approximation

We can approximate a value by satisfying 3 conditions:

• is defined and differentiable
• can be easily calculated
• is very close to

The following cases are first order and second order of Taylor polynomial

Linear approximation

Transclude of linear-approximation

Quadratic approximation

quadratic approximation

The quadratic approximation to a univariate function at the point is given by:

It has the same function value, first derivative and second derivative with at :

For multivariate function, the quadratic approximation (second-order Taylor polynomial) of around is:

where

• is the gradient of the function
• is the Hessian matrix

This includes both the linear and quadratic terms and provides a more accurate approximation by considering the curvature of near .

Note that the quadratic term (order two) of this quadratic approximation will be quadratic form

Example

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