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A saddle surface is a smooth surface containing one or more saddle points. Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid = (which is often referred to as "the saddle surface" or "the standard saddle surface") and the hyperboloid of one sheet.
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is ...
The saddlepoint approximation method, initially proposed by Daniels (1954) [1] is a specific example of the mathematical saddlepoint technique applied to statistics, in particular to the distribution of the sum of independent random variables.
If D(a, b) < 0 then (a, b) is a saddle point of f. If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at (x, y) implies that f xx and f yy ...
A 'saddle point' in mathematics derives its name from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle.
A typical example of a differential equation with a saddle-node bifurcation is: = +. Here is the state variable and is the bifurcation parameter.. If < there are two equilibrium points, a stable equilibrium point at and an unstable one at +.
One can ask whether a minimizer point of the original, constrained optimization problem (assuming one exists) has to satisfy the above KKT conditions. This is similar to asking under what conditions the minimizer x ∗ {\displaystyle x^{*}} of a function f ( x ) {\displaystyle f(x)} in an unconstrained problem has to satisfy the condition ∇ f ...
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.