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Реферат: 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++) { 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.
abОтветы:
00X1 = -12.660899
X2 = -16.107649
X3 = 5.273899
X4 = 66.299137
1X1 = -12.132586
X2 = -14.858407
X3 = 5.186943
X4 = 63.347289
2X1 = -11.604272
X2 = -13.609166
X3 = 5.099988
X4 = 60.39544
3X1 = -11.075957
X2 = -12.359925
X3 = 5.013031
X4 = 57.443595
4X1 = -10.547642
X2 = -11.110685
X3 = 4.926076
X4 = 54.491746
5X1 = -10.019327
X2 = -9.861445
X3 = 4.839121
X4 = 51.539901
10X1 = 13.959632
X2 = -39.106359
X3 = 7.324007
X4 = -27.756765
1X1 = 16.668562
X2 = -46.672114
X3 = 8.73446
X4 = -33.605312
2X1 = 19.377489
X2 = -54.237864
X3 = 10.144913
X4 = -39.453861
3X1 = 22.086416
X2 = -61.803618
X3 = 11.555367
X4 = -45.30241
4X1 = 24.795347
X2 = -69.369373
X3 = 12.96582
X4 = -51.150959
5X1 = 27.504276
X2 = -76.935127
X3 = 14.376274
X4 = -56.999508
20X1 = 1.033843
X2 = -1.696273
X3 = 0.997951
X4 = -0.211727
1X1 = 1.191176
X2 = -2.016845
X3 = 1.183171
X4 = -0.486773
2X1 = 1.348508
X2 = -2.337417
X3 = 1.36839
X4 = -0.761819
3X1 = 1.505841
X2 = -2.657989
X3 = 1.55361
X4 = -1.036865
4X1 = 1.663174
X2 = -2.978561
X3 = 1.73883
X4 = -1.311911
5X1 = 1.820507
X2 = -3.299134
X3 = 1.92405
X4 = -1.586957
30X1 = 0.772977
X2 = -0.794749
X3 = 0.762146
X4 = 0.13016
1X1 = 0.872765
X2 = -0.954303
X3 = 0.902687
X4 = -0.008559
2X1 = 0.972553
X2 = -1.113856
X3 = 1.043229
X4 = -0.147278
3X1 = 1.072341
X2 = -1.27341
X3 = 1.18377
X4 = -0.285998
4X1 = 1.172129
X2 = -1.432964
X3 = 1.324311
X4 = -0.424717
5X1 = 1.271917
X2 = -1.592518
X3 = 1.464853
X4 = -0.563436
40X1 = 0.675128
X2 = -0.476895
X3 = 0.645225
X4 = 0.196021
1X1 = 0.754634
X2 = -0.580642
X3 = 0.763131
X4 = 0.105936
2X1 = 0.83414
X2 = -0.68439
X3 = 0.881037
X4 = 0.015852
3X1 = 0.913647
X2 = -0.788137
X3 = 0.998942
X4 = -0.074233
4 X1 = 0.993153
X2 = -0.891884
X3 = 1.116848
X4 = -0.164317
5X1 = 1.072659
X2 = -0.995631
X3 = 1.234754
X4 = -0.254402