Saddle point

In mathematics is called a saddle point, patio or point Horizontal inflection point a critical point of a function that is not an extreme point. Points of this type, as the latter name it implies, special cases of turning points.

• 2.1 Specification on discharges
• 2.2 specification directly via the function
• 2.3 Examples

One-dimensional case

For functions of one variable with the vanishing of the first derivative at

A condition that a critical point exists. Is the second derivative at that point is not equal to 0, then there is a bending point, and thus no saddle point. For a saddle point, the second derivative must be 0, if it exists. However, this is only a necessary condition ( for twice continuously differentiable functions ), as you can see in the function.

Conversely ( sufficient condition ): Are the first two derivatives equal to 0 and the third derivative equal to 0, then there is a saddle point; So it is a turning point with a horizontal tangent.

This criterion can be generalized: Applies to a

So are the first derivatives equal to 0 and the -th derivative equal to 0, then the graph of has in a saddle point.

However, the above condition is not necessary. Even if a saddle point on the site is available, all derivatives can be equal to 0.

One can interpret a terrace point in the one-dimensional case as a turning point with tangent parallel to the x -axis.

For example for a very rational function ( polynomial ) with two saddle points

From the very rational functions of degree 5 can have two saddle points, as the following example shows:

For the first derivative has two double zeros -2 and 1:

For the 2nd derivative

-2 and 1 are also zero points, however, the third derivative

There equal to zero:

Therefore, and saddle points of the function.

Multi-dimensional case

Specification on discharges

For functions of several variables ( scalar ) with the disappearance of the gradient at the point

A condition that a critical point exists. Means the condition that all the partial derivatives at the point zero.

If, in addition, the Hessian matrix is indefinite, then there is a saddle point.

Specification directly via the function

For the case that the saddle point is aligned with the coordinate axes can also be described a saddle point without leads in a simple manner: A point is a saddle point of the function, if

Is satisfied for all. This clearly means that the function value of in - direction becomes smaller when the saddle point is left, while leaving the saddle point in direction results in an increase of the function result. This description of a saddle point is the origin of the naming: A riding saddle tilts perpendicular to the spine of the horse down, so is the direction, while in -direction, ie, is formed parallel to the spine upwards.

If the saddle-point is not aligned to the coordinate direction, the above relationship is one after a coordinate transformation.

Examples

The function

Has the saddle point: If, then for all. For results

The fact that a saddle point of being, can be proved via the derivation criterion. It is

And after the onset of results. The Hessian matrix is to

And after the onset of the saddle point:

Since an eigenvalue of is positive and one negative, the Hessian matrix is indefinite, which proves that there is a genuine saddle point.