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Решение линейного уравнения методом Гаусса с выбором главной переменной



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Текст программы.

#include <fstream.h>

#include <math.h>

#include <conio.h>

#include <stdlib.h>

const num = 4;

int i,j,I,J;

int c[num+1];

long double x[num+1];

long double max;

long double A[num][num+1];

// -----------------------------------------------------------------void max_el(int sr, int st)

{ max = A[num+1-sr][num+2-st];

I = num+1-sr;

J = num+2-st;

for (i = num+1-sr ; i<=num ; i++)

{

for (j = num+2-st ; j<=num ; j++)

{

if (fabs(A[i][j]) > fabs(max))

{

max = A[i][j];

I = i;

J = j;

}

}

}

cout << "\n\n Max = " << max << " I=" << I<< " J=" << J;

}

// -----------------------------------------------------------------void print(int sr,int st)

{

cout << "\n";

int i,j;

for (i = num+1-sr ; i<=num ; i++)

{

for (j = num+2-st ; j<=num+1 ; j++)

{

if (A[i][j] < 0 ) gotoxy(12*j + j - 1,i+1);

else gotoxy(12*j + j,i+1);

cout << A[i][j];

}

}

}

// -----------------------------------------------------------------void preob(int S)

{

int i,j;

long double temp;

for (j = S; j<=num+1; j++) A[S][j] = A[S][j]/max;

for (i = S + 1; i <= num; i++)

{

temp = A[i][S];

for (j = S; j<= num+1 ; j++) A[i][j] = A[i][j] - A[S][j]*temp;

}

}

// -----------------------------------------------------------------void perestanovka(int sr,int st)

{

if (J != (num+1-sr))

{

for (i = 1; i<=num; i++) {

A[i][J] = A[i][J] + A[i][num+1-sr];

A[i][num+1-sr] = A[i][J] - A[i][num+1-sr];

A[i][J] = A[i][J] - A[i][num+1-sr];

}

c[J] = c[J] + c[num+1-sr];

c[num+1-sr] = c[J] - c[num+1-sr];

c[J] = c[J] - c[num+1-sr];

}

if (I != (num+2-st))

{

for (j = 1; j<=num+1; j++) {

A[I][j] = A[I][j] + A[num+2-st][j];

A[num+2-st][j] = A[I][j] - A[num+2-st][j];

A[I][j] = A[I][j] - A[num+2-st][j]; }

}

}

// -----------------------------------------------------------------void otvet()

{

float temp;

for (i=num; i>=1; i--)

{

temp = A[i][num+1];

for(j = num; j > i; j--) temp = temp - A[i][j]*x[j];

x[i] = temp/A[i][i];

}

}

// -----------------------------------------------------------------void interface()

{

clrscr();

print(num,num+1);

cout << "\n Массив перестановок столбцов ";

for (i = 1; i <= num ;i++) cout << " " << c[i];

}

// -----------------------------------------------------------------void load_file()

{

char ch;

 ifstream in("c:\\gauss\\mat.dat");

 cout << "\n";

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

{

c[i] = i;

while (ch != '|') in >> ch;

ch = 'q';

for (j = 1 ; j<=num+1 ; j++) in >> A[i][j];

}

}

// -----------------------------------------------------------------void main()

{

clrscr();

load_file();

int g;

for(g = num+1; g >= 3; g--)

{

interface(); max_el(g-1,g); getch();

 perestanovka(g-1,g); interface(); getch();

preob(num+2-g);  interface(); getch();

}

clrscr();

print(num,num+1);

otvet();

print(num,num+1);

cout << "\n\n ";

long double X[num];

for (i=1; i<=num; i++) X[c[i]] = x[i];

for (i=1; i<=num; i++) cout << " X" << i << " = " << X[i];

 getch();

}

Тестовые задания.

Задание №1 (найти неизвестные):

4.24x1 + 2.73x2 - 1.55x3 = 1.87

2.34x1 + 1.27x2 + 3.15x3 = 2.16

3.05x1  - 1.05x2 - 0.63x3 = -1.25

1.1 Результат выполнения программы:

x1 = - 0.025461 x2 = 0.915112 x3 = 0.335678

1.2 Расчёт погрешности вычисления:

4.24*(- 0.025461) + 2.73*0.915112 - 1.55*0.335678 = 1,87000022 погрешность: 2,2*10-7

2.34*(- 0.025461) + 1.27*0.915112 + 3.15*0.335678 = 2,1599992 погрешность: 8,0*10-7

3.05*(- 0.025461)  - 1.05*0.915112 - 0.63*0.335678 =  -1,25000079 погрешность: 7,9*10-7

средняя погрешность вычисления: 6,0*10-7

Задание №2 (найти неизвестные):

