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In both the global and local cases, the concept of a strict extremum can be defined. For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we ...
The constrained extrema of f are critical points of the Lagrangian , but they are not necessarily local extrema of (see § Example 2 below). One may reformulate the Lagrangian as a Hamiltonian , in which case the solutions are local minima for the Hamiltonian.
Throughout, it is assumed that is a real or complex vector space.. For any ,,, say that lies between [2] and if and there exists a < < such that = + ().. If is a subset of and , then is called an extreme point [2] of if it does not lie between any two distinct points of .
[e] The extremum [] is called a local maximum if everywhere in an arbitrarily small neighborhood of , and a local minimum if there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema , depending on whether the first derivatives of the continuous functions are respectively ...
In a sequence of distinct elements, the subsequence of local extrema (elements larger than both adjacent elements, or smaller than both adjacent elements) forms a canonical longest alternating sequence. [2] As a consequence, the longest alternating subsequence of a sequence of elements can be found in time (). In sequences that allow ...
The extrema must occur at the pass and stop band edges and at either ω=0 or ω=π or both. The derivative of a polynomial of degree L is a polynomial of degree L−1, which can be zero at most at L−1 places. [3] So the maximum number of local extrema is the L−1 local extrema plus the 4 band edges, giving a total of L+3 extrema.
In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives.
Powell's method, strictly Powell's conjugate direction method, is an algorithm proposed by Michael J. D. Powell for finding a local minimum of a function. The function need not be differentiable, and no derivatives are taken. The function must be a real-valued function of a fixed number of real-valued inputs.