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Characterization of Seeds by a Fuzzy Clustering Algorithm
Published in Don Potter, Manton Matthews, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2020
Younes Chtioui, Dominique Bertrand, Dominique Barba
The scope of the present study was to analyze the strengths and the limitations of FCMA for the discrimination of seeds. The performances of this unsupervised classification technique were studied pertaining to the values of different parameters, such as the number of iterations and the tolerance parameter ε. In previous studies, the fuzziness parameter m was usually set to the value 2.0. In this investigation, the fuzziness parameter m was varied from 1.0 to 9.0 in order to analyze its importance upon the clustering performances. The distance between a pattern and a cluster center was the Euclidean distance. The main advantage of the Euclidean distance is that it is very easily assessed. The parameter v in the matrix norm assessment was set to 2. This matrix norm is called the « Frobenius » norm. The parameter λ defined in the initialization method was set to 10.
Mathematica
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
(Condition numbers) Relative to a matrix norm ||M||, the condition number of an invertible matrix M is ||M||||M−1||. (See Chapter 37 for more information about condition numbers.) A condition number is ≥1; a large condition number indicates sensitivity to round-off errors. The 2-norm condition number can be shown to be the maximum singular value divided by the minimum singular value: cond[m] := With[{s = SingularValueList[m // N,Tolerance → 0]}, First[s]/Last[s]]For example, consider the condition numbers of Hilbert matrices:h[n] := Table[1/(i + j − 1), {i, n}, {j, n}]Table[cond[h[i]], {i, 1, 6}]{1., 19.2815, 524.057, 15513.7, 476607., 1.49511 × 107}
Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
defines a matrix norm, called the Frobenius norm. It can be shown that ‖A‖F=tr(ATA)
A long-step feasible predictor–corrector interior-point algorithm for symmetric cone optimization
Published in Optimization Methods and Software, 2019
S. Asadi, H. Mansouri, Zs. Darvay, G. Lesaja, M. Zangiabadi
From systems (15) and (16), we have Multiplying the first equality with α and adding to the second, we derive Multiplying both sides of this equality by from the left and observing that , we obtain which implies By Proposition 2.2, we have . Note that the matrix norm is the norm induced by the underlying inner product, hence, the 2-norm or spectral norm. Since is self-adjoint, we may write Since , we have . In this case, we certainly have Since , from (27), we derive Since , we also have for all indices i. Hence, (28) gives the following bound for : On the other hand, we have Substitution of (29) and (30) into (26) gives where the last inequality is due to . Finally, since the lemma follows.
Delayed impulsive stabilisation of discrete-time systems: a periodic event-triggering algorithm
Published in International Journal of Control, 2023
Notation. Let denote the set of integers, the set of nonnegative integers, the set of positive integers, the set of real numbers, the set of nonnegative reals, and the n-dimensional real space equipped with the Euclidean norm denoted by . For an matrix A, we use to represent its induced matrix norm. Let denote the diagonal matrix with diagonal entries . A continuous function is said to be of class and we write , if γ is strictly increasing and equals to zero at zero. For a function , we let represent the inverse function of γ. Let denote the discrete-time unit sample (or unit impulse) function defined as Given R>0, denotes the open ball in centred at the origin with radius R, that is, .