Midterm Math

 Name : Ou Visal

Grade :ITE-A2

Midterm Exam

I. ចូរសរសេរកូដ (ក្រៅពី Matlab) ដើម្បីគណនានិងដោះស្រាយ៖
១) ដេទែមីណង់ (Determinant)
២) ម៉ាទ្រីសច្រាស (Inverse matrix)
៣) Gaussian elimination
៤) Iteration method
II. ចូរដោះស្រាយរកឬសប្រព័ន្ធសមីការខាងក្រោម តាមវិធាន៖
១) Cramer's rule
២) Inverse matrix
៣) Gaussian elimination

ចម្លើយ

I. ចូរសរសេរកូដ (ក្រៅពី Matlab) ដើម្បីគណនានិងដោះស្រាយ៖
១) ដេទែមីណង់ (Determinant)

#include<stdio.h> #include<math.h> #include<stdlib.h> #define SIZE 10 int main() { float a[SIZE][SIZE], x[SIZE], ratio, det = 1; int i, j, k, n; printf("Enter Order of Matrix: "); scanf_s("%d", &n); printf("\nEnter Coefficients of Matrix: \n"); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { printf("a[%d][%d]=", i, j); scanf_s("%f", &a[i][j]); } } for (i = 0; i < n; i++) { if (a[i][i] == 0.0) { printf("Mathematical Error!"); exit(0); } for (j = i + 1; j < n; j++) { ratio = a[j][i] / a[i][i]; for (k = 0; k < n; k++) { a[j][k] = a[j][k] - ratio * a[i][k]; } } } printf("\nUpper Triangular Matrix: \n"); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { printf("%0.2f\t", a[i][j]); } printf("\n"); } for (i = 0; i < n; i++) { det = det * a[i][i]; } printf("\n\nDeterminant of given matrix is: %0.3f", det); return 0; }

២) ម៉ាទ្រីសច្រាស (Inverse matrix)

#include<stdio.h> #include<math.h> float determinant(float[][25], float); void cofactor(float[][25], float); void transpose(float[][25], float[][25], float); int main() { float a[25][25], k, d; int i, j; printf("Enter the row and colume of the Matrix : "); scanf_s("%f", &k); printf("Enter the elements of %.0fX%.0f Matrix : \n", k, k); for (i = 0; i < k; i++) { for (j = 0; j < k; j++) { printf("a[%d][%d]=", i, j); scanf_s("%f", &a[i][j]); } } d = determinant(a, k); if (d == 0) printf("\nInverse of Entered Matrix is not possible\n"); else cofactor(a, k); } float determinant(float a[25][25], float k) { float s = 1, det = 0, b[25][25]; int i, j, m, n, c; if (k == 1) { return (a[0][0]); } else { det = 0; for (c = 0; c < k; c++) { m = 0; n = 0; for (i = 0; i < k; i++) { for (j = 0; j < k; j++) { b[i][j] = 0; if (i != 0 && j != c) { b[m][n] = a[i][j]; if (n < (k - 2)) n++; else { n = 0; m++; } } } } det = det + s * (a[0][c] * determinant(b, k - 1)); s = -1 * s; } } return (det); } void cofactor(float num[25][25], float f) { float b[25][25], fac[25][25]; int p, q, m, n, i, j; for (q = 0; q < f; q++) { for (p = 0; p < f; p++) { m = 0; n = 0; for (i = 0; i < f; i++) { for (j = 0; j < f; j++) { if (i != q && j != p) { b[m][n] = num[i][j]; if (n < (f - 2)) n++; else { n = 0; m++; } } } } fac[q][p] = pow(-1, q + p) * determinant(b, f - 1); } } transpose(num, fac, f); } /*Finding transpose of matrix*/ void transpose(float num[25][25], float fac[25][25], float r) { int i, j; float b[25][25], inverse[25][25], d; for (i = 0; i < r; i++) { for (j = 0; j < r; j++) { b[i][j] = fac[j][i]; } } d = determinant(num, r); for (i = 0; i < r; i++) { for (j = 0; j < r; j++) { inverse[i][j] = b[i][j] / d; } } printf("\n\n\nThe inverse of matrix is : \n"); for (i = 0; i < r; i++) { for (j = 0; j < r; j++) { printf("\t%.4f", inverse[i][j]); } printf("\n"); } }

