/* @param i direction of incoming ray, unit vector
* @param r direction of refraction ray, unit vector
* @param normal unit vector
* @param n1 refraction index
* @param n2 refraction index
*
* reference: http://graphics.stanford.edu/courses/cs148-10-summer/docs/2006--degreve--reflection_refraction.pdf
*/
static double fresnel(const point3 r, const point3 l,
const point3 normal, double n1, double n2)
{
/* TIR */
if (length(l) < 0.99)
return 1.0;
double cos_theta_i = -dot_product(r, normal);
double cos_theta_t = -dot_product(l, normal);
double r_vertical_root = (n1 * cos_theta_i - n2 * cos_theta_t) /
(n1 * cos_theta_i + n2 * cos_theta_t);
double r_parallel_root = (n2 * cos_theta_i - n1 * cos_theta_t) /
(n2 * cos_theta_i + n1 * cos_theta_t);
return (r_vertical_root * r_vertical_root +
r_parallel_root * r_parallel_root) / 2.0;
}
/*
* Simple implementation: Prediction r[u,i] = Sigma U[u,k]* Vt[k,i] over f
*/
void matrix_factorization(data_struct *training_set, REAL **U, REAL **Vt, int N, int M, int rank, int training_size){
REAL lambda = 0.00002, regularizer=0.001;
int epochs,i,f,user,item;
REAL error,sq_err,prev_sqerr = 100.0;
REAL pred, tmp, *cached_prods;
cached_prods = safe_malloc(N*M*sizeof(*cached_prods));
for(f=0; f<rank; f++){
for(epochs=0; epochs < 800; epochs++){
for(i=0; i<training_size; i++){
user = training_set[i].u;
item = training_set[i].i;
if (f == 0)
pred = dot_product(U,Vt,user,item,f,rank);
else
pred = cached_prods[user*M+item] + dot_product(U,Vt,user,item,f,rank);
error = training_set[i].rate - pred;
tmp = U[user][f];
U[user][f] += lambda * (error * Vt[f][item] - regularizer * tmp);
Vt[f][item] += lambda * (error * tmp - regularizer * Vt[f][item]);
}
sq_err = 0.0;
for(i=0; i<training_size; i++){
user = training_set[i].u;
item = training_set[i].i;
error = training_set[i].rate - dot_product(U,Vt,user,item,0,rank);
sq_err += pow(error,2);
}
if(fabs(sq_err - prev_sqerr) < 1e-5){
printf("Feat: %d Epochs: %d\n",f,epochs);
break;
}
prev_sqerr = sq_err;
}
for(i=0; i<training_size; i++){
user = training_set[i].u;
item = training_set[i].i;
if (f == 0)
cached_prods[user*M+item] = U[user][f]*Vt[f][item];
else
cached_prods[user*M+item] += U[user][f]*Vt[f][item];
}
}
free(cached_prods);
}
/**
* put matrix vectors to list
*/
template <class ZT, class F> void GaussSieve<ZT, F>::add_mat_list(ZZ_mat<ZT> &B)
{
Z_NR<ZT> t, current_norm;
dot_product(best_sqr_norm, B[0], B[0]);
ListPoint<ZT> *p;
for (int i = 0; i < nr; ++i)
{
p = new_listpoint<ZT>(nc);
matrix_row_to_list_point(B[i], p);
// cout << "# [info] init: additing point ";
// cout << p->v << endl;
if (alg == 3)
current_norm = update_p_3reduce(p);
else if (alg == 2)
current_norm = update_p_2reduce(p);
else if (alg == 4)
current_norm = update_p_4reduce(p);
else
{
cout << " Error, only support 2-, 3- and 4-sieve" << endl;
exit(1);
}
if ((current_norm < best_sqr_norm) && (current_norm > 0))
// if ((current_norm < best_sqr_norm) )
best_sqr_norm = current_norm;
}
}
Int32 fxnTest1(UInt32 size, UInt32 *data)
{
UInt32 start_indx, end_indx ;
#if CHATTER
System_printf("fxnInit : Executing fxnTest1\n");
#endif
FxnArgs *args = (FxnArgs *)((UInt32)data + sizeof(map_info_type));
int* buffer1 = (int*)args->a ;
int* buffer2 = (int*)args->b ;
start_indx = args->start_indx;
end_indx = args->end_indx;
volatile int* result = (int*)BARRIER_CNTR_BASE + REDUCTION_OFFSET ;
int i ;
for(i=0 ; i<NUM_ITER ; i++) {
callBarrier(0, /*lock_id=*/4) ;
result[0] = 0 ;
callBarrier(1, /*lock_id=*/4) ;
dot_product(buffer1, buffer2, result, start_indx, end_indx) ;
callBarrier(2, /*lock_id=*/4) ;
