Метод Гаусса с выбором главной переменной (46224)

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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








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