V. A. MEDVEDEV

APPLICATION OF THE METHOD OF LEAST SQUARES TO EIGENVALUE PROBLEMS

(Presented by Academician G. I. Petrov on 25 I 1963)

Let it be required to find the eigenvalues of the equation

\[ A(\lambda)\varphi=0, \tag{1} \]

i.e., the values \(\lambda\) for which equation (1) has a nontrivial solution; \(A(\lambda)\) is a linear operator depending on the complex parameter \(\lambda\), with domain of definition \(D_A\) in a Hilbert space \(H_1\) and range in a Hilbert space \(H_2\). We shall assume that for each eigenvalue there exists a neighborhood containing no other eigenvalues, and that, if \(\lambda\) is not an eigenvalue, then it is regular, i.e., the operator \(A^{-1}(\lambda)\), inverse to the operator \(A(\lambda)\), is defined on all of \(H_2\) and is bounded. Denote

\[ \varkappa(\lambda)=\inf_{\varphi\in D_A} \frac{\|A(\lambda)\varphi\|_{H_2}^2}{\|\varphi\|_{H_1}^2}. \]

Obviously, \(\varkappa(\lambda)\geq 0\). If \(\lambda=\lambda_i\) is an eigenvalue, then \(\varkappa(\lambda_i)=0\). If \(\lambda\) is a regular value, then

\[ \varkappa(\lambda)=\frac{1}{\|A^{-1}(\lambda)\|^2}>0, \tag{2} \]

where \(\|A^{-1}(\lambda)\|\) is the norm of the operator \(A^{-1}(\lambda)\). Consequently, the characteristic equation for the eigenvalues will be \(\varkappa(\lambda)=0\).

Take a system of linearly independent vectors \(\psi_k\in D_A\) \((k=1,2,\ldots)\), complete in the following sense: for arbitrary fixed \(\lambda\), \(\varphi\in D_A\), and \(\varepsilon>0\) there exists a linear combination \(\sum_{k=1}^{m} a_k\psi_k\) such that

\[ \left\|\varphi-\sum_{k=1}^{m}a_k\psi_k\right\|_{H_1}^{2} + \left\|A(\lambda)\left(\varphi-\sum_{k=1}^{m}a_k\psi_k\right)\right\|_{H_2}^{2} <\varepsilon. \]

A system complete in the indicated sense will henceforth be called \(A\)-complete. Denote

\[ \varkappa_n(\lambda)= \min_{\{a_k\}} \frac{ \left\|A(\lambda)\left(\sum_{k=1}^{n}a_k\psi_k\right)\right\|_{H_2}^{2} }{ \left\|\sum_{k=1}^{n}a_k\psi_k\right\|_{H_1}^{2} }, \quad (n=1,2,\ldots). \]

Obviously, \(\varkappa_n(\lambda)\geq \varkappa(\lambda)\). By virtue of the \(A\)-completeness of the system \(\{\psi_k\}\),

\[ \lim_{n\to\infty}\varkappa_n(\lambda)=\varkappa(\lambda). \]

In particular, \(\varkappa_n(\lambda_i)\to 0\) as \(n\to\infty\), if \(\lambda_i\) is some eigenvalue. Suppose now that the functions \(\varkappa(\lambda)\), \(\varkappa_n(\lambda)\) are continuous. Take a circle \(|\lambda-\lambda_i|=\rho\) such that in the disk \(|\lambda-\lambda_i|\leq\rho\) there are no eigenvalues other than \(\lambda=\lambda_i\). Since \(\lim_{n\to\infty}\varkappa_n(\lambda_i)=0\), it follows that, beginning with some number, the inequality

\[ \varkappa_n(\lambda_i)< \min_{|\lambda-\lambda_i|=\rho}\varkappa(\lambda), \]

will hold.

and, consequently, for such \(n\) the function \(\varkappa_n(\lambda)\) has a local minimum at some point \(\lambda_{in}\),

\[ |\lambda_{in}-\lambda_i|<\rho . \]

Since \(\rho\) is arbitrarily small, the following is true.

