Calculation of discrete semi-bounded operators’ eigenvalues with large numbers

Автор: Kadchenko S.I., Zakirova G.A., Ryazanova L.S., Torshina O.A.

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математика. Механика. Физика @vestnik-susu-mmph

Статья в выпуске: 1 т.11, 2019 года.

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In previous works of the article’s authors on development of the Galerkin method, linear formulas for calculating the approximate eigenvalues of discrete lower semi-bounded operators have been obtained. The formulas allow calculating the eigenvalues of the specified operators of any number, regardless of whether the eigenvalues of the previous numbers are known or not. At that, it is possible to calculate the eigenvalues with large numbers when application of the Galerkin method is becoming difficult. It is shown that eigenvalues of small numbers of various boundary-value problems, generated by discrete lower semi-bounded operators and calculated by linear formulas and by the Galerkin method, are in a good conformity. In this paper we use linear formulas to calculate approximate eigenvalues with large numbers of discrete lower semi-bounded operators. Results of calculation of eigenvalues by linear formulas and by known asymptotic formulas for two spectral problems are given. Comparison of the results of calculations of the approximate eigenvalues shows that they almost coincide for sufficiently large numbers. This proves the fact that linear formulas can be used for the considered spectral problems and sufficiently large numbers of eigenvalues.

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Spectral problem, discrete operators, semi-bounded operators, eigenvalues and eigenfunctions of an operator, galerkin method

Короткий адрес: https://sciup.org/147232799

IDR: 147232799 | DOI: 10.14529/mmph190102

Текст научной статьи Calculation of discrete semi-bounded operators’ eigenvalues with large numbers

It is known that the spectrum of a discrete operator consists of isolated points that have no limit points other than infinity. Moreover, each eigenvalue of a discrete operator has finite multiplicity.

Let L be a discrete semi-bounded from below operator, defined in the separable Hilbert space H . Its eigenvalues µ are determined by finding non-trivial solutions of the equation:

Lu = µu, (1)

which satisfies the given homogeneous boundary conditions. Enumerate them in order of increasing values of eigenvalues, taking into account the multiplicity { u n } n _ 1 .

To find the eigenvalues of the operator L we use the Galerkin method. Consider a sequence { H n } _n _ 1 of finite dimensional spaces H n c H , which is complete in H . Suppose, that the orthonormal basis of space H n is known and consists of functions { ф к } n _ 1 . Wherein the functions ф к must satisfy all boundary conditions of the problem. Following the Galerkin method, we will find the approximate solution of the spectral problem (1) in the form:

U n ^_ f^ k ⁽ П ⁾ Ф k . (2)

к = 1

The following theorems were proved in [1].

Theorem 1 . Let L be a discrete semi-bounded from below operator acting in a separable Hilbert space H. If the system of coordinate functions { ф _к } n _ 1 is a basis in the space H, then the Galerkin method applied to the problem of finding the eigenvalues of the spectral problem (1), constructed on this system of functions, converges.
Theorem 2. Let L be a discrete semi-bounded from below operator acting in a separable Hilbert space H. If the system of coordinate functions { ф _k } n _ 1 is an orthonormal basis in the space H, then

и n ( n ) _ ( L ф _n, ф _п ) + 3 _n , (3)

n - 1

where S n = ^ [ ц k ( n - 1 ) - ц _k ( n ) ] , i _k ( n ) is the Galerkin approximation of order n to the correspond- k = 1

ing eigenvalues µk of the operator L.

Formulas (3) allow, as shown in [1], to calculate the approximate eigenvalues of discrete semibounded operators with high computational efficiency. Unlike classical methods, they drastically reduce the amount of computation, solve the problem of finding the eigenvalues of any matrices of high order. Also formulas (3) allows to find eigenvalues regardless of whether know eigenvalues with lower numbers or not and solve the problem of calculating all necessary points of the spectrum of discrete semibounded operators.

Numerous eigenvalue calculations µ ɶ _n of boundary problems generated by discrete semi-bounded from below operators for n < 50 calculated by formulas (3) and the Galerkin method are in good agreement [1].

In this work, for further verification of the developed methodology for calculating the eigenvalues of discrete semi-bounded operators using formulas (3), we compare the results of their calculation using these formulas with the calculations using known asymptotic formulas for the following spectral problems.

1. Asymptotic formulas for the eigenvalues of the spectral problems under consideration

Consider the classical spectral problem of the form:

-y (x) + q(x)y(x) = цу(x), ц = X2 or ц = S2, 0 < x < n(4)

with boundary conditions

y(0) = 0, y(n) = 0,(5)

y(0) = 0, y (n) - hy(n) = 0(6)

with the requirement that the potential q ( x ) , satisfying the condition:

J xq ( x )| dx <~ .