}
return 1 ;
}
float CriminisiInpainting::ComputeDataTerm(const itk::Index<2>& queryPixel)
{
try
{
FloatVector2Type isophote = this->IsophoteImage->GetPixel(queryPixel);
FloatVector2Type boundaryNormal = this->BoundaryNormals->GetPixel(queryPixel);
if(this->DebugMessages)
{
//std::cout << "Isophote: " << isophote << std::endl;
//std::cout << "Boundary normal: " << boundaryNormal << std::endl;
}
// D(p) = |dot(isophote at p, normalized normal of the front at p)|/alpha
vnl_double_2 vnlIsophote(isophote[0], isophote[1]);
vnl_double_2 vnlNormal(boundaryNormal[0], boundaryNormal[1]);
float dot = std::abs(dot_product(vnlIsophote,vnlNormal));
float dataTerm = dot/this->Alpha;
return dataTerm;
}
catch( itk::ExceptionObject & err )
{
std::cerr << "ExceptionObject caught in ComputeDataTerm!" << std::endl;
std::cerr << err << std::endl;
exit(-1);
}
}
Quaternion
Quaternion::slerp(const Quaternion& o, float t) const
{
/** Matze: I don't understand this code :-/ It's from
* http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It/
* Though the article recommends not to use slerp I see no code for the other
* methods so I'll use slerp anyway
*/
float dot = dot_product(o);
const float DOT_THRESHOLD = 0.995f;
if(dot > DOT_THRESHOLD) {
// quaternions are too close, lineary interpolate them
Quaternion result = *this + (o - *this)*t;
result.normalize();
return result;
}
dot = clamp(dot, -1 ,1); // robustness
float theta_O = acosf(dot);
float theta = theta_O * t;
Quaternion v2 = o - (*this * dot);
v2.normalize();
return (*this * cosf(theta)) + (v2 * sinf(theta));
}
double hexbright::angle_change() {
int* vec1 = vector(0);
int* vec2 = vector(1);
return angle_difference(dot_product(vec1, vec2),
magnitude(vec1),
magnitude(vec2));
}
// Do an iteration of stochastic gradient descent and return the post-update objective function
double stochastic_gradient_descent(double *w, double **features, int *grades, int *num_updates, int num_samples, int num_features) {
double *scores = (double *)malloc(num_samples * sizeof(double));
int sample_ind, other_sample_ind;
double t0;
double step_size, base_step_size;
t0 = pow(num_samples, 2);
base_step_size = C*ETA;
step_size = base_step_size / t0;
for (sample_ind=0; sample_ind<num_samples; ++sample_ind) {
scores[sample_ind] = dot_product(w, features[sample_ind], num_features);
for (other_sample_ind=0; other_sample_ind<sample_ind; ++other_sample_ind) {
if (grades[sample_ind] < grades[other_sample_ind] && scores[sample_ind]+1 > scores[other_sample_ind]) {
// step_size = base_step_size / (t0 + *num_updates);
vec_assign(w, w, 1-ETA/t0, num_features);
vec_add(w, features[sample_ind], -step_size, num_features);
vec_add(w, features[other_sample_ind], step_size, num_features);
++(*num_updates);
} else if (grades[sample_ind] > grades[other_sample_ind] && scores[sample_ind] < 1+scores[other_sample_ind]) {
// step_size = base_step_size / (t0 + *num_updates);
vec_assign(w, w, 1-ETA/t0, num_features);
vec_add(w, features[sample_ind], step_size, num_features);
vec_add(w, features[other_sample_ind], -step_size, num_features);
++(*num_updates);
}
else { vec_assign(w, w, 1-ETA/t0, num_features); }
}
}
free(scores);
return compute_objective(w, features, grades, num_samples, num_features);
}
// Player::pre_collision
void Player::pre_collision (const Manifold& m) {
const Vec2 HORIZ(1.0f, 0.0f);
if ((dot_product (HORIZ, abs(m.normal))) > 0.5f) {
player_body.static_friction = friction2;
player_body.dynamic_friction = friction2;
}
}
/**
* Returns the dot product of the specified arrays of doubles.
