# Linear and Quadratic Taylor Expansions¶

The approximation of the model response around a specific set of input parameters may be of interest. One may then substitute for its Taylor expansion at point . Hence is replaced with a first or second-order polynomial whose evaluation is inexpensive, allowing the analyst to apply the uncertainty propagation methods.
We consider the first and second order Taylor expansions around .

Introducing a vector notation, the previous equation rewrites:

where:

• is the vector model response evaluated at ;

• is the current set of input parameters;

• is the transposed Jacobian matrix evaluated at .

Introducing a vector notation, the previous equation rewrites:

where is the transposed Hessian matrix.

API:

Examples: