C++ da chiziqli tenglamalar sistemasini yechish dasturini ishlab chiqish
![Kramer usulini C++ tilida dasturini tuzish ketma ketligini ko’ramiz.
Kerakli sarlavha fayllarini qo’shamiz.
#include <iostream>
#include <iomanip>
#include <cmath> nXn matritsasining determinantini
hisoblash funksiyasini yozish:
Funksiyani e’lon qilish:
double determinant(double matrix[], int n)
• Bu satr determinant ikkita argumentni oladigan funksiyani
belgilaydi:
• matrix: Kvadrat matritsaning elementlarini ifodalovchi 1D massiv
(foydalanuvchi mos yozuvlar orqali o'tkaziladi []).
• n: Kvadrat matritsaning o'lchami (satr va ustunlar soni).
Funktsiya matritsaning hisoblangan determinanti bo'lgan doubleqiymatni
qaytaradi.
if (n == 1) { return matrix[0]; } else if (n == 2)
{ return matrix[0] * matrix[3] - matrix[1] *
matrix[2]; }
• Funktsiya ikkita asosiy holatni boshqaradi:
o Agar n1 bo'lsa (1x1 matritsa), determinant oddiygina matritsadagi yagona
element ( matrix[0]).
o Agar n2 bo'lsa (2x2 matritsa), determinant formuladan foydalanib
hisoblanadi
matrix[0] * matrix[3] - matrix[1] * matrix[2].
Rekursiv holat (n > 2):
else {
for (int i = 0; i < n; i++)
{ double subMatrix[(n - 1) * (n
12](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_12.png?v=1)
![- 1)]; int subRow = 0; for
(int row = 1; row < n; row++) {
int subCol = 0; for (int col =
0; col < n; col++) { if (col !
= i) {
subMatrix[subRow * (n - 1) + subCol] = matrix[row * n + col];
subCol++;
}
}
subRow++;
}
det += (i % 2 == 0 ? 1 : -1) * matrix[i] * determinant(subMatrix, n -
1);
}
}
Ushbu blok o'lchami n 2 dan katta bo'lgan matritsalar bilan ishlaydi.
• U matritsaning har bir ustuni bo'ylab takrorlanadi .i
• Loop ichida joriy satr ( ) va joriy ustun ( ) ni hisobga olmaganda
olingan submatritsani saqlash uchun o‘lchami bilan vaqtinchalik massiv
subMatrixe’lon
qilinadi .(n - 1) * (n - 1)ii
13](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_13.png?v=1)
![• Yuvalangan halqalar submatritsani qurish uchun matritsaning qolgan
elementlarini takrorlaydi subMatrix. Elementlar nusxalanadi, joriy
qatordagilar bundan mustasno i.
• Keyin o'zgaruvchi detquyidagi bosqichlar yordamida yangilanadi:
o Ifoda (i % 2 == 0 ? 1 : -1)element uchun kofaktorni hisoblab
chiqadi matrix[i]. U ustun indeksiga qarab 1 va -1 o'rtasida almashadi
i(juft ustunlar kofaktori 1 ga, toq ustunlar esa -1 ga ega).
o Keyin kofaktor matrix[i]submatritsaning elementi va
determinanti bilan ko'paytiriladi determinant(subMatrix, n - 1). Ushbu
mahsulot joriy qatordagi har bir elementning hissasini hisobga olish
uchun akkumulyatorga qo'shiladi .
Umumiy determinant hisoblash kodi:
double determinant(double matrix[], int n) {
double det =
0; if (n == 1)
{ return
matrix[0]; }
else if (n == 2) {
return matrix[0] * matrix[3] - matrix[1] * matrix[2];
} else {
for (int i = 0; i < n; i++) {
double subMatrix[(n - 1) * (n - 1)];
int subRow = 0;
for (int row = 1; row < n; row++)
{ int subCol = 0;
for (int col = 0; col < n; col++) {
if (col != i) {
subMatrix[subRow * (n - 1) + subCol] = matrix[row * n
+ col];
subCol++;
}
}
subRow++;
}
det += (i % 2 == 0 ? 1 : -1) * matrix[i] *
determinant(subMatrix, n - 1);
14](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_14.png?v=1)
![}
}
return det;
}
Kramer usuli yordamida nxn chiziqli tenglamalar tizimini yechish
funksiyasini yozish:
Funktsiya e’lon qilish:
void cramersSolution(double matrix[], double constants[], int n)
o Bu satr cramersSolution uchta argumentni oladigan funksiyani
belgilaydi:
o matrix: Koeffitsient matritsasi elementlarini ifodalovchi 1D massiv
(foydalanilgan mos yozuvlar orqali berilgan []). o constants: Tenglamalarning
o'ng tomonidagi doimiy shartlarni ifodalovchi 1D
massiv (foydalanuvchi havola orqali berilgan []).
