Решение линейного уравнения методом Гаусса с выбором главной переменной

Решение линейного уравнения методом Гаусса с выбором главной переменной

ГОСУДАРСТВЕННЫЙ КОМИТЕТ РОССИЙСКОЙ ФЕДЕРАЦИИ

ПО ВЫСШЕМУ И СРЕДНЕСПЕЦИАЛЬНОМУ ОБРАЗОВАНИЮ

КРАСНОЯРСКИЙ ГОСУДАРСТВЕННЫЙ ТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ

Практическое задание

по компьютерной алгебре

тема: метод Гаусса с выбором главной переменной

Выполнил:

студент 2 курса ФИВТ

группы 27-4

Попов А.В.

Проверила:

Быкова Е.Г.

1998 г.

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++)

}

// ------------------------------------------------------------------

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 = |

| | |-12.660|

| | |899 |

| | |X2 = |

| | |-16.107|

| | |649 |

| | |X3 = |

| | |5.27389|

| | |9 |

| | |X4 = |

| | |66.2991|

| | |37 |

| |1 |X1 = |

| | |-12.132|

| | |586 |

| | |X2 = |

| | |-14.858|

| | |407 |

| | |X3 = |

| | |5.18694|

| | |3 |

| | |X4 = |

| | |63.3472|

| | |89 |

| |2 |X1 = |

| | |-11.604|

| | |272 |

| | |X2 = |

| | |-13.609|

| | |166 |

| | |X3 = |

| | |5.09998|

| | |8 |

| | |X4 = |

| | |60.3954|

| | |4 |

| |3 |X1 = |

| | |-11.075|

| | |957 |

| | |X2 = |

| | |-12.359|

| | |925 |

| | |X3 = |

| | |5.01303|

| | |1 |

| | |X4 = |

| | |57.4435|

| | |95 |

| |4 |X1 = |

| | |-10.547|

| | |642 |

| | |X2 = |

| | |-11.110|

| | |685 |

| | |X3 = |

| | |4.92607|

| | |6 |

| | |X4 = |

| | |54.4917|

| | |46 |

| |5 |X1 = |

| | |-10.019|

| | |327 |

| | |X2 = |

| | |-9.8614|

| | |45 |

| | |X3 = |

| | |4.83912|

| | |1 |

| | |X4 = |

| | |51.5399|

| | |01 |

|1 |0 |X1 = |

| | |13.9596|

| | |32 |

| | |X2 = |

| | |-39.106|

| | |359 |

| | |X3 = |

| | |7.32400|

| | |7 |

| | |X4 = |

| | |-27.756|

| | |765 |

| |1 |X1 = |

| | |16.6685|

| | |62 |

| | |X2 = |

| | |-46.672|

| | |114 |

| | |X3 = |

| | |8.73446|

| | |X4 = |

| | |-33.605|

| | |312 |

| |2 |X1 = |

| | |19.3774|

| | |89 |

| | |X2 = |

| | |-54.237|

| | |864 |

| | |X3 = |

| | |10.1449|

| | |13 |

| | |X4 = |

| | |-39.453|

| | |861 |

| |3 |X1 = |

| | |22.0864|

| | |16 |

| | |X2 = |

| | |-61.803|

| | |618 |

| | |X3 = |

| | |11.5553|

| | |67 |

| | |X4 = |

| | |-45.302|

| | |41 |

| |4 |X1 = |

| | |24.7953|

| | |47 |

| | |X2 = |

| | |-69.369|

| | |373 |

| | |X3 = |

| | |12.9658|

| | |2 |

| | |X4 = |

| | |-51.150|

| | |959 |

| |5 |X1 = |

| | |27.5042|

| | |76 |

| | |X2 = |

| | |-76.935|

| | |127 |

| | |X3 = |

| | |14.3762|

| | |74 |

| | |X4 = |

| | |-56.999|

| | |508 |

|2 |0 |X1 = |

| | |1.03384|

| | |3 |

| | |X2 = |

| | |-1.6962|

