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13Oriental Renaissance: Innovative, R VOLUME 1 | ISSUE 11
educational, natural and social sciences ( ) ISSN 2181-1784
Scientific Journal Impact Factor SJIF 2021: 5.423
TEACHING LINE MODELS AND SOLUTIONS WITH THEIR PROGRAMMING SYSTEMS
Kholikulov Bekzod Jovliyevich
Karshi Engineering and Economics Institute Senior Lecturer of the Department of Information Technology E-mail: bekzod1106@umail.uz
ABSTRACT
In this work, the solution of a system of linear algebraic equations using the object-oriented programming language С ++ builder 6 and its solution, consisting of Gauss, Kramer, Jordan and simple iteration methods.
Keywords: С++ builder 6 programming language, mathematical model, optimal solution, exact solution, approximate solution.
AННОТАЦИЯ
В данной работе решение системы линейных алгебраических уравнений с использованием объектно-ориентированного языка программирования C+ + builder 6 и его решение, состоящее из методов Гаусса, Крамера, Джордана и простых итераций.
Ключевые слова: Язык программирования С++ builder 6, математическая модель, оптимальное решение, точное решение, приближенное решение.
INTRODUCTION
It is known that the mathematical model of any object is represented by mathematical relations (equations, inequalities or their systems). One of these relationships is a system of linear algebraic equations. A system of n linear algebraic equations unknown to us be given The numbers given here are s, s are unknown. If (1.1) the main determinant corresponding to the system is different from zero i.e.
I I I aXn
a21 X1 1 a22X2 1 1 a2n ~ b2
(1.1)
an1 X1 + an 2 X2 + + ann = bn
be given The numbers given here are s, s are unknown. If (1.1) the main determinant corresponding to the system is different from zero.
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aii ai2 .. . ain
A = a 2i a22 . . a2 n * 0
an1 an 2. .. ann
DISCUSSION AND RESULTS
Since there are several ways to solve a system of linear algebraic equations, let us consider the stages of building a mathematical model using the Kramer rule, Gaussian, and inverse matrix methods, and methods for using modeling in software development (1.1) for the system.
Method 1: Kramer's rule method in solving a system of linear algebraic equations.
The Kramer rule method is also commonly referred to as the determinants method. We consider this method (1.1) for a system of linear algebraic equations. According to this method, the following (n + 1) units are n-order
The Kramer rule method is also commonly referred to as the determinants method. We consider this method (1.1) for a system of linear algebraic equations. According to this method, the following (n + 1) units are n-order
aii ai2 .. . ain aii ai2 .. . ain aii ai2 .. . ain
A = a2i a22 . . a2 n ? A X = a2i a22 . . a2n ? A X. = a2i a22 . . a2 n
ani an 2. .. ann ani an 2. .. ann ani an 2. .. ann
the values of the determinants are and are unknown. They are found by formula
A* A* A,
X1 ? X2 i ' Xn
A
The disadvantage of the Kramer method is the difficulty associated with calculating high-order determinants. Typically, Kramer's rule is used in a system of equations when the number of equations is small.
1.1- Assignment:
1.2- Create the following system of linear algebraic equations, a program model in the programming language C ++ Builder 6 using the Kramer rule method.
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2 ^r x^ X — 1
x^ 4 x^ ^^ 7 x^ — 14
5 x^
2 x — 4
Once the program is running, you will need 4 StringGrids, 3 Labels, 1 Image, and 1 Button component to program a system of algebraic equations in S ++ Builder 6.
Once the required components are installed in the form window, the application view will look like this.
1-Fig. Software view.
Method 2: Gaussian method for solving a system of linear algebraic equations.
The Gaussian method is a method based on the sequential loss of the unknown, the algorithm of which consists of the following sequence of calculations. (1.1) (if any, can be obtained by substituting the equations in the system). (1.1) is the sum of all terms of the first equation in the system.
