Identification of induction curves
DOI:
https://doi.org/10.24193/subbmath.2023.3.01Keywords:
matrix norms, power norm, \(p\)-norm, induction curves, identification, optimization, \(p\)-eigenvectors, Nelder--Mead methodAbstract
Induction curves (induction surfaces, induction sets in general) were recently introduced to provide a visual aid to examine the fractions defining the norm of a matrix, along with the discovery and description of \(p\)-eigenvectors.In our current investigation we delve into an inverse problem, the identification of induction curves. Namely: could the elements of the matrix and the used power parameter \(p\) be reconstructed given the induction curve, i.e. the case of \(2 \times 2\) matrices is examined. The analytic solution is not possible in most cases already in this planar setting, therefore numerical approximation methods shall be applied.
References
Argyros, I.K., George, S., Senapati, K., Extended local convergence for Newton-type solver under weak conditions,
Stud. Univ. Babec{s}-Bolyai Math., textbf{66}(2021), no. 4, 757-768.
Bokor, J., Schipp, F., Approximate linear $H^infty$ identification in Laguerre and Kautz basis, Automatica J. IFAC, textbf{34}(1998), 463-468.
Cadzow, J.A., Minimum $l_1$, $l_2$ and $l_infty$ norm approximate solutions to an overdetermined system of linear equations,
Digital Signal Processing, textbf{12}(2002), 524-560.
Chang, S., Li, C.K., Certain isometries on $IR^n$,
Linear Algebra Appl., textbf{165}(1992), 251-265.
Csirmaz, L., emph{An optimization problem for continuous submodular functions}, Stud. Univ. Babec{s}-Bolyai Math., textbf{66}(2021), no. 1, 211-222.
Fu{a}rcu{a}c{s}eanu, M., Grecu, A., Mihu{a}ilescu, M., Stancu-Dumitru, D., emph{Perturbed eigenvalue problems: An overview},
Stud. Univ. Babec{s}-Bolyai Math., textbf{66}(2021), no. 1, 55-73.
Fridli, S., L'ocsi, L., Schipp, F., emph{Rational function systems in ECG processing}, Proc. 13th Int. Conf. Computer Aided Systems Theory (EUROCAST), Part I, Springer LNCS 6927 (2011), 88-95.
HegedH{u}s, Cs., emph{The method IRLS for some best $l_p$ norm solutions of under- or overdetermined linear systems},
Ann. Univ. Sci. Budapest. Sect. Comput., textbf{45}(2016), 303-317.
Kov'acs, P., L'ocsi, L., emph{RAIT, the Rational Approximation and Interpolation Toolbox for Matlab, with experiments on ECG signals},
Int. J. of Advances in Telecommunications, Electrotechnics, Signals and Systems
(IJATEStextsuperscript{2}), textbf{1}(2012), no. 2-3, 67-75.
Li, C.K., So, W., emph{Isometries of $ell_p$ norm},
Amer. Math. Monthly, textbf{101}(1994), 452-453.
L'ocsi, L., emph{A hyperbolic variant of the Nelder-Mead simplex method in low dimensions},
Acta Univ. Sapientiae Math., textbf{5}(2013), no. 2, 169-183.
L'ocsi, L., emph{Introducing $p$-eigenvectors, exact solutions for some simple matrices},
Ann. Univ. Sci. Budapest. Sect. Comput., textbf{49}(2019), 325-345.
L'ocsi, L., N'emeth, Zs., emph{On the construction of $p$-eigenvectors},
Ann. Univ. Sci. Budapest. Sect. Comput., textbf{50}(2020), 231-247.
Nelder, J.A., Mead, R., emph{A simplex method for function minimization},
Comput. J., textbf{7}(1965), 308-313.
Samia, K., Djamel, B., emph{Hybrid conjugate gradient-BFGS methods based on Wolfe line search},
Stud. Univ. Babec{s}-Bolyai Math., textbf{67}(2022), no. 4, 855-869.
Schipp, F., emph{On $L^p$-norm convergence of series with respect to product systems},
Anal. Math., textbf{2}(1976), 49-64.
Vinh, N.T., Thuong, N.T., emph{A relaxed version of the gradient projection method for variational inequalities with applications},
Stud. Univ. Babec{s}-Bolyai Math., textbf{67}(2022), no. 1, 73-89.
