Talk:Saddle point

Weak saddle point

what is a weak saddle point? --RyanTMulligan 05:40, 19 February 2007 (UTC)

Citation for a saddle point's significance to game theory

Under "Other uses", the article states that "In a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point." I believe that statement is both true and important; I would hate to see it removed from the article. Unfortunately, it is unsupported. Can anyone find an authoritative reference for the claim? I did a quick search, but found nothing. 70.191.91.18 (talk) 18:11, 19 September 2016 (UTC)

Wrong statement in the article?

quoted: "For example, the Hessian matrix of the function z = x2 − y2 at the stationary point (0,0) is the matrix [2,0,-2,0] which is indefinite. Therefore, this point is a saddle point. "

Is the Hessian matrix of the function not [0,0,0,0] because the Hessian is defined as the matrix of Second not First order derivatives ... which gives us null matrix - not indefinite - thus this test is insufficient to conclude that the stated point is a saddle point ? —Preceding unsigned comment added by 84.110.231.195 (talk) 10:26, 8 July 2008 (UTC)

The second derivative wrt x is 2, not 0. The one wrt y is –2. So the Hessian is [2 0; 0 –2]. Loraof (talk) 14:25, 11 April 2017 (UTC)

Saddle not identifyable from Hessian?

It seems to me that some surfaces have "multiple saddles". Imagine looking down on the x-y plane. Imagine a surface with a ridge ascending in the 3 o'clock direction, another descending in the 1 o'clock direction, and additional ridges and gullies ascending in the 11 o'clock and 7 o'clock and descending at 9 and 6. Since that's an odd function in every direction about the "saddle", won't the Hessian be zero even though it's a saddle? Should this be mentioned as a limitation of the Hessian to identify saddles? —Ben FrantzDale (talk) 15:21, 23 July 2009 (UTC)

Definition

I collected the discussions of the problematic definition in this section. Whether the point $x=0$ of $f(x)=x^{3}$ is classified as a saddle point is controversial. AVM2019 (talk) 06:55, 14 October 2020 (UTC)

x⁴+y⁴+xy & (0,0)

Sorry I can't answer the question posted by a non-signer. I have a different question myself.

I am confused. x⁴+y⁴+xy seems to have a global minimum on (0,0), according to my contour plot. Yet, the hessian (ie [0 1, 1 0]) has a positive and a negative eigenvalue (1 and -1). Hence it should be a saddle point, right? I am confused... I would hope every saddle point looks like the critical point (0,0) in the function x²-y²
--Luzsonriente 12:38, 28 December 2006 (UTC)

Confusingly, not all saddle points look like saddles (MathWorld defines it as a stationary point which is not an extremum). Staring at it long enough, $x^{4}+y^{4}+xy$ will have two minima on the line $x=-y$, substitute in and you get $2x^{4}-x^{2}$, differentiate to get stationary points at $8x^{3}-2x=0$, which has roots at $x=0$ and $4x^{2}=1$. --Elektron 23:53, 22 January 2007 (UTC)

Not all saddle points look like saddles, but...

... the article says: "For a function of two or more variables, the surface at a saddle-point resembles a saddle that curves up in one or more directions, and curves down in one or more other directions (like a mountain pass)", and goes on saying that in general the curve can be reduced to x^2-y^2 (which I don't know if is true). At least the first part is plain false for any saddle point not looking like a saddle, look for instance with a Java-enabled browser at [1]. Or think to a point in a surface, where the restriction of the surface in a given direction has a 1D-saddle point. In the orthogonal direction the restriction can even be a straight line, or another curve with a 1D-saddle point, or it could be everything, but we'll have a saddle point which does not look like a saddle and which does not obey the given sentence; or even, which does not curve up or down in any direction. --Blaisorblade 00:48, 9 July 2007 (UTC)

The "incipit" is plainly wrong

It says:

In mathematics, a saddle point is a point in the domain of a function which is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction (like a horse saddle or a mountain pass).

Now, for x^2 + y^3 the origin is a stationary point, and it is not a local extremum. But the "surface" clearly does not resemble a saddle: it "curves up" in one direction, but it does not "curve down", no hope to find a direction for which it happens. --Fioravante Patrone (talk) 22:13, 7 February 2011 (UTC)

Still plainly wrong... One year after my post, and nearly 5 years from [2] --Fioravante Patrone (talk) 21:58, 18 March 2012 (UTC)

WOW! Eventually fixed! [3]. Thanks to User:TIARABAMUN --Fioravante Patrone (talk) 07:52, 18 January 2013 (UTC)

Definition

The lead sentence says a saddle point is a point in the domain, but the graphs show it as a point on the surface. If only one of these is correct, the other should be clarified (in the text or in the caption). If both are valid uses, the lead sentence should be modified to reflect this. Loraof (talk) 14:31, 11 April 2017 (UTC)

I've corrected it with a source, so the text now coincides with the picture. Loraof (talk) 20:17, 12 April 2017 (UTC)

Definition: saddle point != non-extreme critical point

In mathematics, a saddle point or minimax point[1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function.[2].

This definition is wrong, because it includes critical points that are not extrema (such as inflection points). This mistake is based on a misinterpretation and careless reading of [2]: Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics. On page 312 of [2] the author gives two examples when a critical points is not a local extremum: (i) a saddle point and (ii) a critical inflection point. Someone erroneously thought that a critical inflection point is an example of a saddle point, but this is not what the source says. — Preceding unsigned comment added by AVM2019 (talk • contribs) 06:24, 14 October 2020 (UTC) Update. In K. Lange's [4] book a saddle point is defined in the same way to include critical inflection points. AVM2019 (talk) 20:45, 14 October 2020 (UTC)

To add to this comment, without further clarification the definition sounds like it is suggesting that a plateau would also be a saddle point, which doesn't seem correct. 129.67.116.84 (talk) 20:06, 17 April 2024 (UTC)