Пример выполнения работы. Приложение. Код программы

Код программы z1.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

int main ()

{

float xn, xn1;

float e = (float)0.0001;

ofstream out (" z1.out");

out.precision (4);

out.setf (ios:: fixed);

xn1 = 0.;

xn = acos (xn1*xn1)/3.141592653589;

float q = 2.*fabs(xn1-xn);

int i = 0;

while (q> =e)

{

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < q < < endl;

xn1 = xn;

xn = acos (xn1*xn1)/3.141592653589;

q = 2.*fabs(xn1-xn);

i++;

}

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < q < < endl;

out < < xn;

return 0;

}

Код программы z3.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

int main ()

{

float x [8] = {(float)0.00, (float)0.08, (float)0.13, (float)0.20, (float)0.27,

(float)0.31, (float)0.38, (float)0.44};

float y [8] = {(float)1.00000, (float)1.72539, (float)1.97664, (float)1.92950,

(float)1.45474, (float)1.07307, (float)0.42687, (float)0.09338};

float xx = (float)0.10101918;

float yy = (float)0.;

ofstream out (" z3.out");

out.precision (5);

out.setf (ios:: fixed);

for (int i=1; i< 4; i++)

{

float p = 1.;

for (int j=1; j< 4; j++)

{

if (j! =i)

{

p *= (xx-x[j])/(x[i]-x[j]);

}

}

yy += p*y[i];

}

out < < yy < < endl;

//Оценка погрешности метода

float om = 1;

for (i=1; i< 4; i++)

om*=xx-x[i];

float a = 0;

float p = 1;

for (i=0; i< 4; i++)

{

for (int j=0; j< 4; j++)

{

if (j! =i)

{

p+= 1/(x[i]-x[j]);

}

}

a+= p;

}

float r = fabs(a*om);

out < < " Погрешность метода равна: " < < endl;

out < < r < < endl;

float l =.75;

float eps = (float).00001;

float d = eps*l;

out < < " Погрешность вычислений равна: " < < endl;

out < < d < < endl;

out < < " Суммарная погрешность равна: " < < endl;

out < < r+d;

return 0;

}

Код программы z4.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

int main ()

{

float e = (float)0.001;

float x;

float a = (float)0.8, b = (float)1.2;

float c;

ofstream out (" z4.out");

out.precision (3);

out.setf (ios:: fixed);

//Метод половинного деления

out < < " Метод бисекций: " < < endl;

int i=0;

while (fabs (a-b) > = e)

{

out < < i < < ' ' < < a < < ' ' < < b < < ' ' < < fabs(a-b) < < endl;

c = (a+b)/2;

if ((tan (2*a) + 2*a)*(tan (2*c) + 2*c) < 0.)

b = c;

else

a = c;

i++;

}

out < < i < < ' ' < < a < < ' ' < < b < < ' ' < < fabs(a-b) < < endl;

x = (a+b)/2;

out < < x < < endl;

//Метод Ньютона

out < < " Метод Ньютона: " < < endl;

float xn, xn1 = (float)0.8;

xn = xn1 - (tan (2.*xn1) + 2.*xn1)/(2./(cos (xn1*2)*cos (xn1*2))+2.);

i = 0;

while (fabs (xn-xn1) > = e)

{

out < < i < < ' ' < < xn < < ' ' < < fabs (xn-xn1) < < endl;

xn1 = xn;

xn = xn1 - (tan (2.*xn1) + 2.*xn1)/(2./(cos (xn1*2)*cos (xn1*2))+2.);

i++;

}

out < < i < < ' ' < < xn < < ' ' < < fabs (xn-xn1) < < endl;

out < < xn1 < < endl;

//Метод хорд

out < < " Метод хорд: " < < endl;

float xn2;

xn = (float)0.8;

xn1 = (float)1.2;

xn2 = xn1 - (tan (2.*xn1) + 2.*xn1)*(xn1 - xn)/((tan (2.*xn1) + 2.*xn1)-(tan (2.*xn) + 2.*xn));

i = 0;

while (fabs (xn2-xn1) > = e)

{

out < < i < < ' ' < < xn2 < < ' ' < < fabs (xn2-xn1) < < endl;

xn = xn1;

xn1 = xn2;

xn2 = xn1 - (tan (2.*xn1) + 2.*xn1)*(xn1 - xn)/((tan (2.*xn1) + 2.*xn1)-(tan (2.*xn) + 2.*xn));

i++;

}

out < < i < < ' ' < < xn2 < < ' ' < < fabs (xn2-xn1)< < endl;

out < < xn2;

return 0;

}

Код программы z5.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

int main ()

{

ofstream out (" z5.out");

out.precision (14);

out.setf (ios:: fixed);

float em = 0.5;

while (em + 1. > 1.)