| | |73 |

| | |X3 = |

| | |0.99795|

| | |1 |

| | |X4 = |

| | |-0.2117|

| | |27 |

| |1 |X1 = |

| | |1.19117|

| | |6 |

| | |X2 = |

| | |-2.0168|

| | |45 |

| | |X3 = |

| | |1.18317|

| | |1 |

| | |X4 = |

| | |-0.4867|

| | |73 |

| |2 |X1 = |

| | |1.34850|

| | |8 |

| | |X2 = |

| | |-2.3374|

| | |17 |

| | |X3 = |

| | |1.36839|

| | |X4 = |

| | |-0.7618|

| | |19 |

| |3 |X1 = |

| | |1.50584|

| | |1 |

| | |X2 = |

| | |-2.6579|

| | |89 |

| | |X3 = |

| | |1.55361|

| | |X4 = |

| | |-1.0368|

| | |65 |

| |4 |X1 = |

| | |1.66317|

| | |4 |

| | |X2 = |

| | |-2.9785|

| | |61 |

| | |X3 = |

| | |1.73883|

| | |X4 = |

| | |-1.3119|

| | |11 |

| |5 |X1 = |

| | |1.82050|

| | |7 |

| | |X2 = |

| | |-3.2991|

| | |34 |

| | |X3 = |

| | |1.92405|

| | |X4 = |

| | |-1.5869|

| | |57 |

|3 |0 |X1 = |

| | |0.77297|

| | |7 |

| | |X2 = |

| | |-0.7947|

| | |49 |

| | |X3 = |

| | |0.76214|

| | |6 |

| | |X4 = |

| | |0.13016|

| |1 |X1 = |

| | |0.87276|

| | |5 |

| | |X2 = |

| | |-0.9543|

| | |03 |

| | |X3 = |

| | |0.90268|

| | |7 |

| | |X4 = |

| | |-0.0085|

| | |59 |

| |2 |X1 = |

| | |0.97255|

| | |3 |

| | |X2 = |

| | |-1.1138|

| | |56 |

| | |X3 = |

| | |1.04322|

| | |9 |

| | |X4 = |

| | |-0.1472|

| | |78 |

| |3 |X1 = |

| | |1.07234|

| | |1 |

| | |X2 = |

| | |-1.2734|

| | |1 |

| | |X3 = |

| | |1.18377|

| | |X4 = |

| | |-0.2859|

| | |98 |

| |4 |X1 = |

| | |1.17212|

| | |9 |

| | |X2 = |

| | |-1.4329|

| | |64 |

| | |X3 = |

| | |1.32431|

| | |1 |

| | |X4 = |

| | |-0.4247|

| | |17 |

| |5 |X1 = |

| | |1.27191|

| | |7 |

| | |X2 = |

| | |-1.5925|

| | |18 |

| | |X3 = |

| | |1.46485|

| | |3 |

| | |X4 = |

| | |-0.5634|

| | |36 |

|4 |0 |X1 = |

| | |0.67512|

| | |8 |

| | |X2 = |

| | |-0.4768|

| | |95 |

| | |X3 = |

| | |0.64522|

| | |5 |

| | |X4 = |

| | |0.19602|

| | |1 |

| |1 |X1 = |

| | |0.75463|

| | |4 |

| | |X2 = |

| | |-0.5806|

| | |42 |

| | |X3 = |

| | |0.76313|

| | |1 |

| | |X4 = |

| | |0.10593|

| | |6 |

| |2 |X1 = |

| | |0.83414|

| | |X2 = |

| | |-0.6843|

| | |9 |

| | |X3 = |

| | |0.88103|

| | |7 |

| | |X4 = |

| | |0.01585|

| | |2 |

| |3 |X1 = |

| | |0.91364|

| | |7 |

| | |X2 = |

| | |-0.7881|

| | |37 |

| | |X3 = |

| | |0.99894|

| | |2 |

| | |X4 = |

| | |-0.0742|

| | |33 |

| |4 | X1 = |

| | |0.99315|

| | |3 |

| | |X2 = |

| | |-0.8918|

| | |84 |

| | |X3 = |

| | |1.11684|

| | |8 |

| | |X4 = |

| | |-0.1643|

| | |17 |

| |5 |X1 = |

| | |1.07265|

| | |9 |

| | |X2 = |

| | |-0.9956|

| | |31 |

| | |X3 = |

| | |1.23475|

| | |4 |

| | |X4 = |

| | |-0.2544|

| | |02 |