Метод Гаусса с выбором главной переменной
(
практическая работа по компьютерной алгебре
)
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 |