Let us now combine the results from Subsection 6.4.1 and Subsection 6.4.2 into a backward error result for solving \(A x = y \) via LU factorization and two triangular solves.
Theorem6.4.3.1.
Let \(A \in \Rnxn \)and \(x,y \in \Rn \) with \(A
x = y \text{.}\) Let \(\check x \) be the approximate solution computed via the following steps:
Compute the LU factorization, yielding approximate factors \(\check L\) and \(\check U \text{.}\)
Solve \(\check L z = y \text{,}\) yielding approximate solution \(\check z \text{.}\)
Solve \(\check U x = \check z \text{,}\) yielding approximate solution \(\check x \text{.}\)
Then
\begin{equation*}
( A + \Delta \!\! A) \check x = y
\quad
\mbox{with}
\quad
\vert \Delta \!\! A \vert \leq ( 3 \gamma_n + \gamma_n^2 ) \vert
\check L
\vert \vert \check U \vert.
\end{equation*}
We refer the interested learner to the proof in the previously mentioned papers [6][7].
Homework6.4.3.1.
The question left is how a change in a nonsingular matrix affects the accuracy of the solution of a linear system that involves that matrix. We saw in Subsection 1.4.1 that if
\begin{equation*}
A x = y \mbox{ and } A ( x + \delta\!x ) = y + \delta\!y
\end{equation*}
then
\begin{equation*}
\frac{\| \delta\!x \|}{\| x \|} \leq \kappa( A )
\frac{\| \delta\!y \|}{\| y \|}
\end{equation*}
when \(\| \cdot \| \) is a subordinate norm. But what we want to know is how a change in \(A \) affects the solution:
\begin{equation*}
A x = y \mbox{ and } ( A + \Delta\!A ) ( x + \delta\!x ) = y
\end{equation*}
then
\begin{equation*}
\frac{\| \delta\!x \|}{\| x \|} \leq
\frac{\kappa( A )
\frac{\| \Delta\!A \|}{\| A \|}}
{1 - \kappa( A )
\frac{\| \Delta\!A \|}{\| A \|}}.
\end{equation*}