{

em = em/2.;

}

em = em*2.;

float xx = (float)1.1;

float h = (float).1;

float y0 = (float)3.55975, y1 = (float)3.89537, y2 = (float)4.25216;

float f1 = (y2-y0)/(2.*h);

float f2 = (y2-2.*y1+y0)/(h*h);

float d1 = fabs (2.*y1/h*em*f1);

out < < " f1 = " < < f1 < < ' ' < < d1 < < endl;

float d2 = fabs (4.*y1/(h*h)*em*f2);

out < < " f2 = " < < f2 < < ' ' < < d2 < < endl;

return 0;

}

Код программы z6.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

int main ()

{

ofstream out (" z6.out");

out.precision (4);

out.setf (ios:: fixed);

float x0 = 0., h;

float y [13];

float y1 [25];

h = (float)1./(float)12.;

out < < " Trap: " < < endl;

for (int i=0; i< 13; i++)

{

y[i] = sqrt(x0)*cos(x0);

x0 += h;

}

x0 = 0.;

for (i=0; i< 25; i++)

{

y1[i] = sqrt(x0)*cos(x0);

x0 += h/2;

}

//Формула трапеций

float I = 0., I2 = 0.;

I = h*(y[0]/2 + y [12]/2);

for (i=1; i< 12; i++)

{

I += h*y[i];

}

I2 = (h/2)*(y1[0]/2 + y1[24]/2);

for (i=1; i< 24; i++)

{

I2 += (h/2)*y1[i];

}

out < < I < < ' ' < < I2 < < endl;

float R = fabs (I2-I)/3;

I = I2 - (I2-I)/3;

out < < I < < ' ' < < R < < endl;

//Формула Симпсона

I = 0.;

out < < " Simp: " < < endl;

I = (h/3)*(y [0] + y [12]);

for (i=1; i< 12; i++)

{

if (i%2)

I += 4*(h/3)*y[i];

else

I += 2*(h/3)*y[i];

}

I2 = 0.;

I2 = (h/6)*(y1 [0] + y1 [24]);

for (i=1; i< 24; i++)

{

if (i%2)

I2 += 4*(h/6)*y1[i];

else

I2 += 2*(h/6)*y1[i];

}

out < < I < < ' ' < < I2 < < endl;

R = fabs (I2-I)/15;

I = I2 - (I2-I)/15;

out < < I < < ' ' < < R < < endl;

//формула прямоугольников

out < < " Rectangle: " < < endl;

I = 0.;

for (i=1; i< 12; i++)

{

I += h*(sqrt((float)i*h+h/2)*cos((float)i*h+h/2));

}

I2 = 0.;

for (i=1; i< 24; i++)

{

I2 += (h/2)*(sqrt((float)i*(h/2)+h/4)*cos((float)i*(h/2)+h/4));

}

out < < I < < ' ' < < I2 < < endl;

R = fabs (I2-I)/3;

I = I2 + (I2-I)/3;

out < < I < < ' ' < < R < < endl;

return 0;

}

Код программы z7.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

int main ()

{

ofstream out (" z7.out");

out.precision (11);

out.setf (ios:: fixed);

double I = 0.;

double t [4] = {-0.86113631, -0.33998104, 0.33998104, 0.86113631};

double A [4] = {0.34785484, 0.65214516, 0.65214516, 0.34785484};

for (int i=0; i< 4; i++)

{

I += A[i]*(1/(10+t[i]));

}

double R = (double)40320/(double)(9*9*9*9*9*9*9*9*9)/(double)3472875;

out < < I < < ' ' < < R;

return 0;

}

Код программы z12.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

float f (float q)

{

return q*q - cos (3.141592653589*q);

}

float f1 (float q)

{

return 2*q + 3.141592653589*sin(3.141592653589*q);

}

int main ()

