/** Compute the matrix inverse via Gauss-Jordan elimination.
* This program uses only barriers to 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 <assert.h>
#include <math.h>
#include <limits.h> // PTHREAD_STACK_MIN
#include <pthread.h>
#include <stdio.h>
#include <stdlib.h>
#include <unistd.h> // getopt()
/*********************/
/* Type definitions. */
/*********************/
typedef double elem_t;
struct gj_threadinfo
{
pthread_barrier_t* b;
pthread_t tid;
elem_t* a;
int rows;
int cols;
int r0;
int r1;
};
/********************/
/* Local variables. */
/********************/
static int s_nthread = 1;
/*************************/
/* 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_threadfunc(struct gj_threadinfo* p)
{
int i, j, k;
elem_t* const a = p->a;
const int rows = p->rows;
const int cols = p->cols;
for (i = 0; i < p->rows; i++)
{
if (pthread_barrier_wait(p->b) == PTHREAD_BARRIER_SERIAL_THREAD)
{
// 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];
}
}
}
pthread_barrier_wait(p->b);
// Reduce all rows j != i.
for (j = p->r0; j < p->r1; 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;
}
}
}
}
}
/** Multithreaded Gauss-Jordan algorithm. */
static void gj(elem_t* const a, const int rows, const int cols)
{
int i;
struct gj_threadinfo* t;
pthread_barrier_t b;
pthread_attr_t attr;
int err;
assert(rows <= cols);
t = malloc(sizeof(struct gj_threadinfo) * s_nthread);
pthread_barrier_init(&b, 0, s_nthread);
pthread_attr_init(&attr);
err = pthread_attr_setstacksize(&attr, PTHREAD_STACK_MIN + 4096);
assert(err == 0);
for (i = 0; i < s_nthread; i++)
{
t[i].b = &b;
t[i].a = a;
t[i].rows = rows;
t[i].cols = cols;
t[i].r0 = i * rows / s_nthread;
t[i].r1 = (i+1) * rows / s_nthread;
pthread_create(&t[i].tid, &attr, (void*(*)(void*))gj_threadfunc, &t[i]);
}
pthread_attr_destroy(&attr);
for (i = 0; i < s_nthread; i++)
{
pthread_join(t[i].tid, 0);
}
pthread_barrier_destroy(&b);
free(t);
}
/** 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;
}
int main(int argc, char** argv)
{
int matrix_size;
int silent = 0;
int optchar;
elem_t *a, *inv, *prod;
elem_t eps;
double error;
double ratio;
while ((optchar = getopt(argc, argv, "qt:")) != EOF)
{
switch (optchar)
{
case 'q': silent = 1; break;
case 't': s_nthread = atoi(optarg); break;
default:
fprintf(stderr, "Error: unknown option '%c'.\n", optchar);
return 1;
}
}
if (optind + 1 != argc)
{
fprintf(stderr, "Error: wrong number of arguments.\n");
}
matrix_size = atoi(argv[optind]);
/* Error checking. */
assert(matrix_size >= 1);
assert(s_nthread >= 1);
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;
}