* @param v1 First array.
* @param v2 Second array.
* @throw std::domain_error if the vectors are not the same size.
*/
inline double dot_product(const std::vector<double>& v1,
const std::vector<double>& v2) {
stan::math::check_matching_sizes("dot_product",
"v1", v1,
"v2", v2);
return dot_product(&v1[0], &v2[0], v1.size());
}
static bool find_third_point(int num_points, const double (*points)[3], int a, int b, int* p_c)
{
const double* x1 = points[a];
const double* x2 = points[b];
double x2x1[3] = {x2[0] - x1[0], x2[1] - x1[1], x2[2] - x1[2]};
double ns_x2x1 = norm_squared(x2x1);
int bi = -1;
double max_dist = 0.0;
for (int i = 0;i<num_points;i++)
{
if (i == a || i == b)
continue;
const double* x0 = points[i];
double x1x0[3] = {x1[0] - x0[0], x1[1] - x0[1], x1[2] - x0[2]};
double dot = dot_product(x1x0, x2x1);
double dist = (norm_squared(x1x0) * ns_x2x1 - dot*dot) / ns_x2x1;
if (dist > max_dist)
{
max_dist = dist;
bi = i;
}
}
*p_c = bi;
return max_dist > TOLERANCE;
}
vector<double> Network_t::feed_forward(vector<double> a)
{
int i;
vector<double> tmp;
for (i = 0; i < this->biases.size(); ++i)
{
tmp.clear();
int j;
for (j = 0; j < this->biases[i].size(); ++j)
{
tmp.push_back( sigmoid( dot_product( a, weights[i][j] ) + biases[i][j] ) );
}
a = tmp;
}
// for (i = 0; i < a.size() ; ++i)
// {
// if(a.at(i) < 0.1)
// {
// a.at(i) = 0;
// }
// else if(a.at(i) > 0.9)
// {
// a.at(i) = 1;
// }
// }
return a;
}
void scene_nodes_translation_data::choose_normal_and_reduction(vector3 const& camera_origin)
{
if (m_x_down && !m_y_down && !m_z_down)
{
m_normal = vector3_unit_z();
m_reduction = vector3_unit_x();
}
else if (!m_x_down && m_y_down && !m_z_down)
{
m_normal = vector3_unit_z();
m_reduction = vector3_unit_y();
}
else if (!m_x_down && !m_y_down && m_z_down)
{
m_normal = std::fabsf(dot_product(vector3_unit_x(), camera_origin)) > std::fabsf(dot_product(vector3_unit_y(), camera_origin)) ?
vector3_unit_x() :
vector3_unit_y();
m_reduction = vector3_unit_z();
}
else if (m_x_down && !m_y_down && m_z_down)
{
m_normal = vector3_unit_y();
m_reduction = vector3_unit_y();
}
else if (!m_x_down && m_y_down && m_z_down)
{
m_normal = vector3_unit_x();
m_reduction = vector3_unit_x();
}
else
{
m_normal = vector3_unit_z();
m_reduction = vector3_unit_z();
}
}
int along (struct edge *edg, double axis[3])
{
int orn, k, sgn;
double dt;
double vect[3];
struct vertex *vtx1, *vtx2;
struct arc *a;
struct cept *ex;
a = edg -> arcptr;
vtx1 = a -> vtx[0];
vtx2 = a -> vtx[1];
if (vtx1 == NULL || vtx2 == NULL) {
ex = new_cept (PARAMETER_ERROR, NULL_VALUE, FATAL_SEVERITY);
add_object (ex, VERTEX, "vtx1 or vtx2");
add_function (ex, "along");
add_source (ex, "msrender.c");
return(0);
}
orn = edg -> orn;
sgn = 1 - 2 * orn;
for (k = 0; k < 3; k++)
vect[k] = vtx2 -> center[k] - vtx1 -> center[k];
dt = dot_product (vect, axis) * sgn;
return (dt > 0.0);
}
void player_class::move()
{
//position+=velocity;
//std::cout<<velocity.x<<","<<velocity.y<<"\n";
position=dot_product(position,velocity);
//std::cout<<velocity.x<<","<<velocity.y<<"\n";
}
t_double3 color_diffused(t_scene *scene, t_surface *surface, t_vector ray)
{
t_double3 color_hit;
t_light *light;
int light_nb;
double dot_light;
t_surface *light_intersect;
t_double3 reflected;
color_hit = (t_double3){0, 0, 0};
light = scene->light;
light_nb = 0;
while (light)
{
light_intersect = is_in_light(surface, scene, light, &dot_light);
if (light_intersect->object == NULL || light_intersect->distance > 0)
