blob: cd402ab0f4e9f3917bdf9cb74381e1946137bc74 [file] [log] [blame]
 /** Compute the matrix inverse via Gauss-Jordan elimination. * This program uses OpenMP separate computation steps but no * mutexes. It is an example of a race-free program on which no data races * are reported by the happens-before algorithm (drd), but a lot of data races * (all false positives) are reported by the Eraser-algorithm (helgrind). */ #define _GNU_SOURCE /***********************/ /* Include directives. */ /***********************/ #include #include #include #include #include #include // getopt() /*********************/ /* Type definitions. */ /*********************/ typedef double elem_t; /********************/ /* Local variables. */ /********************/ static int s_trigger_race; /*************************/ /* Function definitions. */ /*************************/ /** Allocate memory for a matrix with the specified number of rows and * columns. */ static elem_t* new_matrix(const int rows, const int cols) { assert(rows > 0); assert(cols > 0); return malloc(rows * cols * sizeof(elem_t)); } /** Free the memory that was allocated for a matrix. */ static void delete_matrix(elem_t* const a) { free(a); } /** Fill in some numbers in a matrix. */ static void init_matrix(elem_t* const a, const int rows, const int cols) { int i, j; for (i = 0; i < rows; i++) { for (j = 0; j < rows; j++) { a[i * cols + j] = 1.0 / (1 + abs(i-j)); } } } /** Print all elements of a matrix. */ void print_matrix(const char* const label, const elem_t* const a, const int rows, const int cols) { int i, j; printf("%s:\n", label); for (i = 0; i < rows; i++) { for (j = 0; j < cols; j++) { printf("%g ", a[i * cols + j]); } printf("\n"); } } /** Copy a subset of the elements of a matrix into another matrix. */ static void copy_matrix(const elem_t* const from, const int from_rows, const int from_cols, const int from_row_first, const int from_row_last, const int from_col_first, const int from_col_last, elem_t* const to, const int to_rows, const int to_cols, const int to_row_first, const int to_row_last, const int to_col_first, const int to_col_last) { int i, j; assert(from_row_last - from_row_first == to_row_last - to_row_first); assert(from_col_last - from_col_first == to_col_last - to_col_first); for (i = from_row_first; i < from_row_last; i++) { assert(i < from_rows); assert(i - from_row_first + to_row_first < to_rows); for (j = from_col_first; j < from_col_last; j++) { assert(j < from_cols); assert(j - from_col_first + to_col_first < to_cols); to[(i - from_row_first + to_col_first) * to_cols + (j - from_col_first + to_col_first)] = from[i * from_cols + j]; } } } /** Compute the matrix product of a1 and a2. */ static elem_t* multiply_matrices(const elem_t* const a1, const int rows1, const int cols1, const elem_t* const a2, const int rows2, const int cols2) { int i, j, k; elem_t* prod; assert(cols1 == rows2); prod = new_matrix(rows1, cols2); for (i = 0; i < rows1; i++) { for (j = 0; j < cols2; j++) { prod[i * cols2 + j] = 0; for (k = 0; k < cols1; k++) { prod[i * cols2 + j] += a1[i * cols1 + k] * a2[k * cols2 + j]; } } } return prod; } /** Apply the Gauss-Jordan elimination algorithm on the matrix p->a starting * at row r0 and up to but not including row r1. It is assumed that as many * threads execute this function concurrently as the count barrier p->b was * initialized with. If the matrix p->a is nonsingular, and if matrix p->a * has at least as many columns as rows, the result of this algorithm is that * submatrix p->a[0..p->rows-1,0..p->rows-1] is the identity matrix. * @see http://en.wikipedia.org/wiki/Gauss-Jordan_elimination */ static void gj(elem_t* const a, const int rows, const int cols) { int i, j, k; for (i = 0; i < rows; i++) { { // Pivoting. j = i; for (k = i + 1; k < rows; k++) { if (a[k * cols + i] > a[j * cols + i]) { j = k; } } if (j != i) { for (k = 0; k < cols; k++) { const elem_t t = a[i * cols + k]; a[i * cols + k] = a[j * cols + k]; a[j * cols + k] = t; } } // Normalize row i. if (a[i * cols + i] != 0) { for (k = cols - 1; k >= 0; k--) { a[i * cols + k] /= a[i * cols + i]; } } } // Reduce all rows j != i. if (s_trigger_race) { # pragma omp parallel for private(j) for (j = 0; j < rows; j++) { if (i != j) { const elem_t factor = a[j * cols + i]; for (k = 0; k < cols; k++) { a[j * cols + k] -= a[i * cols + k] * factor; } } } } else { # pragma omp parallel for private(j, k) for (j = 0; j < rows; j++) { if (i != j) { const elem_t factor = a[j * cols + i]; for (k = 0; k < cols; k++) { a[j * cols + k] -= a[i * cols + k] * factor; } } } } } } /** Matrix inversion via the Gauss-Jordan algorithm. */ static elem_t* invert_matrix(const elem_t* const a, const int n) { int i, j; elem_t* const inv = new_matrix(n, n); elem_t* const tmp = new_matrix(n, 2*n); copy_matrix(a, n, n, 0, n, 0, n, tmp, n, 2 * n, 0, n, 0, n); for (i = 0; i < n; i++) for (j = 0; j < n; j++) tmp[i * 2 * n + n + j] = (i == j); gj(tmp, n, 2*n); copy_matrix(tmp, n, 2*n, 0, n, n, 2*n, inv, n, n, 0, n, 0, n); delete_matrix(tmp); return inv; } /** Compute the average square error between the identity matrix and the * product of matrix a with its inverse matrix. */ static double identity_error(const elem_t* const a, const int n) { int i, j; elem_t e = 0; for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { const elem_t d = a[i * n + j] - (i == j); e += d * d; } } return sqrt(e / (n * n)); } /** Compute epsilon for the numeric type elem_t. Epsilon is defined as the * smallest number for which the sum of one and that number is different of * one. It is assumed that the underlying representation of elem_t uses * base two. */ static elem_t epsilon() { elem_t eps; for (eps = 1; 1 + eps != 1; eps /= 2) ; return 2 * eps; } static void usage(const char* const exe) { printf("Usage: %s [-h] [-q] [-r] [-t] \n" "-h: display this information.\n" "-q: quiet mode -- do not print computed error.\n" "-r: trigger a race condition.\n" "-t: use threads.\n" ": matrix size.\n", exe); } int main(int argc, char** argv) { int matrix_size; int nthread = 1; int silent = 0; int optchar; elem_t *a, *inv, *prod; elem_t eps; double error; double ratio; while ((optchar = getopt(argc, argv, "hqrt:")) != EOF) { switch (optchar) { case 'h': usage(argv[0]); return 1; case 'q': silent = 1; break; case 'r': s_trigger_race = 1; break; case 't': nthread = atoi(optarg); break; default: return 1; } } if (optind + 1 != argc) { fprintf(stderr, "Error: wrong number of arguments.\n"); return 1; } matrix_size = atoi(argv[optind]); /* Error checking. */ assert(matrix_size >= 1); assert(nthread >= 1); omp_set_num_threads(nthread); omp_set_dynamic(0); eps = epsilon(); a = new_matrix(matrix_size, matrix_size); init_matrix(a, matrix_size, matrix_size); inv = invert_matrix(a, matrix_size); prod = multiply_matrices(a, matrix_size, matrix_size, inv, matrix_size, matrix_size); error = identity_error(prod, matrix_size); ratio = error / (eps * matrix_size); if (! silent) { printf("error = %g; epsilon = %g; error / (epsilon * n) = %g\n", error, eps, ratio); } if (isfinite(ratio) && ratio < 100) printf("Error within bounds.\n"); else printf("Error out of bounds.\n"); delete_matrix(prod); delete_matrix(inv); delete_matrix(a); return 0; }