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Assuming n>1, we can add some
amount of the previous step
to the chosen direction
to produce a new search direction
, as
follows:
| ![\begin{displaymath}
{\bf s}_n^{(n-1)} = {\bf c}_n + \beta_n^{(n-1)}\,{\bf s}_{n-1}\;,\end{displaymath}](img24.gif) |
(10) |
where
is an adjustable scalar coefficient. According to
to the fundamental orthogonality principle (7),
| ![\begin{displaymath}
\left({\bf
r}_{n-1},\,{\bf A\,s}_{n-1}\right) = 0\;.\end{displaymath}](img26.gif) |
(11) |
As follows from equation (11), the numerator on the right-hand
side of equation (9) is not affected by the new choice of the
search direction:
| ![\begin{displaymath}
\left({\bf r}_{n-1},\,{\bf A\,s}_n^{(n-1)}\right)^2 = \left[...
...ight)\right]^2 =
\left({\bf r}_{n-1},\,{\bf A\,c}_n\right)^2\;.\end{displaymath}](img27.gif) |
(12) |
However, we can use transformation (10) to decrease the
denominator in (9), thus further decreasing the residual
. We achieve the minimization of the denominator
| ![\begin{displaymath}
\Vert{\bf A\,s}_n^{(n-1)}\Vert^2 = \Vert{\bf A\,c}_n\Vert^2 ...
... +
\left(\beta_n^{(n-1)}\right)^2\,\Vert{\bf A\,s}_{n-1}\Vert^2\end{displaymath}](img28.gif) |
(13) |
by choosing the coefficient
to be
| ![\begin{displaymath}
\beta_n^{(n-1)} = - {{\left({\bf A\,c}_n,\,{\bf A\,s}_{n-1}\right)} \over
{\Vert{\bf A\,s}_{n-1}\Vert^2}}\;.\end{displaymath}](img29.gif) |
(14) |
Note the analogy between (14) and (6). Analogously to
(7), equation (14) is equivalent to the orthogonality condition
| ![\begin{displaymath}
\left({\bf A\,s}_n^{(n-1)},\,{\bf A\,s}_{n-1}\right) = 0\;.\end{displaymath}](img30.gif) |
(15) |
Analogously to (8), applying formula (14) is also equivalent to defining the
minimized denominator as
| ![\begin{displaymath}
\Vert{\bf A\,c}_n^{(n-1)}\Vert^2 = \Vert{\bf A\,c}_n\Vert^2 ...
... A\,s}_{n-1}\right)^2} \over
{\Vert{\bf A\,s}_{n-1}\Vert^2}}\;.\end{displaymath}](img31.gif) |
(16) |
Next: Second step of the
Up: IN SEARCH OF THE
Previous: IN SEARCH OF THE
Stanford Exploration Project
9/11/2000