{

ofstream out (" z12.out");

out.precision (4);

out.setf (ios:: fixed);

out < < " Square: " < < endl;

float x0 = (float)0.2, x1 = (float)0.4, x2 = (float)0.5;

float y0 = f(x0), y1 = f(x1), y2 = f(x2);

float x;

float e = (float)0.0001;

x = (y0*y0*y1*x2+y2*y2*y0*x1+y1*y1*y2*x0-y2*y2*y1*x0-y0*y0*y2*x1-y1*y1*y0*x2)/(y0*y0*y1+y2*y2*y0+y1*y1*y2-y1*y2*y2-y2*y0*y0-y0*y1*y1);

int i = 0;

while (fabs (x-x2) > = e)

{

out < < i < < ' ' < < x0 < < ' ' < < x1 < < ' ' < < x2 < < ' ' < < x < < endl;

x0 = x1;

x1 = x2;

x2 = x;

y0 = f(x0);

y1 = f(x1);

y2 = f(x2);

x = (y0*y0*y1*x2+y2*y2*y0*x1+y1*y1*y2*x0-y2*y2*y1*x0-y0*y0*y2*x1-y1*y1*y0*x2)/(y0*y0*y1+y2*y2*y0+y1*y1*y2-y1*y2*y2-y2*y0*y0-y0*y1*y1);

i++;

}

out < < i < < ' ' < < x0 < < ' ' < < x1 < < ' ' < < x2 < < ' ' < < x< < endl;

out < < x < < endl;

out < < " Newton: " < < endl;

x0 = x1 = (float)0.2;

x1 = x0 - f(x0)/f1(x0);

i=0;

while (fabs (x0-x1) > = e)

{

out < < i < < ' ' < < x0 < < ' ' < < x1 < < endl;

x0 = x1;

x1 = x0 - f(x0)/f1(x0);

i++;

}

out < < i < < ' ' < < x0 < < ' ' < < x1 < < endl;

out < < x1 < < endl;

return 0;

}

. Код программы z13.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

float f(float q)

{

return (2.*q-1)*(q-4.)*(q-4.)*(q-5.)*(q-5.);

}

float f1 (float q)

{

return 10.*q*q*q*q-148.*q*q*q+780.*q*q-1682.*q+1160.;

}

int main ()

{

ofstream out (" z13.out");

out.precision (19);

out.setf (ios:: fixed);

float em = 0.5;

while (em + 1. > 1.)

{

em = em/2.;

}

em = em*2.;

out < < " Epsilon = " < < em < < endl;

out.precision (16);

float a[6] = {-400., 1160., -841., 260., -37., 2.};

//Для простого корня x = 1/2

out < < " x = 1/2" < < endl;

float nu12 [6];

for (int i=0; i< 6; i++)

{

float p = 1.;

for (int j=0; j< i; j++)

p = p*.5;

nu12[i] = 2.*p*fabs(a[i])/(5.*a[5]*0.0625+4.*a[4]*0.125+3.*a[3]*0.25+a[2]+a[1]);

out < < i < < ' ' < < nu12[i] < < endl;

}

float d1 = 0.;

for (i=0; i< 6; i++)

{

d1 += fabs(nu12[i]*em);

}

float d = d1*.5;

out < < " dx = " < < d < < endl;

//Для кратного корня x = 4

out < < " x = 4" < < endl;

float a2[5] = {1160., -1682., 780., -148., 10.};

float nu4 [5];

for (i=0; i< 5; i++)

{

float p = 1.;

for (int j=0; j< i; j++)

p = p*4.;

nu4[i] = p*fabs(a2[i])/(4.*a2[4]*64.+3.*a[3]*16.+2.*a[2]*4.+a[1])/4.;

out < < i < < ' ' < < nu4[i] < < endl;

}

d1 = 0.;

for (i=0; i< 5; i++)

{

d1 += fabs(nu4[i]*2.*em);

}

d = d1*4.;

out < < " dx = " < < d < < endl;

//Для кратного корня x = 5

out < < " x = 5" < < endl;

float nu5 [5];

for (i=0; i< 5; i++)

{

float p = 1.;

for (int j=0; j< i; j++)

p = p*5.;

nu5[i] = p*fabs(a2[i])/(4.*a2[4]*125.+3.*a[3]*25.+2.*a[2]*5.+a[1])/5.;

out < < i < < ' ' < < nu5[i] < < endl;

}

d1 = 0.;

for (i=0; i< 5; i++)

{

d1 += fabs(nu5[i]*2.*em);

}

d = d1*5.;

out < < " dx = " < < d < < endl;

