1 – amaliy ishi Bajardi: Xamzayev Durbek Tekshirdi : Nosirova Namunabonu

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Bajardi: Xamzayev Durbek Tekshirdi : Nosirova Namunabonu 1-laboratoriya ishi topshirig‘i

MUHAMMAD AL-XORAZMIY NOMIDAGI
TOSHKENT AXBOROT TEXNOLOGIYALARI UNIVERSITETI
Televezion Texnalogiyalar fakultite
Elektr Energetika yo’nalishi 520-22 guruh talabasi
Xamzayev Durbekning Algorithmlarni loyihalash fanidan
1 – amaliy ishi

Bajardi: Xamzayev Durbek
Tekshirdi : Nosirova Namunabonu

1-laboratoriya ishi topshirig‘i
Berilgan integral qiymatini to‘g‘ri to‘rtburchaklar, trapetsiyalar, Simpson usullarida ε>0 aniqlikda hisoblang. Aniqlikka erishganlik sharti sifatida |S2n-Sn| < ε tengsizlikdan foydalaning (boshlang’ich n = 10 deb olish mumkin). Natijaga erishish uchun zarur bo’lgan qadamlar soni va integral taqribiy qiymatini chiqariladi. C++ dasturlash tilida natijani oling.

To’g’ri to’rburchaklar usuli
#include
#include

using namespace std;

int main(){
float s,a,b,h,x;
int n,i;
cout<<"a=";cin>>a;
cout<<"b=";cin>>b;
cout<<"n=";cin>>n;
h =(b-a)/n;
s=0;
for(i=0;ix =a+i*h;
s+=pow(sin(x+1),2)*log(pow(x,2)+3);
}
s=s*h;
cout< return 0;
}

Trapetsiya usuli

#include
#include

using namespace std;

double function(double x ) {
return pow(sin(x+1),2)*log(pow(x,2)+3);
}

double trapezoidMethod(double a, double b, int n) {

double h = (b - a) / n;
double sum = function(a) + function(b);

for (int i = 1; i < n; ++i) {

sum += 2*(function(a + i*h));
}

return sum*h/2;

}

int main()

{
double a = 0, b = 1, epsilon = 0.001;
int n = 10;

double ans, result;

int steps = 0;
do
{
ans = trapezoidMethod(a, b, n);
n *= 2;
result = trapezoidMethod(a, b, n);
steps++;
} while (fabs(result - ans) >= epsilon);

cout << "Trapetsiyalar usuli bilan hisoblangan integral qiymati: " << result << endl;

cout << "Zarur bo'lgan qadamlar soni: " << steps << endl;
return 0;
}

Simpson usuli

#include

#include
using namespace std;
float f(float x)
{
return pow(sin(x+1),2)*log(pow(x,2)+3);
}
int main ()
{
float S,a,b,h,x; int n,i;
cin>>a>>b>>n;
h=(b-a)/(2*n);
S=f(a)+f(b);
for (i=1;i<2*n;i++)
{
x=a+i*h;
if (i%2==0) S=S+2*f(x); else S=S+4*f(x);
}
S=S*h/3;
cout< return 0;
}

1.2 Berilgan tenglamaning taqribiy yechimini ε>0 aniqlikda urinmalar (Nyuton) va vatarlar usullarida hisoblang. Aniqlikka erishganlik sharti sifatida |x_n+1-x_n|<ε tengsizlikdan foydalaning. Boshlang‘ich x₀ϵ(a;b) olinadi.
Natijada ildiz taqribiy qiymati va zarur bo‘lgan qadamlar soni chiqarilsin.
С++ dasturida natijani oling.

#include

using namespace std;

// Tenglama x-cos(x)
// Birinchi tartibli hosila 1+sin(x)
// Ikkinchi tartibli hosila cos(x)

//2-Tenglama pow(x, 3) - 3 * x -1 (0, 1)

//Birinchi tartibli hosila 3 * pow(x, 2) - 3
//Ikkinchi tartibli hosila 6 * x

double function(double x) {

// return x-cos(x);
return pow(x, 3) - 3 * x -1;
}

double first_derivative (double x) {

// return 1+sin(x)
return 3 * pow(x, 2) - 3;
}

double second_derivative (double x) {

// return cos(x)
return 6 * x ;
}

double vatarlarMethod(double a, double b, double epsilon) {

double c;
int steps = 0;
do
{
c = a - function(a) * (b - a) / (function(b) - function(a));

(function(a) * function(c) > 0) ? a = c : b = c;

steps++;
} while (fabs(function(c)) >= epsilon);

cout << "Yechim topish uchun zarur bo'lgan qadamlar soni: " << steps << endl;

return c;
}

int main()

{
// double a = 0, b = 1, epsilon = 0.001;
double a = 0, b = 1, epsilon = 0.001;

if (function(a) * function(b) >= 0 || first_derivative(a) * first_derivative(b) <= 0) {

cout << "Tenglama bu oraliqda yechimga ega emas!" << endl;
}

double result = vatarlarMethod(a, b, epsilon);

cout << "Tenglamaning vatarlar usulida hisoblangan ildizi: " << result << endl;
return 0;
}
Urinmalar (Nyuton) usuli
#include
#include

using namespace std;

// Tenglama x -cos(x)=0
// Birinchi tartibli hosila 1+sin(x)
// Ikkinchi tartibli hosila cos(x)

//2-Tenglama pow(x, 3) + 3 * x -1 = 0 (0, 1)

//Birinchi tartibli hosila 3 * pow(x, 2) +3
//Ikkinchi tartibli hosila 6 * x

double function(double x) {

// return x-cos(x);
return pow(x, 3) + 3 * x -1;
}

double first_derivative (double x) {

// return 1+sin(x);
return 3 * pow(x, 2) +3;
}

double second_derivative (double x) {

// return cos(x);
return 6 * x ;
}

double NyutonMethod(double a, double b, double epsilon) {

double c;
int steps = 0;
if (function(a) * second_derivative(a) > 0)
c = a;
else
c = b;

do
{

c = c - function(c) / first_derivative(c);
steps++;

} while (fabs(function(c)) >= epsilon);

cout << "Yechim topish uchun zarur bo'lagan qadamlar soni: " << steps << endl;
return c;
}

int main()

{
// double a = 0, b = 1, epsilon = 0.001;
double a = 0, b = 1, epsilon = 0.001;

if (function(a) * function(b) >= 0 || first_derivative(a) * first_derivative(b) <= 0) {

cout << "Tenglama bu oraliqda yechimga ega emas!" << endl;
}

double result = NyutonMethod(a, b, epsilon);

cout << "Tenglamaning Nuyuton usulida hisoblangan ildizi: " << result << endl;
return 0;
}

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