{
color_hit = v_plus_v(color_hit, color_mix(scale_v(light->color,
dot_light), surface->object->gloss,
// scale_v(surface->object->color, dot_light)));
scale_v(surface->color, dot_light)));
reflected = reflect(scale_v(normalize(v_minus_v(light->pos, surface->point)), -1), surface->normal);
color_hit = v_plus_v(color_hit, scale_v(light->color, pow(max_double(0, -dot_product(reflected, ray.dir) * surface->object->gloss), 2)));
}
free(light_intersect);
light_nb++;
light = light->next;
}
if (light_nb > 1)
color_hit = scale_v(color_hit, (1.0 / (double)light_nb));
return (color_hit);
}
vnl_vector_fixed<float,3> TrackingHandlerPeaks::ProposeDirection(const itk::Point<float, 3>& pos, std::deque<vnl_vector_fixed<float, 3> >& olddirs, itk::Index<3>& oldIndex)
{
// CHECK: wann wird wo normalisiert
vnl_vector_fixed<float,3> output_direction; output_direction.fill(0);
itk::Index<3> index;
m_DummyImage->TransformPhysicalPointToIndex(pos, index);
vnl_vector_fixed<float,3> oldDir; oldDir.fill(0.0);
if (!olddirs.empty())
oldDir = olddirs.back();
float old_mag = oldDir.magnitude();
if (!m_Interpolate && oldIndex==index)
return oldDir;
output_direction = GetDirection(pos, m_Interpolate, oldDir);
float mag = output_direction.magnitude();
if (mag>=m_PeakThreshold)
{
output_direction.normalize();
float a = 1;
if (old_mag>0.5)
a = dot_product(output_direction, oldDir);
if (a>=m_AngularThreshold)
output_direction *= mag;
else
output_direction.fill(0);
}
else
output_direction.fill(0);
return output_direction;
}
void dot_product_v(vector_t * vec1, vector_t * vec2, double * results, int count)
{
for (int i = 0; i < count; ++i)
{
results[i] = dot_product(&(vec1[i]), &(vec2[i]));
}
}
/* Compute how many planes of the hull of the given set of points
the given witness point lies outside of.
*/
static int num_outside( int npts, double errin, double * errout,
REAL (*pts)[DIM], struct transf * t, REAL witness[DIM])
{
double val, worst;
int f, nf, nv, nbad;
double trans_pts[MAX_POINTS][3];
double faces[2*MAX_POINTS][4];
if ( t!=0 )
transform_hull( npts, pts, trans_pts, t);
/* construct the hull */
nv = sac_qhull( npts, ( t==0 ) ? pts : trans_pts,
(double (*)[3]) 0, (int *) 0, (int *) 0,
&nf, faces);
nbad = 0;
for ( f=0 ; f<nf ; f++ )
{
val = dot_product( witness, faces[f]) + faces[f][3];
if ( val > errin + ZETA ) {
if ( ( nbad==0 ) || ( val > worst ) )
worst = val;
nbad++;
}
}
*errout = ( nbad>0 ) ? worst : 0.0;
return nbad;
}
t_ray find_refract_vect(t_ray *start_ray, t_hit drawn_pixel, double c_r, int test)
{
double ref_refract;
double new_ref_index;
t_ray res;
t_vec new_norm;
t_vec inv_dir;
res.pos = vec_add(start_ray->pos, scalar_product(start_ray->dir, drawn_pixel.t));
inv_dir = scalar_product(start_ray->dir, -1);
drawn_pixel.point_norm = scalar_product(drawn_pixel.point_norm, test);
ref_refract = dot_product(drawn_pixel.point_norm, inv_dir);
ref_refract /= (get_length(drawn_pixel.point_norm) * get_length(inv_dir));
new_ref_index = 1 - c_r * c_r * (1 - ref_refract * ref_refract);
if (new_ref_index > 0)
{
new_ref_index = sqrt(new_ref_index);
new_ref_index = c_r * ref_refract - new_ref_index;
new_norm = scalar_product(drawn_pixel.point_norm, new_ref_index);
res.dir = scalar_product(inv_dir, c_r);
res.dir = vec_sub(res.dir, new_norm);
}
else
res.dir = init_vector(0, 0, 0);
return (res);
}
// Do an iteration of gradient descent and return the post-update objective function.
