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10
MSC 47A10
DOI: 10.14529/mmph190102
CALCULATION OF DISCRETE SEMI-BOUNDED OPERATORS' EIGENVALUES WITH LARGE NUMBERS
12 11 S.I. Kadchenko , G.A. Zakirova , L.S. Ryazanova , O.A. Torshina
1 Magnitogorsk State Technical University of G.I. Nosova, Magnitogorsk, Russian Federation
2 South Ural State University, Chelyabinsk, Russian Federation E -mail: kadchenko@masu.ru
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.
Keywords: spectral problem; discrete operators; semi-bounded operators; eigenvalues and eigenfunctions of an operator; Galerkin method.
Introduction
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 j are determined by finding non-trivial solutions of the equation:
Lu = ju, (1)
which satisfies the given homogeneous boundary conditions. Enumerate them in order of increasing values of eigenvalues, taking into account the multiplicity {mn }J=1 .
To find the eigenvalues of the operator L we use the Galerkin method. Consider a sequence {Hn }¥=1 of finite dimensional spaces Hn c H, which is complete in H. Suppose, that the orthonormal
basis of space Hn is known and consists of functions {f }n=1 . Wherein the functions fk 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:
n
un = Z ak (n)fk. (2)
k=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 {f }"k=l is a basis in the space H, then the Galerkin
method applied to the problem offinding 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 {fk }"k=1 is an orthonormal basis in the space H, then _m (n) = ( Lf ) + Sn,_(3)
Kadchenko S.I., Zakirova G.A., Calculation of Discrete Semi-Bounded Operators'
Ryazanova L.S., Torshina O.A. Eigenvalues with Large Numbers
n—1
where dn = fik (n — 1) — flk (n)], tk (n) is the Galerkin approximation of order n to the correspond-k=1
ing eigenvalues ¡j.k of the operator L.
Formulas (3) allow, as shown in [1], to calculate the approximate eigenvalues of discrete semi-bounded 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 semi-bounded operators.
Numerous eigenvalue calculations fln 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) =my(x), m = 12 or m = S2, 0< x<p (4)
with boundary conditions
y(0) = 0, y(p) = 0, (5)
or
y(0) = 0, y(p) — hy(p) = 0 (6)
with the requirement that the potential q( x), satisfying the condition:
|x|q(x)|dx < ¥. 0
In the thesis of Z.M. Gasimov [2] it was shown, that for eigenvalues ¡j.n of spectral problems (4), (5) and (4), (6) the following asymptotic formulas:
1 p
m =12, 1 = n + — fq(t)sin2(nt)dt + O(r„2)), (7)
pn 0
h 1 P
m = Si, Sn = n — 0,5 — + fq(t)sin2 [(n — 0,5)t]dt + O(r„2)) (8)
p(n — 0,5) p(n — 0,5)0
are true respectively. Here:
1 2/n i P
t = - + rn, rn = f t|q(t)|dt + - f |q(t)|dt.
nn
0 1/ 2n
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
-f( x) = bf( x), 0 < x <p,
f(0) = 0, f(p) = 0.
r 2 "I ¥
It is not difficult to show, that the spectral problem (9) has a set of eigenvalues {n } , which cor-
L J n=1
responds to an orthogonal system of eigenfunctions {Cn sin(nx)|J=1 . Constants Cn 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:
-f (x) = yf(x), 0 < x <p,
(10)
f(0) = 0, f (p)- hy(p) = 0.
The set of eigenvalues {yn 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
Jycos (pyfy)- h sin (p<Jy^) = 0, (11)
and the corresponding system of eigenfunctions is orthogonal and have the form \Cn sin(*/5n"x)j .
L J n=1
Constants Cn are found from the normalization conditions.
In case you need to find the eigenvalue yn with a sufficiently large number it is difficult to use the transcendental equation (11), because it is necessary to consistently find all the values yn 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:
gn = Si Sn = n - 0,5 - * + O(rn), rn = - (12)
p(n - 0,5) v ' n
2. Numerical experiments
Denote by ¡1 the approximate eigenvalues of spectral problems (4), (5) and (4), (6), found by the Galerkin method, by 1 the eigenvalues found by formulas (3), by i the eigenvalues found by asymptotic formulas (7) or (8). In all the above calculations it was assumed that dn = 0 .
