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 << »

Max = » << max << » I=» << I<< » J=» << J;
}

// ——————————————————————

void print(int sr,int st)
{
cout << »
«;
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 << »
Массив перестановок столбцов «;
for (i = 1; i <= num ;i++) cout << » » << c[i];
}

// ——————————————————————

void load_file()
{
char ch;
ifstream in(«c:\gauss\mat.dat»);
cout << »
«;
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 << »

«;

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):
4.24×1 + 2.73×2 — 1.55×3 = 1.87
2.34×1 + 1.27×2 + 3.15×3 = 2.16
3.05×1 — 1.05×2 — 0.63×3 = -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):

3.81×1 + 0.25×2 + 1.28×3 + (0.75+a)x4 = 4.21
2.25×1 + 1.32×2 + (4.5+a)x3 + 0.49×4 = 6.47+b
5.31×1 + (0.28+a) x2 + 0.98×3 + 1.04×4 = 2.38
(9.39+a)x1 + 2.45×2 + 3.35×3 + 2.28×4 = 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 |