4.1.4 Newton Direction

For the Newton direction a quadratic approximation of the target function (4.5) is used. For existing second derivatives of $f(\vec{x})$ the minimum of the target function is found by

$${\mathop{\nabla }\nolimits f(\vec{x}_0)} + {\mathop{\nabla }\nolimits ^2 f(\vec{x}_0)} \vec{p} = 0 .$$ (4.9)

So the Newton search direction is given by the equation

$${\mathop{\nabla }\nolimits ^2 f(\vec{x}_0)} \vec{p} = - {\mathop{\nabla }\nolimits f(\vec{x}_0)}$$ (4.10)

using the gradient and the Hessian of the target function.

These algorithms need the derivatives of the model function $f(\vec{x})$. If the function $f(\vec{x})$ is given by an analytical formula, the first and second derivatives can be calculated directly. But for a model function formed by results of several simulations, it is impossible to find analytical expressions for the derivatives. For this reason approximation methods have to be used.