//Найдем корни методом Ньютона

out < < " Newton: " < < endl;

out < < " Простой корень: " < < endl;

float xn1 = (float)0., xn;

float e = em*.5;

xn = xn1 - f(xn1)/f1(xn1);

i=0;

while (fabs (xn-xn1) > = e)

{

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < fabs (xn-xn1)/em < < endl;

xn1 = xn;

xn = xn1 - f(xn1)/f1(xn1);

i++;

}

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < fabs (xn-xn1)/em < < endl;

out < < " x1 = " < < xn1 < < endl;

out < < " Первый кратный корень: " < < endl;

xn1 = (float)3.5;

e = em*4.;

xn = xn1 - f(xn1)/f1(xn1);

i=0;

while (fabs (xn-xn1) > = e)

{

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < fabs (xn-xn1)/em < < endl;

xn1 = xn;

xn = xn1 - f(xn1)/f1(xn1);

i++;

}

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < fabs (xn-xn1)/em < < endl;

out < < " x2, 3 = " < < xn1 < < ' ' < < 4-xn1< < endl;

out < < " Второй кратный корень: " < < endl;

xn1 = (float)5.5;

e = em*5.;

xn = xn1 - f(xn1)/f1(xn1);

i=0;

while (fabs (xn-xn1) > = e)

{

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < fabs (xn-xn1)/em < < endl;

xn1 = xn;

xn = xn1 - f(xn1)/f1(xn1);

i++;

}

out < < i < < ' ' < < xn1 < < ' ' < < xn < < ' ' < < fabs (xn-xn1)/em < < endl;

out < < " x4, 5 = " < < xn1 < < ' ' < < 5.-xn1< < endl;

return 0;

}

. Код программы z14.cpp

#include < iostream>

#include < fstream>

#include < math.h>

using namespace std;

float f (float r)

{

return r*r - cos(3.141592653589*r);

}

float f1 (float r)

{

return 2.*r + 3.141592653589*sin(3.141592653589*r);

}

float f2 (float r)

{

return 2. + 3.141592653589*3.141592653589*cos(3.141592653589*r);

}

int main ()

{

ofstream out (" z14.out");

out.precision (18);

out.setf (ios:: fixed);

float em = 0.5;

while (em + 1. > 1.)

{

em = em/2.;

}

em = em*2.;

out < < " Epsilon = " < < em < < endl;

float em3 = (float)0.00000605545444;

out < < " k = -10" < < endl;

float h = em3*.0000000001;

float x = (float)0.26;

float d1, d2;

float y1, y2, y21, y11;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = -9" < < endl;

h = em3*.000000001;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = -2" < < endl;

h = em3*.01;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = -1" < < endl;

h = em3*.1;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = 0" < < endl;

h = em3;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = 1" < < endl;

h = em3*10.;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = 2" < < endl;

h = em3*100.;

out < < " h = " < < h < < endl;;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = 9" < < endl;

h = em3*1000000000.;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

out < < " k = 10" < < endl;

h = em3*10000000000.;

out < < " h = " < < h < < endl;

y1 = (f(x+h) - f (x-h))/(2.*h);

y2 = (f(x+h)-2.*f(x)+f(x-h))/(h*h);

y11 = f1 (x);

y21 = f2 (x);

d1 = fabs (y1 - y11);

d2 = fabs (y2 - y21);

out < < y1 < < ' ' < < y11 < < ' ' < < y2 < < ' ' < < y21 < < endl;

out < < " d1 = " < < d1 < < ' ' < < " d2 = " < < d2 < < endl;

return 0;

}

Литература

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3. Бахвалов Н.С., Лапин А.В., Чижонков Е.В. Численные методы в задачах и упражнениях. М., Высшая школа, 2000. -254 с.

4. В.М. Вербжицкий. Основы численных методов. М., Высшая школа, 2002. -847 с.

5. Волков Е.А. Численные методы. СПб., Лань, 2004. -244 с.

6.. Калиткин Н.Н. Численные методы. М., Наука, 1978. -246с.

7. Крылов В.И., Бобков В.В., Монастырский П.И. Вычислительные методы.. Наука,. 1976. – 300с..(есть др. издания)

8. Научно-практический журнал " Exponenta Pro. Математика в приложениях" https://rrc.dgu.ru/res/exponenta/educat/systemat/systemat.asp.htm

9. Пирумов У.Г. Численные методы. М., Дрофа, 2003. -224 с.

10. Турчак Л.И., Плотников П.В. Основы численных методов. М., Физматлит, 2002. -356 с.