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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 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.
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
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.
Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. [1]
[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 ...
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.
There are four possibilities, the first two cases where c is an extremum, the second two where c is a (local) saddle point: If n is odd and (+) <, then c is a local maximum. If n is odd and (+) >, then c is a local minimum.