Abstract
A dynamic method which produces estimates of real eigenvectors and eigenvalues is presented. More generally, the technique can be applied to estimate eigenspectra of real n-dimensional k-forms. The proposed approach is based on a spectral splicing property of the line manifolds often found in solutions of polynomial differential equations. As such, it defines an artificial continuous time neural network with stored memories determined by the eigenspectra locations. This paradigm provides a good insight into an analog behavior of large scale neural structures which provide auto- or hetero-associative memories. Consequently, it has applications not only in computational sciences but also as an information processor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York
Coleman C (1984) Boundedness and unboundedness in polynomial differential systems. Nonlin Anal Theor Method Appl 8/11:1287–1294
Golub GH, Van Loan CF (1989) Matrix computations. Johns Hopkins University Press, Baltimore
Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New York Berlin Heidelberg
Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA 79:2554–2556
Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092
Hostetter GH, Savant CJ, Stefani RT (1982) Design of feedback control systems. Holt, Rinehart and Winston, New York
Lee YC, Doolen G, Chen HH, Sun GZ, Maxwell T, Lee HY, Giles CL (1986) Machine learning using a higher order correlation network. Physica 22D:276–306
Lippmann RP (1987) An introduction to computing with neural nets. IEEE ASSP Mag 3:4–22
Luenberger DG (1973) Inlroduclion to linear and nonlinear programming. Addison-Wesley, Reading, Mass
Personnaz L, Guyon I, Dreyfus G (1985) Intormation storage and retrival in spin-glass like neural networks. J Phys Lett 46:L359-L365
Personnaz L, Guyon I, Dreyfus G (1986) Collective computational properties of neural networks: new learning mechanism. Phys Rev A 34:4217–4228
Röhrl H (1977) A theorem on non-associative algebras and its applications to differential equations. Manus Math 21:181–187
Samardzija N (1983) Slabilily properties of homogeneous polynomial differential syslems. J Differ Eq 48:60–70
Samardzija N (1984) Controllability, pole placement, and stabilizability for homogeneous polynomial systems. IEEE Trans Auto Cont 29/11:1042–1045
Samardzija N (1990) Information storage matrices in neural networks. Biol Cybern 63:81–89
Scheffé H (1959) The analysis of variance. Wiley, New York
Venkatesh SS, Psaltis D (1989) Linear and logarithmic capacities in associative neural networks. IEEE Trans Inf Theory 35:558–567
Wilkinson JH (1965) The algebraic eigenvalue problem. Clarendon Press, Oxford
Wilkinson JH, Reinsch C (1971) Handbook for automatic computation, vol 2: Linear algebra. Springer, New York Berlin Heidelberg
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Samardzija, N., Waterland, R.L. A neural network for computing eigenvectors and eigenvalues. Biol. Cybern. 65, 211–214 (1991). https://doi.org/10.1007/BF00206218
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00206218