៣) Gaussian elimination

#include<stdio.h> int main() { int i, j, k, n; float A[20][20], c, x[10], sum = 0.0; printf("Enter the order of matrix: "); scanf_s("%d", &n); printf("\nEnter the elements of augmented matrix row-wise:\n\n"); for (i = 1; i <= n; i++) { for (j = 1; j <= (n + 1); j++) { printf("A[%d][%d] : ", i, j); scanf_s("%f", &A[i][j]); } } for (j = 1; j <= n; j++) { for (i = 1; i <= n; i++) { if (i > j) { c = A[i][j] / A[j][j]; for (k = 1; k <= n + 1; k++) { A[i][k] = A[i][k] - c * A[j][k]; } } } } x[n] = A[n][n + 1] / A[n][n]; for (i = n - 1; i >= 1; i--) { sum = 0; for (j = i + 1; j <= n; j++) { sum = sum + A[i][j] * x[j]; } x[i] = (A[i][n + 1] - sum) / A[i][i]; } printf("\nThe solution is: \n"); for (i = 1; i <= n; i++) { printf("\nx%d=%.4f\t", i, x[i]); } return(0); }

៤) Iteration method មានចំនួនពីរគឺ 

#include<stdio.h> float coeff[10][10]; float Dinv[10][10]; float approx[10][1]; float R[10][10]; float matrixRes[10][1]; float b[10][1]; float temp[10][1]; int row, column, size, navigate; void multiply(float matrixA[][10], float matrixB[][1]) { int ctr, ictr; for (ctr = 0; ctr < size; ctr++) { matrixRes[ctr][0] = 0; for (navigate = 0; navigate < size; navigate++) matrixRes[ctr][0] = matrixRes[ctr][0] + matrixA[ctr][navigate] * matrixB[navigate][0]; } } int main() { printf("Enter the number of unknown(below 10)\n"); scanf_s("%d", &size); printf("Enter the coefficent matrix\n"); for (row = 0; row < size; row++) for (column = 0; column < size; column++) scanf_s("%f", &coeff[row][column]); printf("Enter the first approximation\n"); for (row = 0; row < size; row++) scanf_s("%f", &approx[row][0]); printf("Enter the RHS coefficient\n"); for (row = 0; row < size; row++) scanf_s("%f", &b[row][0]); for (row = 0; row < size; row++) for (column = 0; column < size; column++) { if (row == column) Dinv[row][column] = 1 / coeff[row][column]; else Dinv[row][column] = 0; } for (row = 0; row < size; row++) for (column = 0; column < size; column++) { if (row == column) R[row][column] = 0; else if (row != column) R[row][column] = coeff[row][column]; } int iter; printf("Enter the number of iterations:\n"); scanf_s("%d", &iter); int ctr = 1; int octr; while (ctr <= iter) { multiply(R, approx); for (row = 0; row < size; row++) temp[row][0] = b[row][0] - matrixRes[row][0]; multiply(Dinv, temp); for (octr = 0; octr < size; octr++) approx[octr][0] = matrixRes[octr][0]; printf("The Value after iteration %d is\n", ctr); for (row = 0; row < size; row++) printf("%.3f\n", approx[row][0]); ctr++; } getch(); }

៤.២)Gauss Seidel

#include<stdio.h> int main() { float a[10][10], b[10], x[10], y[10]; int n = 0, m = 0, i = 0, j = 0; printf("Enter size of 2d array(Square matrix) : "); scanf_s("%d", &n); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { printf("Enter values no :( %d , %d ) ", i, j); scanf_s("%d", &a[i][j]); } } printf("\nEnter Values to the right side of equation\n"); for (i = 0; i < n; i++) { printf("Enter values no :( %d , %d ) ", i, j); scanf_s("%d", &b[i]); } printf("Enter initial values of x\n"); for (i = 0; i < n; i++) { printf("Enter values no. :( %d ):", i); scanf_s("%d", &x[i]); } printf("\nEnter the no. of iteration : "); scanf_s("%d", &m); while (m > 0) { for (i = 0; i < n; i++) { y[i] = (b[i] / a[i][i]); for (j = 0; j < n; j++) { if (j == i) continue; y[i] = y[i] - ((a[i][j] / a[i][i]) * x[j]); x[i] = y[i]; } printf("x%d = %f ", i + 1, y[i]); } printf("\n"); m--; } return 0; }

II. ចូរដោះស្រាយរកឬសប្រព័ន្ធសមីការខាងក្រោម តាមវិធាន៖