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

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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)

or

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

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

П

J xq ( x )| dx <~ .

In the thesis of Z.M. Gasimov [2] it was shown, that for eigenvalues µn 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

П

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

( 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

( 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 Cn are found from the normalization conditions.

In case you need to find the eigenvalue γn with a sufficiently large number it is difficult to use the transcendental equation (11), because it is necessary to consistently find all the values γn 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 ex 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' µn

µn

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

µn

µn

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 ) + ex are given.

Table 3

n

^ µn

µn

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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