In the thesis of Z.M. Gasimov [2] it was shown, that for eigenvalues µ_n of spectral problems (4), (5) and (4), (6) the following asymptotic formulas:

1 *

Ц = X2, Xn = n + — [q (t )sin2( nt) dt + O (r2)),(7)

ⁿ ⁿ 0

Ц = Sn2, Sn = n - 0,5 - + fq(t)sin2 [(n - 0,5)t]dt + O(rn2))(8)

n ( n - 0,5) n ( n - 0,5) 0

are true respectively. Here:

1 2/n1

r ^ n =- + r n , r n = J tq ( t )| dt + - J q ( t )| dt . n 0 n 1/2 n

To find the approximate eigenvalues of the spectral problem (4), (5) we construct a system of coordinate functions, each function of which is an eigenfunction of the spectral problem

-ф ( x ) = вф (x ), 0 < x < n , ф (0) = 0, ф ( п ) = 0.

It is not difficult to show, that the spectral problem (9) has a set of eigenvalues {n2 } , which cor- n=1

responds to an orthogonal system of eigenfunctions { C n sin( nx ) } n = 1 . Constants C n are found from the normalization conditions.

To find the approximate eigenvalues of the spectral problem (4), (6) we construct a system of coordinate functions, each function of which is an eigenfunction of the spectral problem:

Математика

-ф ( x ) = уф (x ), 0 < x < п , ф (0) = 0, ф ( п ) - hy( п ) = 0.

The set of eigenvalues { y n } n = 1 of the spectral problem (10) has no finite limit points. All the eigenvalues are real, non-negative, simple. They are the roots of the transcendental equation

Y c cos (W Y ) - h sin ( ПУ ) = 0, (11)

and the corresponding system of eigenfunctions is

orthogonal and have the form {Cn sin( Jynx)} • n=1

Constants C_n are found from the normalization conditions.

In case you need to find the eigenvalue γ_n with a sufficiently large number it is difficult to use the transcendental equation (11), because it is necessary to consistently find all the values γ_n with smaller numbers. This leads to a sharp increase in computational calculations. Therefore, in such cases it is necessary to use asymptotic formulas, which can be easily obtained from formulas (8), assuming that q ⁽ t ) = 0 :

Y n = S n , S n = n — 0,5 — —h + O ( r n ) , r = 1. (12)

п ( n - 0,5) ^v ’ n

2. Numerical experiments

Denote by µ ɶ the approximate eigenvalues of spectral problems (4), (5) and (4), (6), found by the Galerkin method, by µ ˆ the eigenvalues found by formulas (3), by µ the eigenvalues found by asymptotic formulas (7) or (8). In all the above calculations it was assumed that 5 _n = 0 .

In tables 1 and 2 the eigenvalues of problem (4), (5), found by formulas (3), and asymptotic formulas (7) with potential q ( x ) = x ² - 5 x + 13 - sin(6 x ) + 2 e^x are given.

Table 1

n	^** ^µ n	^µ n	^µ n	/ < n ^- Л п	/ < n ^- Л п
12	166,712	166,503	167,382	6,705·10^–1	8,792·10^–1
13	191,685	191,507	192,257	5,719·10^–1	7,494·10^–1
14	218,663	218,511	219,157	4,935·10^–1	6,463·10^–1
15	247,646	247,513	248,076	4,302·10^–1	5,632·10^–1
16	278,632	278,515	279,010	3,784·10^–1	4,951·10^–1
17	311,620	311,517	311, 956	3,354·10^–1	4,386·10^–1
18	346,611	346,519	346,910	2,993·10^–1	3,913·10^–1
19	383,602	383,520	383,871	2,687·10^–1	3,512·10^–1
20	422,595	422,521	422,838	2,426·10^–1	3,170·10^–1

43	1871,545	1871,529	1871,598	5,261·10^–2	6,863·10^–2
44	1958,544	1958,529	1958,595	5,025·10^–2	6,554·10^–2
45	2047,544	2047,529	2047,592	4,804·10^–2	6,266·10^–2
46	2138,543	2138,529	2138,589	4,597·10^–2	5,997·10^–2
47	2231,543	2231,529	2231,587	4,404·10^–2	5,744·10^–2
48	2326,542	2326,529	2326,584	4,222·10^–2	5,508·10^–2
49	2423,542	2423,529	2423,582	4,052·10^–2	5,285·10^–2
50	2522,541	2522,529	2522,580	3,892·10^–2	5,076·10^–2
51	2623,541	2623,530	2623,578	3,740·10^–2	4,879·10^–2

