The connection between the length of a geodesic which is given in local coordinates as in , and the length function $d^2$ defined by equation is considered in section 2. This way the problem of finding the length of a geodesic is reduced to solving a nonlinear first-order partial differential equation in some domain of Euclidean space.

The computation of $d^2$ is still far from trivial, however. Even if the data $g_{ij}$ are analytic functions, power series expansion typically lead to power series solutions for $d^2$ with small radius of convergence. Hence, the question is how we can approximate the function $d^2$ globally. Moreover, for some applications such as the accurate computation of diffusions we need the approximation of $d^2$ in strong norms (Sobolev norms of form $H^{s,p}$ for possibly any positive real $s$. For that matter recall that $H^{0,p}\left( {\mathbb R}^n\right) =L^p\left( {\mathbb R}^n\right)$ and that for any $s\in {\mathbb R}$ we may define $H^{s,p}$ to be the set of all tempered distributions $\phi\in {\cal S}'$ such that $I_{-s}\phi$ is a function in $L^p\left( {\mathbb R}^n\right)$, where $I_s$ is the pseudo-differential operator with symbol $\sigma_s(\xi)=\left(1+|\xi|^2\right)^{-\frac{s}{2}}$, i.e. $$I_s\phi ={\cal F}^{-1}\sigma_s{\cal F} \phi,~~\phi\in {\cal S}',$$ ${\cal F}$ denoting the Fourier transform. 