Mathematics

I. P. Mysovskikh

On the Computation of Eigenvalues of an Integral Equation by Means of Traces of Iterated Kernels

(Presented by Academician V. I. Smirnov on 21 I 1957)

Let \(K(s,t)\) be a real, symmetric, and continuous kernel in the square \(a \leq s,t \leq b\). For simplicity we shall assume the kernel to be positive in the sense of integral equations. We shall denote the traces of the iterated kernels by

\[ k_j=\int_a^b K_j(s,s)\,ds,\qquad j=1,2,\ldots \]

It is known that the logarithmic derivative of the Fredholm determinant \(D(\lambda)\) of the kernel \(K(s,t)\) is represented by the power series

\[ \frac{D'(\lambda)}{D(\lambda)}=-k_1-k_2\lambda-k_2\lambda^2<\cdots, \tag{1} \]

convergent for \(|\lambda|<\lambda_1\), where \(\lambda_1\) is the smallest eigenvalue of the kernel \(K(s,t)\).

We shall consider the analytic function \(D(\lambda)/D'(\lambda)\) and its expansion in a power series

\[ \frac{D(\lambda)}{D'(\lambda)}=a_1+a_2\lambda+a_3\lambda^2+\cdots \tag{2} \]

The radius of convergence of the series on the right-hand side of (2), \(\lambda'_1\), is greater than \(\lambda_1\). This follows from the following easily proved proposition: the roots of the derivative of the Fredholm determinant of a symmetric kernel are real; if the kernel is positive, then none of the roots of \(D'(\lambda)=0\) can lie to the left of \(\lambda_1\).

The fact that the radius of convergence of the series (2) is greater than \(\lambda_1\) makes it possible to find \(\lambda_1\) as a root of the equation

\[ f(\lambda)\equiv a_1+a_2\lambda+a_3\lambda^2+\cdots=0. \tag{3} \]

The series (2) can be obtained as the result of dividing unity by the series (1); in this case the coefficients \(a_k\) are determined by the recurrence relations

\[ k_1a_1=-1;\qquad k_1a_2+k_2a_1=0;\qquad k_1a_3+k_2a_2+k_3a_1=0;\ldots \tag{4} \]

The principal result used in what follows is that the coefficients \(a_k\) satisfy the inequalities

\[ a_k a_{k+2}-a_{k+1}^{\,2}>0\qquad (k\geq 2). \tag{5} \]

In particular, from this and from the fact that \(a_2\) and \(a_3>0\), it follows that \(a_k>0\) for \(k\geq 2\).

Let the traces of the iterated kernels \(k_1, k_2, \ldots, k_m\) be known. By means of relations (4) one can compute \(a_1, a_2, \ldots, a_m\). Assuming \(m \geq 3\), introduce the notation \(q=a_m/a_{m-1}\). From inequalities (5) it follows that

\[ a_{m+1}>qa_m;\qquad a_{m+2}>q^2a_m;\ldots \tag{6} \]

Along with equation (3), consider the equation

\[ S_m(\lambda)=a_1+a_2\lambda+\cdots+a_m\lambda^{m-1} +\lambda^mqa_m(1+\lambda q+\lambda^2q^2+\cdots)=0 \]

or

\[ S_m(\lambda)=a_1+a_2\lambda+\cdots+a_{m-2}\lambda^{m-3} +\frac{\lambda^{m-2}a_{m-1}^2}{a_{m-1}-\lambda a_m}=0. \tag{7} \]

The smallest positive root \(\lambda^{(m)}\) of this equation is taken as an approximation to \(\lambda_1\). It is clear that \(\lambda^{(m)}>\lambda_1\), since, by virtue of (6), \(f(\lambda)>S_m(\lambda)\) for \(\lambda>0\).

The well-known trace method (1) makes it possible to indicate approximations from below and from above for \(\lambda_1\):

\[ \sqrt[m]{\frac{p_1}{k_m}}<\lambda_1<\frac{k_{m-1}}{k_m}. \tag{8} \]

Here \(p_1\) is the unknown multiplicity of \(\lambda_1\). It can be proved that for sufficiently large \(m\)

\[ \lambda^{(m)}<k_{m-1}/k_m, \]

i.e., \(\lambda^{(m)}\) approximates \(\lambda_1\) more accurately than \(k_{m-1}/k_m\).

Equation (7) is naturally solved by Newton’s method, taking \(k_{m-1}/k_m\) as the initial approximation to its root. In this process one can determine the multiplicity \(p_1\). Indeed, the following approximate equality holds:

\[ \frac{1}{p_1}=S_m'\left(\frac{k_{m-1}}{k_m}\right). \]

Knowing \(p_1\), we can also indicate the lower bound in (8).

We note that \(S_3(k_2/k_3)=0\), so that the proposed method is meaningful to apply for \(m\geq 4\).