In tables 1 and 2 the eigenvalues of problem (4), (5), found by formulas (3), and asymptotic formulas (7) with potential q(x) = x2 - 5x +13 - sin(6x) + 2ex are given.
Table 1
n h fin fin fin - fin fin - fin
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 f n И n fn fin - fn f n - fn
67 4511,538 4511,530 4511,558 2,045-10-2 2,827-10-2
68 4646,539 4646,530 4646,558 1,849-Ю-2 2,744-10-2
69 4783,543 4783,530 4783,557 1,348-Ю-2 2,665-Ю-2
70 4922,577 4922,530 4922,556 2,053-Ю-2 2,590-Ю-2
71 5063,898 5063,530 5063,555 3,426-Ю-1 2,517-Ю-2
Table 2
n fn fn fn - fn
1000 1000022,531 1000022,531 1,269-10-4
1001 1002023,531 1002023,531 1,267-Ю-4
1002 1004026,531 1004026,531 1,264-Ю-4
1003 1006031,531 1006031,531 1,262-10-4
1004 1008038,531 1008038,531 1,259-Ю-4
10000 100000022,531 100000022,531 1,269-Ю-6
10001 100020023,531 100020023,531 1,267-Ю-6
10002 100040026,531 100040026,531 1,264-Ю-6
10003 100060031,531 100060031,531 1,268-Ю-6
10004 100080038,531 100080038,531 1,268-Ю-6
100000 10000000022,531 10000000022531 1,269-Ю-8
100001 10000200023,531 10000200023,531 1,269-Ю-8
100002 10000400026,531 10000400026,531 1,269-Ю-8
100003 10000600031,531 10000600031,531 1,269-Ю-8
100004 10000800038,531 10000800038,531 1,269-Ю-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 jin and jn 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) = x3 - 4x + 5 - cos(3x) + ex are given.
Table 3
n И n fn fn f n - fn f n - fn
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-Ю-4
38 1488,574 1488,571 1488,580 5,653-10-3 8,248-Ю-4
39 1566,574 1566,571 1566,579 5,371-10-3 7,836-Ю-4
40 1646,574 1646,571 1646,579 5,110-10-3 7,454-10-3
End of the Table 3
n К ß n ßn ßn - ßn ßn - ßn
66 4428,572 4428,571 4428,574 1,771-10"3 2,764-10-3
67 4562,573 4562,571 4562,574 1,561-10"3 2,683-10-3
68 4698,573 4698,571 4698,574 1,104-10-3 2,605-10-3
69 4836,577 4836,571 4836,574 3,285-10-3 2,530-10-3
70 4976,588 4976,571 4976,574 1,422-10-2 2,459-10-3
Table 4
n ßn ßn ß n - ßn
1000 99906,571 999006,571 1,221-10"5
1001 1001006,571 1001006,571 1,219-10-5
1002 1003008,571 1003008,571 1,216-10-5
1003 1005012,571 1005012,571 1,214-10-5
1004 1007018,571 1007018,571 1,211-10"5
10000 99990006,571 99990006,571 1,220-10-6
10001 100010006,571 100010006,571 1,220-10-6
10002 100030008,571 100030008,571 1,219-10-6
10003 100050012,571 100050012,571 1,219-10-6
10004 100070018,571 100070018,571 1,219-10-6
100000 9999900006,571 9999900006,571 1,220-10-9
100001 10000100006,571 10000100006,571 1,220-10-9
100002 10000300008,571 10000300008,571 1,220-10-9
100003 10000500012,571 10000500012,571 1,220-10-9
100004 10000700018,571 10000700018,571 1,220-10-9
The results of calculations of the approximate eigenvalues of the spectral problem (4), (6), given in Tables 3 and 4 are in good agreement.
Conclusion
Comparison of the results of calculations of the approximate eigenvalues of the spectral problems (4), (5) and (4), (6), carried out according to formulas (3) and asymptotic formulas (7) and (8), show that for sufficiently large numbers the results are almost the same.