3.81x1 + 0.25x2 + 1.28x3 + (0.75+a)x4 = 4.21

2.25x1 + 1.32x2 + (4.5+a)x3 + 0.49x4 = 6.47+b

5.31x1 + (0.28+a) x2 + 0.98x3 + 1.04x4 = 2.38

(9.39+a)x1 + 2.45x2 + 3.35x3 + 2.28x4 = 10.48+b

a = (0,1,2,3,4) b = (0,1,2,3,4,5)

2.1 Таблица значений.

a

b

Ответы:

0

0

X1 = -12.660899

 

 

X2 = -16.107649

 

 

X3 = 5.273899

 

 

X4 = 66.299137

 

1

X1 = -12.132586

 

 

X2 = -14.858407

 

 

X3 = 5.186943

 

 

X4 = 63.347289

 

2

X1 = -11.604272

 

 

X2 = -13.609166

 

 

X3 = 5.099988

 

 

X4 = 60.39544

 

3

X1 = -11.075957

 

 

X2 = -12.359925

 

 

X3 = 5.013031

 

 

X4 = 57.443595

 

4

X1 = -10.547642

 

 

X2 = -11.110685

 

 

X3 = 4.926076

 

 

X4 = 54.491746

 

5

X1 = -10.019327

 

 

X2 = -9.861445

 

 

X3 = 4.839121

 

 

X4 = 51.539901

1

0

X1 = 13.959632

 

 

X2 = -39.106359

 

 

X3 = 7.324007

 

 

X4 = -27.756765

 

1

X1 = 16.668562

 

 

X2 = -46.672114

 

 

X3 = 8.73446

 

 

X4 = -33.605312

 

2

X1 = 19.377489

 

 

X2 = -54.237864

 

 

X3 = 10.144913

 

 

X4 = -39.453861

 

3

X1 = 22.086416

 

 

X2 = -61.803618

 

 

X3 = 11.555367

 

 

X4 = -45.30241

 

4

X1 = 24.795347

 

 

X2 = -69.369373

 

 

X3 = 12.96582

 

 

X4 = -51.150959

 

5

X1 = 27.504276

 

 

X2 = -76.935127

 

 

X3 = 14.376274

 

 

X4 = -56.999508

2

0

X1 = 1.033843

 

 

X2 = -1.696273

 

 

X3 = 0.997951

 

 

X4 = -0.211727

 

1

X1 = 1.191176

 

 

X2 = -2.016845

 

 

X3 = 1.183171

 

 

X4 = -0.486773

 

2

X1 = 1.348508

 

 

X2 = -2.337417

 

 

X3 = 1.36839

 

 

X4 = -0.761819

 

3

X1 = 1.505841

 

 

X2 = -2.657989

 

 

X3 = 1.55361

 

 

X4 = -1.036865

 

4

X1 = 1.663174

 

 

X2 = -2.978561

 

 

X3 = 1.73883

 

 

X4 = -1.311911

 

5

X1 = 1.820507

 

 

X2 = -3.299134

 

 

X3 = 1.92405

 

 

X4 = -1.586957

3

0

X1 = 0.772977

 

 

X2 = -0.794749

 

 

X3 = 0.762146

 

 

X4 = 0.13016

 

1

X1 = 0.872765

 

 

X2 = -0.954303

 

 

X3 = 0.902687

 

 

X4 = -0.008559

 

2

X1 = 0.972553

 

 

X2 = -1.113856

 

 

X3 = 1.043229

 

 

X4 = -0.147278

 

3

X1 = 1.072341

 

 

X2 = -1.27341

 

 

X3 = 1.18377

 

 

X4 = -0.285998

 

4

X1 = 1.172129

 

 

X2 = -1.432964

 

 

X3 = 1.324311

 

 

X4 = -0.424717

 

5

X1 = 1.271917

 

 

X2 = -1.592518

 

 

X3 = 1.464853

 

 

X4 = -0.563436

4

0

X1 = 0.675128

 

 

X2 = -0.476895

 

 

X3 = 0.645225

 

 

X4 = 0.196021

 

1

X1 = 0.754634

 

 

X2 = -0.580642

 

 

X3 = 0.763131

 

 

X4 = 0.105936

 

2

X1 = 0.83414

 

 

X2 = -0.68439

 

 

X3 = 0.881037

 

 

X4 = 0.015852

 

3

X1 = 0.913647

 

 

X2 = -0.788137

 

 

X3 = 0.998942

 

 

X4 = -0.074233

 

4

 X1 = 0.993153

 

 

X2 = -0.891884

 

 

X3 = 1.116848

 

 

X4 = -0.164317

 

5

X1 = 1.072659

 

 

X2 = -0.995631

 

 

X3 = 1.234754

 

 

X4 = -0.254402



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