# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Continuous methods for extreme and interior eigenvalue problems. (English) Zbl 1092.65029
The authors suggest to apply some continuous methods (actually their discrete analogs) for a possible numerical solution of the partial eigenvalue problems $Ax=\lambda x$ with a symmetric matrix $A$; the eigenvalues are numerated in increasing order and are assumed to be smaller than a constant $c$; as an example they make use of $c=\|A\|_1+1$. Instead of the standard minimization of the Rayleigh quotient on the unit sphere for approximations of $\lambda_1$, they consider the minimization of the merit function $f(x)\equiv (Ax,x)-c\|x\|^2$ on the unit ball (a closed convex set). They write: “Therefore, it is much easier to solve”. In reality they construct a dynamical system such that $f(x)$ is monotonically nonincreasing along the solution of the system. More precisely, the system $x'(t)=-[x-P(x-\nabla f(x))]$ is applied where $P$ is projection operator on the unit ball (the use of such equations for minimization problems is a very old idea in mathematics but which can not replace the theory of iterative methods). The authors concentrate on the applicability of classical theorems for the differential systems under consideration including the study of the asympotical behavior of the solutions. Similar topics are discussed when the minimization of the merit function $F(x)\equiv ((A-aI)(A-bI)x,x)$ is applied (it helps to deal with eigenvalues in $[a,b]$). Numerical examples are given for model matrices of order $n\in [1000,7500]$. The ordinary differential equation solver used is ODE45 (a nonstiff medium order method). The computational time “grows at a rate of $n^{2+\varepsilon}$ ”. The authors write that “our new methods are very effective and attractive” but also that “the investigation of the method is still not enough for a side-by-side comparison with the existing linear algebra methods”.

##### MSC:
 65F15 Eigenvalues, eigenvectors (numerical linear algebra)
Full Text:
##### References:
 [1] Chu, M. T.: On the continuous realization of iterative processes. SIAM rev. 30, 375-387 (1988) · Zbl 0657.65075 [2] Chu, M. T.: Matrix differential equations: a continuous realization process for linear algebra problems. Nonlinear anal. Theory methods appl. 18, 1125-1146 (1992) · Zbl 0773.34007 [3] Cichocki, A.; Unbehauen, R.: Neural networks for computing eigenvalues and eigenvectors. Biolog. cybernet. 68, 155-164 (1992) · Zbl 0769.65015 [4] Golub, G. H.; Van Loan, C. F.: Matrix computation. (1996) · Zbl 0865.65009 [5] He, B. S.: Inexact implicit methods for monotone general variational inequalities. Math. prog. 86, 199-217 (1999) · Zbl 0979.49006 [6] Hopfield, J. J.: Neural networks and physical systems with emergent collective computational ability. Proc. natl. Acad. sci. 79, 2554-2558 (1982) [7] Hopfield, J. J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. natl. Acad. sci. 81, 3088-3092 (1984) [8] Hopfield, J. J.; Tank, D. W.: Neural computation of decisions in optimization problems. Biolog. cybernet. 52, 141-152 (1985) · Zbl 0572.68041 [9] Liao, L. -Z.; Qi, H. D.; Qi, L. Q.: Neurodynamical optimization. J. global optim. 28, 175-195 (2004) · Zbl 1058.90062 [10] Rudin, W.: Principles of mathematical analysis. (1976) · Zbl 0346.26002 [11] Slotine, J. -J.E.; Li, W.: Applied nonlinear control. (1991) · Zbl 0753.93036 [12] Van Der Vorst, H. A.; Golub, G. H.: 150 years old and still alive: eigenproblems. The state of the art in numerical analysis, 93-119 (1997) · Zbl 0881.65026