1. A.Abhari CPS1251 Multidimensional Arrays Multidimensional array is the array with two or more dimensions. For example: char box [3] [3] defines a two-dimensional array and box[2][1] is an element in row 2, column 1 and char box[][3] can be used in the function prototype note that only the first dimension can be omitted

2. A.Abhari CPS1252 Multidimensional Arrays For example : double table [NROWS][NCOLS]; Can be used as parameter in the function prototype as: void process_martrix( int in[ ][4], int out[ ][4], int nrows)

3. A.Abhari CPS1253 Two Dimensional Array Char box [3] [3] 0 1 2 012 Row Column box [1] [2]

4. A.Abhari CPS1254 /* * Checks whether a box is completely filled */ int filled(char box[3][3]) /* input - box to check*/ { int r,c, /* row and column subscripts*/ ans; /* whether or not box is filled.*/ /* Assumes box is filled until blank is found*/ ans = 1; /* Resets ans to zero if a blank is found*/ for (r = 0; r < 3; ++r) for (c = 0; c < 3; ++c) if (box[r][c] == ' ') ans = 0; return (ans); }

5. A.Abhari CPS1255 Arrays with Several Dimensions int soil_type[4] [7] [MAXDEPTH]

6. A.Abhari CPS1256 Case Study: Cellular Telephone System Problem: Finding the best way to build a cellular network. There is some marketing data that predicts the demand will be at tree time of interest. There are only 10 transmitters and there is a need for a program to help analyzing call demand data.

7. A.Abhari CPS1257 Case Study: Cellular Telephone System Analysis: There should be three matrices shows traffic density for each time of the day: Input: int commuters[GRID_SIZE][GRID_SIZE] int salesforce[GRID_SIZE][GRID_SIZE] int weekend[GRID_SIZE][GRID_SIZE] Output: int summed_data[GRID_SIZE][GRID_SIZE] int location_i, location_j

8. A.Abhari CPS1258 Case Study: Cellular Telephone System Design: initial algorithm: 1.Get traffic data for three time period 2.Get the weights from user 3.Multiply weight by each matrix entry and store the sum in the summed data 4.Find highest valued cells in the summed data and display them as the pair of location_i and location_j Implementation

9. Filling the multidimensional array /* Fills 3 GRID_SIZE x GRID_SIZE arrays with traffic data from TRAFFIC_FILE*/ void get_traffic_data(int commuters[GRID_SIZE][GRID_SIZE], /* output */ int salesforce[GRID_SIZE][GRID_SIZE], /* output */ int weekend[GRID_SIZE][GRID_SIZE]) /* output */ { int i, j; /* loop counters */ FILE *fp; /* file pointer */ fp = fopen(TRAFFIC_FILE, "r"); for (i = 0; i < GRID_SIZE; ++i) for (j = 0; j < GRID_SIZE; ++j) fscanf(fp, "%d", &commuters[i][j]); for (i = 0; i < GRID_SIZE; ++i) for (j = 0; j < GRID_SIZE; ++j) fscanf(fp, "%d", &salesforce[i][j]); for (i = 0; i < GRID_SIZE; ++i) for (j = 0; j < GRID_SIZE; ++j) fscanf(fp, "%d", &weekend[i][j]); fclose(fp); }

10. A.Abhari CPS12510 /* Computes and displays the weighted, summed_data */ for (i = 0; i < GRID_SIZE; ++i) for (j = 0; j < GRID_SIZE; ++j) summed_data[i][j] = commuter_weight* commuters[i][j] + salesforce_weight * salesforce[i][j] + weekend_weight * weekend[i][j]; Modifying the multidimensional array

11. /* Finds the NUM_TRANSMITTERS highest values in the summed_data matrix.Temporarily stores the coordinates in location_i and location_j, and then displays the resulting locations */ printf("\n\nLocations of the %d transmitters:\n\n", NUM_TRANSMITTERS); for (tr = 1; tr <= NUM_TRANSMITTERS; ++tr) { current_max = SELECTED; /* Starts off our search with a value that is known to be too low. */ for (i = 0; i < GRID_SIZE; ++i) { for (j = 0; j < GRID_SIZE; ++j) { if (current_max < summed_data[i][j]) { current_max = summed_data[i][j]; location_i = i; location_j = j; } Searching in the multidimensional array

12. A.Abhari CPS12512 Printing the contents of multidimensional array /* * Displays contents of a GRID_SIZE x GRID_SIZE matrix of integers */ void print_matrix(int matrix[GRID_SIZE][GRID_SIZE]) { int i, j; /* loop counters */ for (i = 0; i < GRID_SIZE; ++i) { for (j = 0; j < GRID_SIZE; ++j) printf("%3d ", matrix[i][j]); printf("\n"); }

13. A.Abhari CPS12513 Vectors Vector: a mathematical object consisting of a sequence of numbers. /* a vector */ int vect[3] = {4, 12, 19}; Differences between vector and array: 1- an n_dimensional vector is represented in C as a one dimensional array of size n. 2- vect 3 is vect[2] in C

14. A.Abhari CPS12514 Vectors Calculating scalar product:. = 1*2 + 2*3 +4*1=12 In C: sum_prod = 0; for (k=0; k<n; k++) sum_prod += x[k] * w[k];