Theorem. For any eigenvalue \(\lambda_i\) of equation (1) there exists a sequence of points \(\lambda_{in}\) at which the corresponding functions \(\varkappa_n(\lambda)\) have a local minimum, \(\lim\limits_{n\to\infty}\lambda_{in}=\lambda_i\), and moreover \(\lim\limits_{n\to\infty}\varkappa_n(\lambda_{in})=0\).

Thus, as approximate eigenvalues one must take those values of \(\lambda\) at which the function \(\varkappa_n(\lambda)\) has a local minimum.

Let us note that if there exists a point \(\lambda_0\) at which the function \(\varkappa(\lambda)\) has a local minimum different from zero, then there exists a convergent sequence of points \(\lambda_{0n}\) at which the corresponding functions \(\varkappa_n(\lambda)\) have a local minimum, although \(\lambda_0=\lim\limits_{n\to\infty}\lambda_{0n}\) is not an eigenvalue. The points \(\lambda_{0n}\) are extraneous solutions, which must be rejected. To do this, it is necessary to verify that \(\lim\limits_{n\to\infty}\varkappa_n(\lambda_{0n})\ne 0\). In practical application of the method such a check may prove impossible; therefore it is important to determine whether the function \(\varkappa(\lambda)\) has local minima different from zero. Suppose that at regular points the operator \(A^{-1}(\lambda)\) is a holomorphic function of \(\lambda\), i.e., in some neighborhood of an arbitrary regular point \(\lambda_0\) it can be represented by the norm-convergent series

\[ A^{-1}(\lambda)=A^{-1}(\lambda_0)+(\lambda-\lambda_0)A_1+(\lambda-\lambda_0)^2A_2+(\lambda-\lambda_0)^3A_3+\cdots . \]

Then the mean-value theorem holds,

\[ A^{-1}(\lambda_0)=\frac{1}{2\pi\rho}\int_C A^{-1}(\lambda)\,ds, \]

where \(C\) is a circle of sufficiently small radius \(\rho\) with center at the point \(\lambda_0\). Hence the inequality follows

\[ \|A^{-1}(\lambda_0)\|\leq \frac{1}{2\pi\rho}\int_C \|A^{-1}(\lambda)\|\,ds, \]

and, consequently, \(\|A^{-1}(\lambda)\|\) cannot have a local maximum at the regular point \(\lambda_0\). From (2) it follows that \(\varkappa(\lambda)\) cannot have a local minimum different from zero.

The equation satisfied by \(\varkappa_n(\lambda)\) has the form

\[ |a_{ij}-\varkappa_n\beta_{ij}|=0, \tag{3} \]

where \(\varkappa_n(\lambda)\) is the smallest root of equation (3), with

\[ a_{ij}=(A(\lambda)\psi_j,A(\lambda)\psi_i)_{H_2}, \qquad \beta_{ij}=(\psi_j,\psi_i)_{H_1}\quad (i,j=1,2,3,\ldots,n). \]

If \(\{\psi_k\}\) is an orthonormal system in \(H_1\), then equation (3) is simplified, and \(\varkappa_n(\lambda)\) is the smallest characteristic number of a Hermitian matrix.

Let us dwell on the choice of an \(A\)-complete system. Suppose the operator \(A(\lambda)\) can be represented in the form

\[ A(\lambda)=T(\lambda)A_0, \]

where \(A_0\) is a linear operator with domain coinciding with \(D_A\), with values in \(H_2\), and having a bounded inverse operator \(A_0^{-1}\); the operator \(T(\lambda)\), defined in \(H_2\), is bounded. Take a system \(\{f_k\}\), complete in \(H_2\). Then the system \(\psi_k=A_0^{-1}f_k\) \((k=1,2,\ldots)\) will be \(A\)-complete. Indeed, let \(\varphi\in D_A\). Then \(f=A_0\varphi\in H_2\), and for arbitrary \(\varepsilon>0\) there will be

such a linear combination \(\sum_{k=1}^{m} a_k f_k\) that

\[ \left\| f-\sum_{k=1}^{m} a_k f_k \right\|_{H_2}^{2}<\varepsilon . \]

