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\bf{Numerical methods of approximation connected with\\
the navigation problem}\\
{\rm V.I.Berdyshev (Russia, Ekaterinburg)}

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Let function $f: Q\to {\bf R}^m$ characterizes the region $Q\subset {\bf
R}^n, t \in Q$ is unknowing point, $\Delta \in {\bf R}^n, \Delta+t
\subset Q$ and $\varphi(x)=\varphi(x,t)=f(x+t)\quad (x\in \Delta)$ is
the fragment of $f.$ It is supposed the information on the function $f$
is stored by means of approximating function ("polynomial") $p, p\approx
f, p$ belongs to some liner class ${\cal P}=\{ p\}.$ The navigation
problem is a problem of determining the coordinates of the point $t$ from
the function $p$ and the fragment $\varphi.$ This problem is reduced to
the next one:
$$
d(p,t)=\inf\{ \| p(x+T)-\varphi(x)\|_{\Delta}:\Delta+T\subset Q\}.
$$
If $T=T(p,t)\in \arg d(p,t),$ then euclidean distance $|t-T|$ is a
navigation error, and
$$
{\cal D}(p)=\sup \{ |t-T(p,t)|:\Delta_t \subset Q\}
$$
is a maximal error, where $T(p,t)$ is taken such that $|t-T(p,t)|=\sup
\{ |t-T|:T\in \arg d(p,t)\}.$

We want to find the best navigation function $p^* \in {\cal P}$ that
ensures the minimal navigation error: ${\cal D}(p^*)=\inf \{ {\cal
D}(p): p\in {\cal P} \}.$ So we need the derivatives of ${\cal D}(p).$
Let
$$
\| f\|_{\Delta}= \left( \sum_{j=1}^m \| f_j\|_{\Delta}^{\alpha}
\right)^{\frac{1}{\alpha}},\quad \| f_j\|_{\Delta}=\left( \int_{\Delta}
|f_j(x)|^{\alpha}\ dx \right)^{\frac{1}{\alpha}}\quad (2\le \alpha <
\infty).
$$
We suppose that $\partial^2p/ \partial T_i^2\quad (i=1,\ldots,n)$ are
continues for every $p\in {\cal P}$ and $p\mapsto T(p,t)$ is one-to-one
map. Then
$$
\lim_{\lambda\to 0}\frac{{\cal D}(p+\lambda q)-{\cal
D}(p)}{\lambda}=\sup_{t\in {\bf t}} \sum_{i=1}^n (t_i-T_i(p,t)) \gamma_i
(p,t,q)
$$
with $t=(t_1,\ldots,t_n),\ T=T(p,t)=(T_1 \ldots, T_n),\
{\bf t}=\arg {\cal D}(p),\ q\in {\cal P},$
$$
\gamma_i(p,t,q)=-\sum_{j=1}^{m} \int\limits_{\Delta} [\delta_i(x,T)q_i
(x+T)]_{T_i}'\ dx\times
$$
$$
\times \left( \sum_{j=1}^m \int\limits_{\Delta}
\left[ \delta_i(x,T)(p_j)_{T_i}' (x+T)\right]_{T_i}'\ dx \right)^{-1},
$$
$$
\delta_j(x,T)=|p_j(x+T)-\varphi_j(x)|^{\alpha-1} \mbox{sign}
[p_j(x+T)-\varphi_j(x)],\quad \varphi=(\varphi_1,\ldots, \varphi_m).
$$

The work has been supported by RSFI grant \No~96-01-00121.
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