In previous papers in the development of the Galerkin method linear formulas for calculating the approximate eigenvalues of discrete semibounded from below operators were obtained by the authors of the article. Formulas allow you to calculate the eigenvalues of the specified operators with any of their numbers, regardless of whether the eigenvalues with the previous numbers are known or not. In this case, it is possible to calculate the eigenvalues with large numbers, when the application of the Galerkin method becomes difficult. To test the new method for calculating the eigenvalues of discrete semi-bounded operators, computational experiments were conducted, which showed that the eigenvalues of small numbers of various boundary-value problems calculated by linear formulas and the Galerkin method are in good agreement. For further verification of the obtained linear formulas, it became necessary to find out how they behave when calculating eigenvalues with large numbers when asymptotic formulas begin to work. In this paper we use linear formulas to calculate approximate eigenvalues with large numbers of discrete semi-bounded from below operators. The results of calculating the eigenvalues by linear formulas and by known asymptotic formulas for two spectral problems are given. Comparison of the results of the calculations of the approximate eigenvalues show that for sufficiently large numbers they almost coincide. This confirms the fact that linear formulas can be used for the considered spectral problems and sufficiently large numbers of eigenvalues.
In the spectral problems considered, linear formulas give the same result as asymptotic formulas. This confirms the possibility of applying linear formulas to the approximate calculation of any eigenvalue of a discrete semi-bounded operator. By virtue of the linearity of formulas, finding eigenvalues becomes computationally efficient compared to any classical method.
Kadchenko S.I., Zakirova G.A., Calculation of Discrete Semi-Bounded Operators'
Ryazanova L.S., Torshina O.A. Eigenvalues with Large Numbers
References
1. Kadchenko S.I., Zakirova G.A. A numerical method for inverse spectral problems. Bulletin of South Ural State University. Series of "Mathematical Modelling, Programming & Computer Software ", 2015, Vol. 8, no. 3, pp. 116-126. DOI: 10.14529/mmp150307
2. Gasymov Z.M. Reshenie obratnoy zadachi po dvum spektram dlya singulyarnogo uravneniya Shturma-Liuvillya: dis. ... kand. fiz.-mat. nauk (Solution of an inverse problem by two spectra for the singular Sturm-Liouville equation. Cand. phys. and math. sci. diss.), Baku, 1992, 121 p. (in Russ.).
Received December 10, 2018
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2019, vol. 11, no. 1, pp. 10-15
УДК 519.642.8 DOI: 10.14529/mmph190102
ВЫЧИСЛЕНИЕ СОБСТВЕННЫХ ЗНАЧЕНИЙ С БОЛЬШИМИ НОМЕРАМИ ДИСКРЕТНЫХ ПОЛУОГРАНИЧЕННЫХ ОПЕРАТОРОВ
12 11 С.И. Кадченко, Г. А. Закирова2, Л.С. Рязанова', О. А. Торшина'
Магнитогорский государственный технический университет, г. Магнитогорск, Российская Федерация
2 Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: kadchenko@masu.ru
В предыдущих работах авторов статьи в развитии метода Галеркина получены линейные формулы для вычислений приближенных собственных значений дискретных полуограниченных снизу операторов. Формулы позволяют вычислять собственные значения указанных операторов любого номера независимо от того, известны ли собственные значения с предшествующими номерами или нет. При этом можно вычислять собственные значения и с большими номерами, когда применение метода Галеркина становится затруднительным. Показано, что собственные значения небольших номеров различных краевых задач, порожденных дискретными полуограниченными снизу операторами, вычисленные по линейным формулам и методом Галеркина, хорошо согласуются.
В работе применены линейные формулы для вычисления приближенных собственных значений с большими номерами дискретных полуограниченных снизу операторов. Приведены результаты вычислений собственных значений по линейным формулам и по известным асимптотическим формулам для двух спектральных задач. Сравнение результатов проведенных вычислений приближенных собственных значений показывает, что для достаточно больших номеров они практически совпадают. Это подтверждает тот факт, что для рассматриваемых спектральных задач и достаточно больших номеров собственных значений можно использовать линейные формулы.
Ключевые слова: спектральная задача; дискретные операторы; полуограниченные операторы; собственные числа и собственные функции оператора; метод Галеркина.
Литература
1. 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.
2. 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.
Поступила в редакцию 10 декабря 2018 г.