o n: Kvadrat matritsaning o'lchami (satr va ustunlar soni).
o Funktsiya voidqaytish turiga ega, ya'ni u hech qanday qiymat
qaytarmaydi,
lekin to'g'ridan-to'g'ri konsolga echimlarni chop etadi.
Aniqlovchini hisoblash:
double det = determinant(matrix, n); if (det == 0) { std::cout << "The
system of equations has no unique solution." << std::endl; return;
}
• Funktsiya birinchi navbatda determinantkoeffitsient matritsasi
determinantini hisoblash uchun funktsiyani chaqiradi (boshqa joyda
aniqlangan deb hisoblanadi).
• Agar determinant ( det) nolga teng bo'lsa, bu tenglamalar tizimining
cheksiz ko'p yechimga ega ekanligini yoki umuman yechim yo'qligini
ko'rsatadi. Bunday holda, funksiya xabarni chop etadi va dan foydalanib, erta
qaytib keladi.
15](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_15.png?v=1)
![Yechimlar qatorini taqsimlash:
double* solutions = new double[n];
• Har bir o'zgaruvchi (x1, x2, ..., xn) uchun hisoblangan yechimlarni
saqlash uchun nomli dinamik ravishda ajratilgan massiv
solutionsyaratiladi .new double[n] Har bir o‘zgaruvchi uchun yechish:
for (int i = 0; i < n; i++)
{ double subMatrix[(n - 1) * (n
- 1)]; int subRow = 0; for (int
row = 0; row < n; row++)
{ int subCol = 0; for (int
col = 0; col < n; col++) { if
(col != i) {
subMatrix[subRow * (n - 1) + subCol] = matrix[row * n + col];
subCol++;
}
}
subMatrix[subRow * (n - 1) + i] = constants[row];
subRow++;
}
solutions[i] = determinant(subMatrix, n) /
det; }
• Kod itenglamalar tizimidagi har bir o'zgaruvchi ( ) orqali takrorlanadi.
o Loop ichida koeffitsient matritsasining o'zgartirilgan versiyasini saqlash
uchun o'lchamli vaqtinchalik massiv e'lon qilinadi.subMatrix(n - 1) * (n - 1)
o Yana bir ichki o'rnatilgan halqalar to'plami quyidagilarni yaratadi subMatrix:
16](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_16.png?v=1)
![▪ matrixU joriy ustundan ( ) tashqari asl koeffitsient matritsasi ( ) elementlarini
ko'chiradi i.
▪ Joriy ustunda ( i) u elementlarni massivdan mos keladigan doimiy shartlar
bilan almashtiradi constants.
o Ushbu o'zgartirilgan submatritsaning determinanti yordamida hisoblab
chiqiladi determinant(subMatrix, n).
o Nihoyat, o'zgaruvchining yechimi x[i]submatritsaning determinantini
dastlabki koeffitsient matritsasi ( det) ning determinantiga bo'lish yo'li bilan
hisoblanadi. Bu natija massivda saqlanadi solutions.
Chop etish yechimlari:
std::cout << "The solutions are: " << std::endl; for (int i = 0;
i < n; i++) { std::cout << "x" << i + 1 << " = " <<
solutions[i] << std::endl;
}
• Agar determinant nolga teng bo'lmasa, funktsiya yechimlarni ko'rsatuvchi
xabarni chop etadi.
• U massiv bo'ylab takrorlanadi solutionsva har bir o'zgaruvchining qiymatini
(x1, x2, ..., xn) tegishli indeks bilan birga chop etadi.
Xotirani ajratish:
delete[] solutions;
• Solutionsdan foydalanib dinamik ravishda ajratilganligi sababli new, funktsiya
delete[] solutionsxotira oqishini oldini olish uchun xotirani bo'shatadi.
