Let h: \Rset^d \rightarrow \Rset^q be a function, let \ux \in \Rset^{n_X} be an input point and let \uy=h(\ux) be the corresponding output.

First-order Taylor expansion

The first-order Taylor expansion of h at the point \ux_0 is the function \widehat{h}: \Rset^d \rightarrow \Rset^q defined for each marginal function h_k of h by the equation:

\widehat{h}_k(\ux) = h_k(\ux_0) + \sum_{i=1}^{d} \left(\frac{\partial h_k}{\partial x_i}\right)(\ux_0)\left(x_i - x_{0,i} \right)

for k \in \{1, ..., q\} which can be written as:

\widehat{h}(\ux) = h(\ux_0) + \mat{L} (\ux-\ux_0)

where \mat{L} = (L_{ij})_{1 \leq i \leq q, 1\leq j \leq d} is the Jacobian matrix evaluated at the point \ux_0:

L_{ij} = \left(\frac{\partial h_i}{\partial x_i}\right)(\ux_0)

for i \in \{1, ..., q\} and j \in \{1, ..., d\}.

Second-order Taylor expansion

The second-order Taylor expansion of h at the point \ux_0 is the function \widehat{h}: \Rset^d \rightarrow \Rset^q defined for each marginal function h_k of h by the equation:

\widehat{h}_k(\ux) = h_k(\ux_0) + \sum_{i=1}^{d}  \left(\frac{\partial h_k}{\partial x_i}\right)(\ux_0)
\left(x_i - x_{0,i} \right) +  \frac{1}{2} \sum_{i,j = 1}^d \left( \frac{\partial^2 h_k}{\partial x_i
\partial x_j}\right)(\ux_0)(x_i-x_{0,i})(x_j-x_{0,j})

which can be written as:

\widehat{h}(\ux) = h(\ux_0) + \mat{L} (\ux-\ux_0) +  \frac{1}{2}  \left\langle \left\langle\mat{Q},\ux-
\ux_0 \right \rangle, \ux-\ux_0 \right \rangle

where \mat{Q} = (Q_{ijk})_{1 \leq i,j \leq d, 1\leq k \leq q} is the Hessian tensor of order 3 evaluated at \ux_0:

Q_{ijk} = \left(\frac{\partial^2 h_k}{\partial x_i\partial x_j}\right)(\ux_0)

for i, j \in \{1, ..., d\} and k \in \{1, ..., q\}.

The first and second order Taylor expansions are used in the following cases: