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The MINRES method iteratively calculates an approximate solution of a linear system of equations of the form =, where is a symmetric matrix and a vector. For this, the norm of the residual ():= in a -dimensional Krylov subspace = + {, …,} is minimized.
The minimum can be computed using a QR decomposition: find an (n + 1)-by-(n + 1) orthogonal matrix Ω n and an (n + 1)-by-n upper triangular matrix ~ such that ~ = ~. The triangular matrix has one more row than it has columns, so its bottom row consists of zero.
In control theory, the minimum energy control is the control () that will bring a linear time invariant system to a desired state with a minimum expenditure of energy. Let the linear time invariant (LTI) system be
The generalized version was popularized by Hoffmeister & Bäck [3] and Mühlenbein et al. [4] Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. On an -dimensional domain it is defined by:
For the expansion, if the reflection point is the new minimum along the vertices, we can expect to find interesting values along the direction from to . Concerning the contraction , if f ( x r ) > f ( x n ) {\displaystyle f(\mathbf {x} _{r})>f(\mathbf {x} _{n})} , we can expect that a better value will be inside the simplex formed by all the ...
Now, applying h min to both A and B, and assuming no hash collisions, we see that the values are equal (h min (A) = h min (B)) if and only if among all elements of , the element with the minimum hash value lies in the intersection . The probability of this being true is exactly the Jaccard index, therefore:
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.
For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point. A non-degenerate critical point is a local maximum if and only if the index is n , or, equivalently, if the Hessian matrix is negative definite ; it is a local minimum if the index is zero, or ...