Asosiy qism:
17](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_17.png?v=1)
![int n; std::cout << "Enter the size of the
matrix (n): "; std::cin >> n;
Foydalanuvchi matritsa o’lchamini kiritishni so’raydi.
double* matrix = new double[n * n];
double* constants = new double[n];
Kod dinamik ravishda koeffitsient matritsasi ( matrix) va doimiy
atamalar vektori ( constants) uchun xotira ajratadi new. Matritsaning o'lchami
n x n, shuning uchun matrixmassiv elementlarni talab qiladi n * n(er-xotin
qiymatlar). Massiv har bir tenglamaning doimiy shartlarini saqlash uchun
elementlarni constantstalab qiladi.
std::cout << "Enter the coefficients of the
matrix:\n"; for (int i = 0; i < n; i++) { for (int
j = 0; j < n; j++) { std::cin >> matrix[i * n
+ j];
}
} std::cout << "Enter the constant
terms:\n"; for (int i = 0; i < n; i++) {
std::cin >> constants[i];
}
U foydalanuvchidan koeffitsient matritsasi ( ) elementlarini kiritishni taklif
qiladi matrix.
O'rnatilgan tsikllar matritsaning har bir satri ( i) va ustuni ( ) bo'ylab
takrorlanadi va foydalanuvchidan tegishli qiymatni kiritish so'raladi. Ifoda satr
va ustundagi element uchun 1D massivida to'g'ri indeksni hisoblab chiqadi.
Funksiyaga murojaat qilish:
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![cramersSolution(matrix, constants, n);
Xotirani tozalash:
delete[] matrix;
delete[] constants;
Dasturdan chiqish:
return 0;
Kramer usulida chiziqli tenglamani yechishning c++ tilidagi dasturi.
#include <iostream>
#include <iomanip>
#include <cmath>
double determinant(double matrix[], int n) {
double det =
0; if (n == 1)
{ return
matrix[0]; }
else if (n == 2) {
return matrix[0] * matrix[3] - matrix[1] * matrix[2];
} else {
for (int i = 0; i < n; i++) {
double subMatrix[(n - 1) * (n - 1)];
int subRow = 0;
for (int row = 1; row < n; row++)
{ int subCol = 0;
for (int col = 0; col < n; col++) {
if (col != i) {
subMatrix[subRow * (n - 1) + subCol] = matrix[row * n
+
col];
subCol++;
} }
subRow++;
}
det += (i % 2 == 0 ? 1 : -1) * matrix[i] *
determinant(subMatrix, n - 1);
} }
return det;
19](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_19.png?v=1)
![}
void cramersSolution(double matrix[], double constants[], int n) {
double det = determinant(matrix, n);
if (det == 0) {
std::cout << "The system of equations has no unique solution."
<<
std::endl;
return;
}
double* solutions = new
double[n]; for (int i = 0; i < n;
i++) { double subMatrix[(n -
1) * (n - 1)]; int subRow = 0;
for (int row = 0; row < n; row++)
{ int subCol = 0;
for (int col = 0; col < n; col++) {
if (col != i) {
subMatrix[subRow * (n - 1) + subCol] = matrix[row * n +
col]; subCol++;
}
}
subMatrix[subRow * (n - 1) + i] = constants[row];
subRow++;
}
solutions[i] = determinant(subMatrix, n) / det;
}
std::cout << "The solutions are: " << std::endl;
for (int i = 0; i < n; i++) {
std::cout << "x" << i + 1 << " = " << solutions[i] << std::endl;
}
delete[] solutions;
} int
main(
)
{ in
t n;
std::cout << "Enter the size of the matrix (n): ";
std::cin >> n;
double* matrix = new double[n * n];
20](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_20.png?v=1)
![double* constants = new double[n];
std::cout << "Enter the coefficients of the matrix:\n";
for (int i = 0; i < n; i++)
{ for (int j = 0; j < n;
j++) {
std::cin >> matrix[i * n + j];
}
}
std::cout << "Enter the constant terms:\n";
for (int i = 0; i < n; i++) {
std::cin >> constants[i];
}
cramersSolution(matrix, constants, n);
delete[]
matrix;
delete[]
constants;
return 0;
}
21](https://docx.uz/documents/5f7267d6-41fc-4660-b927-9a0a9997357c/page_21.png?v=1)
C++ da chiziqli tenglamalar sistemasini yechish dasturini ishlab chiqish