Further,

\[ \left\| \varphi-\sum_{k=1}^{m} a_k\psi_k \right\|_{H_1}^{2} = \left\| A_0^{-1}\left(f-\sum_{k=1}^{m} a_k f_k\right) \right\|_{H_1}^{2} < \left\| A_0^{-1}\right\|^{2}\varepsilon, \]

\[ \left\| A(\lambda)\left(\varphi-\sum_{k=1}^{m} a_k\psi_k\right) \right\|_{H_2}^{2} = \left\| T(\lambda)\left(f-\sum_{k=1}^{m} a_k\psi_k\right) \right\|_{H_2}^{2} < \left\| T(\lambda)\right\|^{2}\varepsilon . \]

The last two inequalities prove, by virtue of the arbitrariness of \(\varepsilon\), that the system \(\{\psi_k\}\) is \(A\)-complete. If \(H_1=H_2\), and the system of eigenvectors of the equation

\[ A_0\psi-\lambda\psi=0 \]

is complete in \(H_1=H_2\), then as an \(A\)-complete system one may take the system of eigenvectors of this equation. In some problems such a choice of the system may substantially facilitate the computation of the elements of the determinant in equation (3).

Let us consider in more detail the case when \(H_1=H_2\) and equation (1) has the form

\[ A\varphi-\lambda\varphi=0, \tag{4} \]

where \(A\) is a self-adjoint linear operator having a complete system of eigenvectors \(\{\varphi_k\}\) in \(H_1=H_2\), with the corresponding system of eigenvalues \(\{\lambda_k\}\). In this case, as is known, all \(\lambda_k\) are real, and the system \(\{\varphi_k\}\) may be regarded as orthonormal. For arbitrary \(\varphi\in D_A\) and real \(\lambda\) we obtain, using the closure equation,

\[ \frac{\|A\varphi-\lambda\varphi\|^{2}}{\|\varphi\|^{2}} = \frac{\sum_{k=1}^{\infty} |(A\varphi-\lambda\varphi,\varphi_k)|^{2}}{\|\varphi\|^{2}} = \frac{\sum_{k=1}^{\infty} |(\varphi,A\varphi_k-\lambda\varphi_k)|^{2}}{\|\varphi\|^{2}} = \]

\[ = \frac{\sum_{k=1}^{\infty}(\lambda_k-\lambda)^2 |(\varphi,\varphi_k)|^{2}}{\|\varphi\|^{2}} \ge (\lambda_i-\lambda)^2 \frac{\sum_{k=1}^{\infty}|(\varphi,\varphi_k)|^{2}}{\|\varphi\|^{2}} = (\lambda_i-\lambda)^2, \]

where \(\lambda_i\) is the eigenvalue nearest to \(\lambda\). Putting \(\varphi=\varphi_i\), we obtain

\[ \frac{\|A\varphi_i-\lambda\varphi_i\|^{2}}{\|\varphi\|^{2}} = (\lambda_i-\lambda)^2, \]

and, consequently,

\[ \varkappa(\lambda)=(\lambda_i-\lambda)^2 . \tag{5} \]

The above reasoning repeats the reasoning from work \((^2)\) for a second-order differential equation. In contrast to work \((^2)\), we do not fix \(\lambda\). Since \(\varkappa_n(\lambda)\ge \varkappa(\lambda)\), from (5) we obtain

\[ (\lambda_i-\lambda)^2 \le \varkappa_n(\lambda) \quad\text{or}\quad |\lambda_i-\lambda|\le \sqrt{\varkappa_n(\lambda)} . \]

In particular,

\[ |\lambda_i-\lambda_{in}|\le \sqrt{\varkappa_n(\lambda_{in})}, \tag{6} \]

where \(\lambda_{in}\) is the point at which \(\varkappa_n(\lambda)\) has a local minimum. Since

\[ \lim_{n\to\infty}\varkappa_n(\lambda_{in})=0, \]

simultaneously with an approximate eigenvalue we obtain from formula (6) an estimate of the accuracy.

Research Institute of Mechanics
Moscow State University
named after M. V. Lomonosov

Received
24 I 1963

CITED LITERATURE

1. S. G. Mikhlin, Direct Methods in Mathematical Physics, Moscow—Leningrad, 1950.
2. N. M. Krylov, N. N. Bogolyubov, Izv. AN SSSR, OMEN, No. , 471 (1929).