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A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). 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]
Theorem — (sufficiency) If there exists a solution to the primal problem, a solution (,) to the dual problem, such that together they satisfy the KKT conditions, then the problem pair has strong duality, and , (,) is a solution pair to the primal and dual problems.
The example function (,) = (+) illustrates that the equality does not hold for every function. A theorem giving conditions on f , W , and Z which guarantee the saddle point property is called a minimax theorem .
For saddle point problems, however, many discretizations are unstable, giving rise to artifacts such as spurious oscillations. The LBB condition gives criteria for when a discretization of a saddle point problem is stable. The condition is variously referred to as the LBB condition, the Babuška–Brezzi condition, or the "inf-sup" condition.
An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.
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Cayley–Hamilton theorem (Linear algebra) Cayley–Salmon theorem (algebraic surfaces) Cayley's theorem (group theory) Central limit theorem (probability) Cesàro's theorem (real analysis) Ceva's theorem ; Chasles' theorem, Chasles' theorem ; Chasles' theorem (algebraic geometry) Chebotarev's density theorem (number theory)
If at least one has a positive real part, the point is unstable. If at least one eigenvalue has negative real part and at least one has positive real part, the equilibrium is a saddle point and it is unstable. If all the eigenvalues are real and have the same sign the point is called a node.
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