The case \(m=2\), which is practically the most important, does not fit into the scheme given above, and we shall consider it separately. Thus, let \(k_1\) and \(k_2\) be known. From inequality (2)

\[ k_1^3-3k_1k_2+2k_3= \int_a^b\int_a^b\int_a^b K\begin{pmatrix} t_1, t_2, t_3\\ t_1, t_2, t_3 \end{pmatrix} \,dt_1dt_2dt_3\geq 0 \]

we find

\[ k_3\geq \frac{1}{2}(3k_1k_2-k_1^3). \]

Since, by (4), \(a_3=k_1^{-3}(k_1k_3-k_2^2)\), it follows that

\[ a_3\geq k_1^{-3}\left[k_1\cdot\frac{1}{2}(3k_1k_2-k_1^3)-k_2^2\right] = k_1^{-3}(k_1^2-k_2)(k_2-\tfrac{1}{2}k_1^2)=a_3^*. \tag{9} \]

Obviously, \(k_1^2-k_2\geq 0\), and the equality sign occurs if and only if the kernel \(K(s,t)\) has a single simple eigenvalue. Excluding this uninteresting case, we obtain for \(a_3\) the positive lower bound \(a_3^*\), if

\[ \frac{k_2}{k_1^2}>\frac{1}{2}. \tag{10} \]

Suppose that condition (10) is satisfied. Then \(a_3 \geqslant a_3^*>0\). Denote \(q^*=a_3^*/a_2\). Obviously, \(q^* \leqslant q=a_3/a_2\). From inequalities (6) it follows that

\[ a_4>q^*a_3^*;\qquad a_5>q^{*}a_3^*;\ldots \]

Let us form the equation

\[ S_3^*(\lambda)=a_1+a_2\lambda+a_2^*\lambda^2+q^*a_3^*\lambda^3+q^{*}a_3^*\lambda^4+\ldots=0. \]

By virtue of the preceding inequalities, for \(\lambda>0\),

\[ f(\lambda)>S_3^*(\lambda)>a_1+a_2\lambda, \]

therefore the root of the equation \(S_3^*(\lambda)=0\), equal to

\[ \lambda^{(2)*}=\frac{2k_2}{k_1(3k_2-k_1^2)}, \tag{11} \]

is an upper bound for \(\lambda_1\), more accurate than \(k_1/k_2\).

Condition (10) is not always satisfied. Let us indicate one case in which this condition is satisfied. Suppose that the traces \(k_r\) and \(k_{2r}\) are known to us, where \(r\) is a positive integer. Such a situation occurs, for example, when the method of traces is applied, computing \(k_p\) for \(p=1,2,4,8,\ldots\). Considering \(K_r(s,t)=L(s,t)\) as the initial kernel and denoting the traces of its iterated kernels by \(l_1,l_2,l_3,\ldots\), we have \(l_1=k_r,\ l_2=k_{2r},\ l_3=k_{3r}\), etc.

From the relation

\[ \frac{l_2}{l_1^2}=\frac{k_{2r}}{k_r^2} = \left(\frac{p_1}{\lambda_1^{2r}}+\frac{p_2}{\lambda_2^{2r}}+\ldots\right) \left(\frac{p_1}{\lambda_1^{r}}+\frac{p_2}{\lambda_2^{r}}+\ldots\right)^{-2} \]

it is seen that, for sufficiently large \(r\), the ratio \(l_2/l_1^2\) will be close to \(1/p_1\), and condition (10) for \(L(s,t)\) proves to be satisfied if \(\lambda_1\) is a simple eigenvalue \((p_1=1)\). To find \(p_1\), it is sufficient to calculate the ratio \(k_r^2/k_{2r}\).

In the case of arbitrary multiplicity \(p_1\) one can prove that, for sufficiently large \(r\), the inequalities

\[ \lambda_1^r<2k_{2r}\left(k_r-\frac{p_1-1}{\sqrt{p_1}}\sqrt{k_{2r}}\right)^{-1} \left[3k_{2r}-p_1\left(k_r-\frac{p_1-1}{\sqrt{p_1}}\sqrt{k_{2r}}\right)^2\right]^{-1} <\frac{k_r}{k_{2r}} \]

hold.

Inequalities (8) with \(m=2\), when applied to the kernel \(L(s,t)\), are written as follows:

\[ \sqrt{\frac{p_1}{k_{2r}}}<\lambda_1^r<\frac{k_r}{k_{2r}}. \tag{12} \]

The significance of these inequalities is that they make it possible to give an error estimate for the approximate values found for \(\lambda_1\). But, on the other hand, it is clear that the estimate obtained will be too large, since, obviously, the upper bound in (12) is cruder than the lower one. If the upper bound is replaced by the right-hand side of formula (11) as applied to the kernel \(L(s,t)\) (for example, when \(p_1=1\)), then we obtain bounds of the same order of closeness to \(\lambda_1\):

\[ \sqrt{\frac{p_1}{k_{2r}}}<\lambda_1^r<\frac{2k_{2r}}{k_r(3k_{2r}-k_r^2)}. \tag{13} \]

Example. Consider the kernel

\[ K(s,t)= \begin{cases} 10s(1-t), & \text{for } 0\leqslant s\leqslant t\leqslant 1,\\ 10t(1-s), & \text{for } 0\leqslant t\leqslant s\leqslant 1. \end{cases} \]

We have \(k_4=1.0582010;\quad k_8=1.1107304.\) Since \(k_8/k_4^2=0.9919\), it follows that \(p_1=1\).

Inequalities (12) give

\[ 0.9869596<\lambda_1<0.9879612, \]

and for the error of the approximate value found we obtain the bound \(10^{-3}\).

Let us now use inequalities (13):

\[ 0.9869596<\lambda_1<0.9869647. \]

Hence we obtain the error bound \(0.51\cdot 10^{-5}\).

The exact value is

\[ \lambda_1=10^{-1}\pi^2=0.9869604\ldots . \]

Leningrad State University
named after A. A. Zhdanov

Received
19 I 1957

REFERENCES CITED

1. S. G. Mikhlin, Integral Equations, 1949, p. 106.
2. Encyklopädie der mathematischen Wissenschaften, II\(_3\), H. 9, Leipzig, 1923—1927, S. 1510.