Next: Induction
Up: IN SEARCH OF THE
Previous: First step of the
Now let us assume n > 2 and add some amount of the step from
the (n-2)-th iteration to the search direction, determining the new
direction
, as follows:
| ![\begin{displaymath}
{\bf s}_n^{(n-2)} = {\bf s}_n^{(n-1)} + \beta_n^{(n-2)}\,{\bf s}_{n-2}\;.\end{displaymath}](img33.gif) |
(17) |
We can deduce that after the second change, the value of numerator in
equation (9) is still the same:
| ![\begin{displaymath}
\left({\bf r}_{n-1},\,{\bf A\,s}_n^{(n-2)}\right)^2 = \left[...
...ight)\right]^2 =
\left({\bf r}_{n-1},\,{\bf A\,c}_n\right)^2\;.\end{displaymath}](img34.gif) |
(18) |
This remarkable fact occurs as the result of transforming the dot product
with the help of equation
(4):
| ![\begin{displaymath}
\left({\bf r}_{n-1},\,{\bf A\,s}_{n-2}\right) =
\left({\bf r...
..._{n-1}\,\left({\bf A\,s}_{n-1},\,{\bf A\,s}_{n-2}\right) = 0\;.\end{displaymath}](img36.gif) |
(19) |
The first term in (19) is equal to zero according to formula
(7); the second term is equal to zero according to formula
(15). Thus we have proved the new orthogonality equation
| ![\begin{displaymath}
\left({\bf r}_{n-1},\,{\bf A\,s}_{n-2}\right) = 0\;,\end{displaymath}](img37.gif) |
(20) |
which in turn leads to the numerator invariance (18). The
value of the coefficient
in (17) is defined
analogously to (14) as
| ![\begin{displaymath}
\beta_n^{(n-2)} = -
{{\left({\bf A\,s}_n^{(n-1)},\,{\bf A\,...
...bf A\,s}_{n-2}\right)} \over
{\Vert{\bf A\,s}_{n-2}\Vert^2}}\;,\end{displaymath}](img39.gif) |
(21) |
where we have again used equation (15). If
is
not orthogonal to
, the second step of the improvement leads
to a further decrease of the denominator in (8) and,
consequently, to a further decrease of the residual.
Next: Induction
Up: IN SEARCH OF THE
Previous: First step of the
Stanford Exploration Project
11/12/1997