Метод Гаусса с выбором главной переменной

(практическая работа по компьютерной алгебре)

Текст программы.

#include

#include

#include

#include

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