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Continuing by induction the process of adding a linear combination of
the previous steps to the arbitrarily chosen direction
(known in mathematics as the Gram-Schmidt orthogonalization
process), we finally arrive at the complete definition of the new
step
, as follows:
| ![\begin{displaymath}
{\bf s}_n = {\bf s}_n^{(1)} =
{\bf c}_{n} + \sum_{j=1}^{j=n-1}\,\beta_n^{(j)}\,{\bf s}_{j}\;.\end{displaymath}](img42.gif) |
(22) |
Here the coefficients
are defined by equations
| ![\begin{displaymath}
\beta_n^{(j)} =
- {{\left({\bf A\,c}_n,\,{\bf A\,s}_{j}\right)} \over
{\Vert{\bf A\,s}_{j}\Vert^2}}\;,\end{displaymath}](img44.gif) |
(23) |
which correspond to the orthogonality principles
| ![\begin{displaymath}
\left({\bf A\,s}_n,\,{\bf A\,s}_{j}\right) = 0\;,\;\;1 \leq j \leq n-1\end{displaymath}](img45.gif) |
(24) |
and
| ![\begin{displaymath}
\left({\bf r}_{n},\,{\bf A\,s}_{j}\right) = 0\;,\;1 \leq j \leq n\;.\end{displaymath}](img46.gif) |
(25) |
It is these orthogonality properties that allowed us to optimize the
search parameters one at a time instead of solving the n-dimensional
system of optimization equations for
and
.
Next: ALGORITHM
Up: IN SEARCH OF THE
Previous: Second step of the
Stanford Exploration Project
11/12/1997