63	3991,538	3991,530	3991,562	2,448·10^–2	3,197·10^–2
64	4118,537	4118,530	4118,561	2,366·10^–2	3,098·10^–2
65	4247,537	4247,530	4247,560	2,294·10^–2	3,004·10^–2
66	4378,537	4378,530	4378,559	2,208·10^–2	2,913·10^–2

Kadchenko S.I., Zakirova G.A., Calculation of Discrete Semi-Bounded Operators’

Ryazanova L.S., Torshina O.A. Eigenvalues with Large Numbers

End of the Table 1

n	'-S' µ_n	µ_n	µ _n	^Л n ^- Л п	^Л n ^- Л п
67	4511,538	4511,530	4511,558	2,045·10^–2	2,827·10^–2
68	4646,539	4646,530	4646,558	1,849·10^–2	2,744·10^–2
69	4783,543	4783,530	4783,557	1,348·10^–2	2,665·10^–2
70	4922,577	4922,530	4922,556	2,053·10^–2	2,590·10^–2
71	5063,898	5063,530	5063,555	3,426·10^–1	2,517·10^–2

Table 2

n	µ_n	µ_n	^Л n ^- Л п
1000	1000022,531	1000022,531	1,269·10^–4
1001	1002023,531	1002023,531	1,267·10^–4
1002	1004026,531	1004026,531	1,264·10^–4
1003	1006031,531	1006031,531	1,262·10^–4
1004	1008038,531	1008038,531	1,259·10^–4

10000	100000022,531	100000022,531	1,269·10^–6
10001	100020023,531	100020023,531	1,267·10^–6
10002	100040026,531	100040026,531	1,264·10^–6
10003	100060031,531	100060031,531	1,268·10^–6
10004	100080038,531	100080038,531	1,268·10^–6

100000	10000000022,531	10000000022531	1,269·10^–8
100001	10000200023,531	10000200023,531	1,269·10^–8
100002	10000400026,531	10000400026,531	1,269·10^–8
100003	10000600031,531	10000600031,531	1,269·10^–8
100004	10000800038,531	10000800038,531	1,269·10^–8

Numerical calculations showed that the results of calculations of eigenvalues in three ways are in good agreement. As the number of eigenvalues increases, the difference between them decreases.

The results of calculations for sufficiently large numbers of the eigenvalues of the spectral problem (4), (5) are given in the table 2. The calculation of eigenvalues with such numbers by the Galerkin method causes difficulties due to the large dimensions of the matrices with which you have to work. Therefore, a comparison is made between the approximate eigenvalues found by formulas (3) and the asymptotic formulas (8). For n > 100 000 the values µ ˆ _n and µ _n are almost the same.

In Tables 3 and 4 the approximate eigenvalues of the spectral problem (4), (6) calculated by formulas (3) and asymptotic formulas (8) with h = 0,5 and q ( x ) = x ³ - 4 x + 5 - cos(3 x ) + e^x are given.

Table 3

n	^ µ_n	µ_n	µ _n	~Лn ^- Л п	^Л n ^- Л п
8	78,633	78,572	78,742	1,097·10^–1	1,740·10^–1
9	96,619	96,572	96,708	8,864·10^–2	1,362·10^–1
10	116,610	116,572	116,683	7,314·10^–2	1,114·10^–1
11	138,603	138,572	138,665	6,137·10^–2	9,280·10^–2
12	162,598	162,572	162,650	5,221·10^–2	7,849·10^–2

36	1338,574	1338,571	1338,581	6,287·10^–3	9,177·10^–3
37	1412,574	1412,571	1412,580	5,957·10^–3	8,694·10^–4
38	1488,574	1488,571	1488,580	5,653·10^–3	8,248·10^–4
39	1566,574	1566,571	1566,579	5,371·10^–3	7,836·10^–4
40	1646,574	1646,571	1646,579	5,110·10^–3	7,454·10^–3

Список литературы Calculation of discrete semi-bounded operators’ eigenvalues with large numbers

Kadchenko, S.I. A numerical method for inverse spectral problems / S.I. Kadchenko, G.A. Zakirova // Вестник Южно-Уральского государственного университета. Серия «Математическое моделирование и программирование». - 2015. - Т. 8, № 3. - C. 116-126.
Гасымов, З.М. Решение обратной задачи по двум спектрам для сингулярного уравнения Штурма-Лиувилля: дис.... канд. физ.-мат. наук / З.М. Гасымов. - Баку, 1992. - 121 с.

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