View source for Eigen values of integral operators, numerical methods

Numerical methods for computing the complete spectrum of an integral operator or a part of it (usually one is required to find one or two eigen values of minimal or maximal modulus).

This is often accompanied by the problem of finding numerical approximations of the eigen functions or, more generally, of the root functions, of a given integral operator, corresponding to the desired eigen values. The most important problem is that of finding the eigen values (and eigen functions) of a Fredholm linear integral operator (cf. [[Fredholm-operator(2)|Fredholm operator]]).

==Numerical methods for determining the eigen values of Fredholm integral operators.==
The eigen value and eigen function problems for a Fredholm integral operator consist of finding the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351701.png" /> for which there is a non-trivial solution (in a given class of functions) of the integral equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351703.png" /> is a function (or matrix function) of two groups of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351705.png" /> such that the integral operator with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351706.png" /> is a Fredholm operator on the given class of functions, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351707.png" /> is a domain in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351708.png" />. The class of functions may be the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e0351709.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517010.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517011.png" /> of square-integrable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517012.png" /> or some other function space.

The basic approximation method for solving the eigen value problem (1) is as follows. One chooses some approximation of the integral operator in (1) (see [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]]), e.g., the integral is replaced by a quadrature formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517014.png" /> are nodes of the quadrature formula and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517015.png" /> are its weights (see [[#References|[3]]]–[[#References|[5]]]).

Instead of (1), one considers the problem of finding the eigen values and corresponding root spaces of some matrix naturally connected with the approximation (2). Namely,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

To solve (3), one can use any of the methods of linear algebra for finding eigen values and eigen vectors, or, more generally, root spaces (see [[Linear algebra, numerical methods in|Linear algebra, numerical methods in]]). The resulting eigen values and eigen vectors of the algebraic problem (3) will be close to certain eigen values and eigen vectors of the original problem (1), provided that the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517018.png" /> are close in some specific sense. Instead of (2), other approximations of the integral operator can also be used. The original problem (1) is thus reduced to an algebraic problem analogous to (3). The investigation of the distance between the solutions of the problems (1) and (3) is carried out by methods of functional analysis within the framework of the general theory of approximation methods. The eigen value problem (1) is then treated as a problem of finding the eigen values of some completely-continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517019.png" /> acting on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517020.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

Problem (3) is treated as the eigen value problem for an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517022.png" /> which is close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517023.png" /> but, generally speaking, acts on another space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517024.png" /> (related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517025.png" />):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

In the general theory of approximation methods, one can prove various theorems about the distance between solutions of problems (4) and (5). As an example of such a statement one can mention the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517027.png" /> be a sequence of operators acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517028.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517029.png" /></td> </tr></table>

Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517031.png" /> is the spectrum of the corresponding operator. In this case, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517032.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517033.png" />.

The majority of general estimates of the distance between the eigen values and eigen vectors of the approximation problem (5) to those of (4) are not effective: they contain constants whose values are usually not known. To control the precision in such cases one can use a sequence of eigen values (vectors) approximating the desired eigen value (vector) in (1) (or (4)). It is advisable to construct such a sequence without appealing directly to a sequence of problems of the type (5) with successive refinements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517034.png" />, since this course leads to very cumbersome calculations. Instead of this, one can apply various refinement algorithms (for example, based on [[Perturbation theory|perturbation theory]]).

==Generalized eigen value problems.==
In applications one also encounters a more general problem than (4), namely that of finding critical parameters of eigen value type. Such problems can be formulated in abstract form as follows.

One has to find values of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517035.png" /> for which the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

has more than one solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517037.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517038.png" /> is some non-linear integral operator on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517039.png" /> which depends on the complex parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517040.png" />).

In problem (6) there can be further restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517041.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517042.png" /> (for example, one can ask for only those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517043.png" /> that satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517045.png" /> is given, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517046.png" />).

Problem (6) is closely connected with various problems about bifurcation points in non-linear integral equations. A case of interest is that in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517047.png" /> is linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517048.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517049.png" /> does not enter the equation multiplicatively. The general problem on bifurcation points can be reduced to this form. Moreover, the problem of determining the eigen values of the linear operator (1) that lie in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517050.png" />, for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517051.png" />, reduces to the more general problem (6), in which the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517052.png" /> is linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517053.png" /> and has a finite-dimensional range. In fact, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517054.png" /> be an integral operator with a degenerate kernel and close in norm to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517055.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517056.png" />. Equation (1) can be rewritten as:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517057.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517059.png" /> is invertible, and the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517060.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517061.png" /> can be found from the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517063.png" />. Equation (7) is equivalent (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517064.png" />) to some system of linear algebraic equations. By putting its determinant equal to zero, one obtains an equation whose roots are eigen values of the integral operator (1). This argument is valid in general for an arbitrary completely-continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517065.png" /> on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517066.png" />, provided that it admits an approximation in norm by operators with a finite-dimensional range. The construction (7) can also be used to obtain a refinement of an approximate eigen value (and eigen function).

The general problem (6) can (by approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517067.png" />) be reduced by approximation to a finite-dimensional problem of type (6). In the case of more complex problems of this type, the Monte-Carlo method can be used (see [[#References|[7]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,   V.I. Krylov,   "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Krylov,   V.V. Bobkov,   P.I. Monastyrnyi,   "Numerical methods" , '''2''' , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.P. Mysovskikh,   "On interpolation by polynomials in two variables"  ''Metod. Vychisl.'' , '''8'''  (1973)  pp. 3–114  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.I. Marchuk,   V.I. Lebedev,   "Numerical methods in the theory of neutron transport" , Harwood  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.S. Vladimirov,   I.M. Sobol',   "Computation of the least eigen value of the Peierls equation by the Monte Carlo method"  ''Vychisl. Mat.'' , '''3'''  (1958)  pp. 130–137  (In Russian)</TD></TR></table>



====Comments====
Instead of  "generalized eigen value problem" , the phrase  "non-linear eigenvalue problem for an integral operator47J10non-linear eigen value problem"  is also used. Moreover, what is called  "eigen value"  in the article is also called  "characteristic value of an integral operator45C05characteristic value" , while the term  "eigen value"  is (more frequently) used for complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517068.png" /> such that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517069.png" /> has a non-trivial solution.

For many concrete methods of approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517070.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517071.png" />, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517072.png" /> used above is too strong. Instead, one typically has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517073.png" /> converges pointwise to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517074.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517075.png" /> is collectively compact (i.e., for all bounded sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517077.png" /> is compact). For convergence results for eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517078.png" /> to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035170/e03517079.png" /> in this framework see, e.g., [[#References|[a1]]], Chapt. 4.

In the Western literature, detailed accounts of the topic of the article are given, e.g., in [[#References|[a2]]], Chapt. 3, and [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Anselone,   "Collectively compact operator approximation theory and applications to integral equations" , Prentice-Hall  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.H. Baker,   "The numerical treatment of integral equations" , Clarendon Press  (1977)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Chatelin,   "Spectral approximation of linear operators" , Acad. Press  (1983)</TD></TR></table>