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The secant equation
Let us define a general polynomial of degree 2:
![$\displaystyle q(x)=q(0)+<g(0),x>+\frac{1}{2}<x,H(0) x>$](img1395.png) |
(13.27) |
where
are constant. From the rule for differentiating a
product, it can be verified that:
if
and
depend on
. It therefore
follows from 13.27 (using
) that
![$\displaystyle \nabla
q(x)= g(x) = H(0) x + g(0)$](img1399.png) |
(13.28) |
A consequence of 13.28 is that if
and
are two given points and if
and
(we simplify the notation
), then
![$\displaystyle g_{(2)}-g_{(1)}=H(x_{(2)}-x_{(1)})$](img1406.png) |
(13.29) |
This is called the ``Secant
Equation''. That is the Hessian matrix maps the differences in
position into differences in gradient.
Frank Vanden Berghen
2004-04-19