mentioned in §1.2, it is usually necessary to precondition an interval linear system to obtain meaningful bounds on the solution set. This is clear for the interval Gauss-Seidel method, since it reduces to the classical Gauss-Seidel method when the coefficients of the system of equations are thin, and since the classical Gauss-Seidel method in general does not converge without preconditioning. However, preconditioning is also necessary and useful for the Krawczyk method and interval Gaussian elimination. For example, if interval Gaussian elimination, i.e. Algorithm 1 is applied to the system $AX = B$ of (1.19) on page 20, the result is

$$ \left( {\frac{{\left[ {2,4} \right]\left| {\left[ { - 2,1} \right]} \right|\left[ { - 2,2} \right]}}{{\left[ { - 1,2} \right]\left| {\left[ {2,4} \right]} \right|\left[ { - 2,2} \right]}}} \right)\left( {\frac{{\left[ {2,4} \right]\left| {\left[ { - 2,1} \right]} \right|\left[ { - 2,2} \right]}}{{0\left| {\left[ {2,4} \right] - \frac{{\left[ { - 2,1} \right]}}{{\left[ {2,4} \right]}}\left[ {2,4} \right]} \right|\left[ { - 2,2} \right] - \frac{{\left[ { - 2,2} \right]}}{{\left[ {2,4} \right]}}\left[ { - 2,2} \right]}}} \right) = \left( {\frac{{\left[ {2,4} \right]\left| {\left[ { - 2,1} \right]} \right|\left[ { - 2,2} \right]}}{{\left[ { - 1,2} \right]\left| {\left[ { - 4,4} \right]} \right|\left[ { - 6,6} \right]}}} \right) $$

which gives only ${x_2} \in \frac{{\left[ { - 6,6} \right]}}{{\left[ { - 4,4} \right]}} = {\kern 1pt} and{\kern 1pt} {x_1} \in$, whereas, as is seen in Gigure 1.1, the smallest possible interval enclosure of the solution set is $\sum {(A,B) = {{\left( {\left[ { - 4,4} \right],\left[ { - 4,4} \right]} \right)}^T}}$. Similarly, applying the interval Gauss-Seidel method results in $\overline {{x_1}} \leftarrow \frac{{\left[ { - 2,2} \right] - \left[ { - 2,1} \right]\left[ { - 10,10} \right]}}{{\left[ {2,4} \right]}} = \left[ { - 11,11} \right] \supset \left[ { - 10,10} \right]{\kern 1pt} and{\kern 1pt} \overline {{x_2}} \leftarrow \frac{{\left[ { - 2,2} \right] - \left[ { - 1,2} \right]\left[ { - 10,10} \right]}}{{\left[ {2,4} \right]}} = \left[ { - 11,11} \right] \supset \left[ { - 10,10} \right]$. Finally, the Krawczyk method only makes sense in the context of presconditioners.