15. A.Abhari CPS12515 Matrices Matrix: a mathematical object consisting of a rectangular arrangement of numbers called the element of matrix.. /* a matrix 3 6 4 5 int x[2][2] = {{3, 6}, {4, 5}}; 36 45 x

16. A.Abhari CPS12516 Matrices Multiplying a matrix by a vector A * X = V 1 1 1 5 2 3 1 1 10 multiplication 1 -1 -1 * 2 = -3 on 0 1 2 2 6 the right In C for each member of V: v[i] = 0; for (k=0; k<n; k++) v[k] += a[i][k] * x[k];

17. A.Abhari CPS12517 /* Computes the product of M-by-N matrix a and the N- dimensional vector x. The result is stored in the output parameter v, an M-dimensional vector.*/ void mat_vec_prod(double v[], /* M-dimensional vector */ double a[M][N], /* M-by-N matrix */ double x[]) /* N-dimensional vector */ { int i, k; for (i = 0; i < M; ++i) { v[i] = 0; for (k = 0; k < N; ++k) { v[i] += a[i][k] * x[k]; }

18. A.Abhari CPS12518 Matrix Multiplication 1 1 1 2 0 1 6 0 0 2 3 1 * 1 -1 0 = 10 -2 1 1 -1 -1 3 1 -1 -2 0 2 for ( i=0; i< m, ++i) { for (j=0; j<p; ++j) { …….. compute c[i][j]…. }

19. A.Abhari CPS12519 /* Multiplies matrices A and B yielding product matrix C */ void mat_prod(double c[M][P], /* output - M by P matrix */ double a[M][N], /* input - M by N matrix */ double b[N][P]) /* input - N by P matrix */ { int i, j, k; for (i = 0; i < M; ++i) { for (j = 0; j < P; ++j) { c[i][j] = 0; for (k = 0; k < N; ++k) c[i][j] += a[i][k] * b[k][j]; }

20. A.Abhari CPS12520 Solving System of Linear Equations To solve many problems such as force equation in three-dimensional system we need to solve a three linear equations: A X = Y 1 1 1 x 1 4 2 3 1 * x 2 = 9 1 -1 -1 x 3 -2 It is multiplication of a matrix by a vector on the right

21. A.Abhari CPS12521 Gaussian Elimination Gaussian elimination can be used to solve a linear equation. The algorithm for Gussian elimination is: 1.Transform the original system into scaled triangular form. 2.Solve for x i by back substitution

22. A.Abhari CPS12522 Gaussian Elimination triangular form 1 1 1 x 1 4 0 1 -1 * x 2 = 1 0 0 1 x 3 1 back substitution x 1 + x 2 + x 3 = 4 x 2 - x 3 = 1 x 3 = 1

23. A.Abhari CPS12523 Gaussian Elimination For doing that we need to triangularizing the augmented matrix by following operations: 1.Multiply any row of aug by nonzero number 2.Add to any row of aug a multiple of other rows 3.Swap any two rows If system has a unique solution, we can get the system into desired form by this three operations.

24. A.Abhari CPS12524 /* * Performs pivoting with respect to the pth row and the pth column * If no nonzero pivot can be found, FALSE is sent back through piv_foundp */ void pivot(double aug[N][N+1], /* input/output - augmented matrix */ int p, /* input - current row*/ int *piv_foundp) /* output - whether or not nonzero pivot found*/ { double xmax, xtemp; int j, k, max_row; /* Finds maximum pivot*/ xmax = fabs(aug[p][p]); max_row = p; for (j = p+1; j < N; ++j) { if (fabs(aug[j][p]) > xmax) { xmax = fabs(aug[j][p]); max_row = j; }

25. A.Abhari CPS12525 /* Swaps rows if nonzero pivot was found*/ if (xmax == 0) { *piv_foundp = FALSE; } else { *piv_foundp = TRUE; if (max_row != p) { /* swap rows */ for (k = p; k < N+1; ++k) { xtemp = aug[p][k]; aug[p][k] = aug[max_row][k]; aug[max_row][k] = xtemp; }

26. /* * Performs back substitution to compute a solution vector to a system of * linear equations represented by the augmented matrix aug. Assumes that * the coefficient portion of the augmented matrix has been triangularized, * and its diagonal values are all 1. */ void back_sub(double aug[N][N+1], /* input - scaled, triangularized augmented matrix*/ double x[N]) /* output - solution vector */ { double sum; int i, j; x[N - 1] = aug[N - 1][N]; for (i = N - 2; i >= 0; --i) { sum = 0; for (j = i + 1; j < N; ++j) sum += aug[i][j] * x[j]; x[i] = aug[i][N] - sum; }

27. A.Abhari CPS12527 Common Programming Errors Use constants for each dimension’s size when declaring multidimensional array When declaring the array as a parameter of a function if you omit the first dimension all other dimensions must be supplied Since access to the elements of a multidimensional array requires nested counting loops it is easy to make out-of-range error. Since using multidimensional arrays as local variables requires large memory space, you may need to tell to operating system to increase stack size when the program is running