Mustaqil ishi mavzu: Gauss usuli yordamida tenglamalar sistemasini yechish algoritmi Tekshirdi

Yechish. Birinchi tenglamani ketma – ket 1,3,2 sonlarga ko`paytirib, so`ngra ikkinchi, uchinchi va to`rtinchi tenglamalardan birinchi tenglamani ayirsak,
𝑥<sub>1</sub> + 2𝑥<sub>2</sub> − 3𝑥<sub>3</sub> + 5𝑥<sub>4</sub> = 1
{ 𝑥2 − 10𝑥<sub>3</sub> + 17𝑥<sub>4</sub> = −2
−𝑥<sub>2</sub> + 10𝑥<sub>3</sub> − 17𝑥<sub>4</sub> = 2
−𝑥<sub>2</sub> + 10𝑥<sub>3</sub> − 17𝑥<sub>4</sub> = 2
sistemaga ega bo`lamiz.
Endi ikkinchi tenglamani uchinchi va to`rtinchi tenglamalarga qo`shib, natijada quyidagi sistemani hosil qilamiz:
𝑥₁ + 2𝑥₂ − 3𝑥₃ + 5𝑥₄ = 1
{ 𝑥2 − 10𝑥₃ + 17𝑥₄ = −2
0 = 0
0 = 0
Oxirgi ikkita tenglama
0 ∙ 𝑥₁ + 0 ∙ 𝑥₂ + 0 ∙ 𝑥₃ + 0 ∙ 𝑥₄ = 0
ko`rinishdagi tenglama bo`lib, u noma’lumning har qanday qiymatida ham o`rinli bo`lgani uchun uni tashlab yuboramiz.
Ikkinchi tenglamani qanoatlantiradigan noma`lumning qiymatini topsh uchun 𝑥<sub>3</sub> va 𝑥<sub>4</sub> larga ixtiyoriy qiymatlarni beramiz. Masalan, 𝑥<sub>3</sub> = 𝛼, 𝑥<sub>4</sub> = 𝛽 bo`lsin, u holda 𝑥₂ = 10𝛼 − 17𝛽 − 2 bo`ladi. Bu 𝑥<sub>2</sub>, 𝑥<sub>3</sub>, 𝑥<sub>4</sub> larning qiymatlarini birinchi tenglamaga qo`yib 𝑥₁ = −17𝛼 + 29𝛽 + 5 ni topamiz. Sistemaning yechimi
𝑥₁ = −17𝛼 + 29𝛽 + 5;
𝑥₂ = 10𝛼 − 17𝛽 − 2; 𝑥₃ = 𝛼; 𝑥₄ = 𝛽
bo`lib 𝛼 va 𝛽 ning ixtiyoriy qiymatlarida berilgan sistemaning hamma yechimlarini beradi.
6 – misol. Quyidagi chiziqli tenglamalar sistemasini Gauss usuli bilan yeching:
𝑥<sub>1</sub> + 𝑥<sub>2</sub> − 3𝑥<sub>3</sub> = 7
{2𝑥<sub>1</sub> + 3𝑥<sub>2</sub> + 𝑥<sub>3</sub> = 1
2𝑥<sub>1</sub> − 𝑥<sub>2</sub> + 2𝑥<sub>3</sub> = 6
Yechish. Birinchi tenglamaning barcha hadlarini 2 ga ko`paytirib, undan ikkinchi va uchinchi tenglamalarni ayiramiz. Natijada quyidagi ko`rinishdagi sistemaga ega bo`lamiz:
𝑥₁ + 𝑥₂ − 3𝑥₃ = 7
{ −𝑥₂ − 7𝑥₃ = 13
3𝑥₂ − 8𝑥₃ = 8
Ikkinchi va uchinchi tenglamalar faqat 𝑥₂ va 𝑥₃ nomalumlarga ega. Ikkinchi tenglamaning hadlarini 3 ga ko`paytirib, uchinchi tenglamaga qo`shamiz. Natijada quyidagi sistema hosil bo`ladi:
𝑥<sub>1</sub> + 𝑥<sub>2</sub> − 3𝑥<sub>3</sub> = 7
{ −𝑥<sub>2</sub> − 7𝑥<sub>3</sub> = 13
−𝑥<sub>3</sub> = 13
Uchinchi tenglamadan: 𝑥<sub>3</sub> = −13, buni ikkinchi tenglamaga qo`yib 𝑥₂ noma`lumni topamiz:
−𝑥<sub>2</sub> − 7 ∙ (−13) = 13,
𝑥<sub>2</sub> = 78
𝑥<sub>3</sub> va 𝑥<sub>2</sub> noma’lumlarning qiymatlarini birinchi tenglamaga qo`yib 𝑥₁ noma’lumni topamiz.
𝑥₁ + 78 − 3 ∙ (−13) = 7, 𝑥₁ = −110
Javob: (-110, 78, -13)

Gauss usulining C# dasturlash tizimida ko’rinishi

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Forms;


namespace Equations_Solver
{
public partial class Form1 : Form
{
public Form1()
{
InitializeComponent();
}


private void button1_Click(object sender, EventArgs e)
{
textBox2.Clear();
double[][] rows = new double[textBox1.Lines.Length][];
for (int i = 0; i < rows.Length; i++)
{
rows[i] = (double[])Array.ConvertAll(textBox1.Lines[i].Split(' '), double.Parse);
}


int length = rows[0].Length;
for (int i = 0; i < rows.Length - 1; i++)
{
for (int j = i; j < rows.Length; j++)
{
double[] d = new double[length];
for (int x = 0; x < length; x++)
{
if (i == j && rows[j][i] == 0)
{
bool changed = false;
for (int z = rows.Length - 1; z > i; z--)
{
if (rows[z][i] != 0)
{
double[] temp = new double[length];
temp = rows[z];
rows[z] = rows[j];
rows[j] = temp;
changed = true;
}
}
if (!changed)
{
textBox2.Text += "No Solution\r\n";
return;
}
}
if (rows[j][i] != 0)
{
d[x] = rows[j][x] / rows[j][i];
}
else
{
d[x] = rows[j][x];
}
}
rows[j] = d;
}
for (int y = i + 1; y < rows.Length; y++)
{
double[] f = new double[length];
for (int g = 0; g < length; g++)
{
if (rows[y][i] != 0)
{
f[g] = rows[y][g] - rows[i][g];
}
else
{
f[g] = rows[y][g];
}
}
rows[y] = f;
}
}
double val = 0;
int k = length - 2;
double[] result = new double[rows.Length];
for (int i = rows.Length - 1; i >= 0; i--)
{
val = rows[i][length - 1];
for (int x = length - 2; x > k; x--)
{
val -= rows[i][x] * result[x];
}
result[i] = val / rows[i][i];
if (result[i].ToString() == "NaN" || result[i].ToString().Contains("Infinity"))
{
textBox2.Text += "No Solution Found!\n";
return;
}
k--;
}
for (int i = 0; i < result.Length; i++)
{
textBox2.Text += string.Format("X{0} = {1}\r\n", i + 1, Math.Round(result[i], 10));
}
}
}
}