X1 + C12 X2 + ••• + C1nXn — d1
1.2
Here
C ; —
a
CL
j — 2,3,...n, d —
11
C11
(1.2) by helping this equation from (1.2) sytem x we lose the unknown. For this (1.2) equation an,a31anl, multiplying in series (1.1) the second, third, etc. of the system of equations, respectively. Dividing the n-equations by depending on the unknown
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a22X2 ^ ^ a2nXn b2
a\x7 + ••• + a* x = b1
n 2 2 nn n n
(1.3)
We have a system of linear algebraic equations. Here :
al=aij -ancij,bi = di -aA, uj = 2,...,n
Now from (1.3) system x2 we should lose unknown. For this (1.3) system's
first
equition a\2 ^ 0 ( If a\2 =0 will be = 0, (1.3) equations in the system exchange th place al2 ^ 0
X2 ^ C23X3 ^ ••• ^ C2nXn = d2
(1.4)
Here
a\j Ь C2 j = 1 , d2 = Ï-, j = •••, n
a
22
a
22
Repeat once this process n -1
xn-l + Cn-l,nXn = An-l
We will have equality. Here
(1.5)
'n—1 ,n
an~2 bn~2
an—1,ni _ bn-1 an-1
a
„П - 2
n—1 ,n—1
a
- 2 n—1,n—1
(1.5) from the previous system using the equation xw-1 we lose unknown, xw for
find
Xn=dn
(2.6)
We will have. Here d =
b
a'
n—1 n_
..n—1
So (1.1) syste^o. s , I, ,
unknowns for clarify Гаусс ways (1.2), (1.4), (1.5) ва (1.6) We have the following solution algorithm based on
<
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X — d
n'
Xn-1 — dn-1 cn-1,nxn
This
formula
%2
X — d can be
C23 X3
C12 X2
C2nXn,
Ci tX •
1n n
summarized
in the following
view x—dk
ckjxj,k
n, n
1,...,1
j—k+1
2.1- Task. Following
X| 2X2 x3 — 1
2 X| 3 X2 4 x3 — 13
3 X| X2
- 2 x3 — - 6
Create a system of linear algebraic equations, a program model in the Gaussian method C ++ Builder 6 programming language. Once the program is running, you will need 2 StringGrid, 3 Label, 1 Image, and 1 Button components to program in a S ++ Builder 6 visual environment to solve a system of algebraic equations.
Once the required components are installed in the form window, the application view will look like this.
Figure 2. Software view. Method 3: Inverse matrix method for solving systems of linear algebraic
equations.
Let us consider the inverse matrix method in solving a system of linear algebraic equations (1.1). To do this, (1.1) is an n-dimension composed of coefficients in front of the unknowns in the system
n
<
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a11 a12 .. . a1n
A — a 21 a22 . . a2 n
an1 an 2. .. ann
Consider a square matrix. Consider a square matrix.
A"1* A — E
would be appropriate. Where E is the unit matrix, namely 1 0 ... 0
E —
0 1 ... 0 0 0 ... 1
Theorem. If the determinant value composed of the elements of the matrix A is different from zero, i.e., there is an inverse matrix to the matrix A. If there is an inverse matrix to matrix A, it is calculated using the following formula
a11 a21 .. . an1
A—1 — I a12 a22 . . an 2
À
a1 n a2 n . .. ann
Here À = det A, A. — a algebraic fillers of elements
A, — (—1)
i+j
a11 a21 .. a1,—1 a1, +1 .. .. a1n
a21 a21 ... a2j—1 a1 j+1 . ... a1n , i,j —1,2,3,
ai—11 ai- -12 ... ai—1 j—1 ai- -1 j+1 ai—1n
3.1- Task. Following
2 x^ ^b x^ ^b 2 — 4
Xi
x2 ^b 2 x3 — 1
3X| ^r x2 2 x3 — 3
system of linear algebraic equations, creation of a program model in the programming language C ++ Builder 6 using the inverse matrix method.
Oriental Renaissance: Innovative, R VOLUME 1 | ISSUE 11
educational, natural and social sciences ( ) ISSN 2181-1784
Scientific Journal Impact Factor SJIF 2021: 5.423
Once the program is running, you will need 4 StringGrids, 3 Labels, 1 Image, and 1 Button component to program a system of algebraic equations in S ++ Builder 6. Once the required components are installed in the form window, the application view will look like
this.
Figure 3. Software view.
(2,3,4) -As can be seen in the figures, in solving a system of linear algebraic equations, the Kramer method, the Gaussian method, the methods of solving inverse matrix methods are mentioned.
CONCLUSION
Denmak can solve any obvious problem in several ways. If the real process in question can be expressed with sufficient accuracy through mathematical relations, it will be possible to solve this problem by constructing a mathematical model. Solving a problem in this way is called the process of mathematical modeling.
REFERENCES:
1. Эшматов X., Абдуллаев З.С., Акбаров У.Й. «Математическое моделирование нелинейных несвязанных задач динамики термовязкоупругих систем».-Тошкент, 2004.
2. Эшматов X., Бобаназаров Ш.П., Ахмеров И.С., Абдикаримов Р.А. «Математическое моделирование нелинейных задач о колебаниях и динамической устойчивости вязкоупругих систем». -Тошкент, 2006.
3. Эшматов X., Юсупов М., Айнакулов Ш., Ходжаев Д. «Математик моделлаштириш укув кулланма» -Тошкент, 2004