// Possibly update the optimal step size in place.
double gradient_descent(double *w, double *step_size, double **features, int *grades, int num_samples, int num_features) {
double *scores = (double *)malloc(num_samples * sizeof(double));
double *grad = (double *)malloc(num_features * sizeof(double));
int sample_ind, other_sample_ind;
double slack_coeff = C / pow(num_samples, 2);
vec_assign(grad, w, 1, num_features);
for (sample_ind=0; sample_ind<num_samples; ++sample_ind) {
scores[sample_ind] = dot_product(w, features[sample_ind], num_features);
for (other_sample_ind=0; other_sample_ind<sample_ind; ++other_sample_ind) {
if (grades[sample_ind] < grades[other_sample_ind] && scores[sample_ind]+1 > scores[other_sample_ind]) {
vec_add(grad, features[sample_ind], slack_coeff, num_features);
vec_add(grad, features[other_sample_ind], -slack_coeff, num_features);
} else if (grades[sample_ind] > grades[other_sample_ind] && scores[sample_ind] < 1+scores[other_sample_ind]) {
vec_add(grad, features[sample_ind], -slack_coeff, num_features);
vec_add(grad, features[other_sample_ind], slack_coeff, num_features);
}
}
}
// vec_add(w, grad, -step_size, num_features);
free(scores);
free(grad);
// return compute_objective(w, features, grades, num_samples, num_features);
return take_gradient_step(w, grad, step_size, features, grades, num_samples, num_features);
}
/*
* The viscosity is computed using the Wilke mixture rule.
* \f[
* \mu = \sum_k \frac{\mu_k X_k}{\sum_j \Phi_{k,j} X_j}.
* \f]
* Here \f$ \mu_k \f$ is the viscosity of pure species \e k,
* and
* \f[
* \Phi_{k,j} = \frac{\left[1
* + \sqrt{\left(\frac{\mu_k}{\mu_j}\sqrt{\frac{M_j}{M_k}}\right)}\right]^2}
* {\sqrt{8}\sqrt{1 + M_k/M_j}}
* \f]
* @see updateViscosity_T();
*/
doublereal LiquidTransport::viscosity() {
update_temp();
update_conc();
if (m_visc_mix_ok) return m_viscmix;
// update viscSpecies_[] and m_phi[] if necessary
if (!m_visc_temp_ok) {
updateViscosity_temp();
}
if (!m_visc_conc_ok) {
updateViscosities_conc();
}
if (viscosityModel_ == LVISC_CONSTANT) {
return m_viscmix;
} else if (viscosityModel_ == LVISC_MIXTUREAVG) {
m_viscmix = dot_product(viscSpecies_, m_molefracs);
} else if (viscosityModel_ == LVISC_WILKES) {
multiply(m_phi, DATA_PTR(m_molefracs), DATA_PTR(m_spwork));
m_viscmix = 0.0;
for (int k = 0; k < m_nsp; k++) {
m_viscmix += m_molefracs[k] * viscSpecies_[k]/m_spwork[k];
}
}
return m_viscmix;
}
// Player::post_collision
void Player::post_collision (const Manifold& m) {
const Vec2 JUMP_VEC(0.0f, -1.0f);
if (dot_product (JUMP_VEC, ((m.a == &player_body) ? -m.normal : m.normal)) > 0.5f)
can_jump = true;
player_body.static_friction = friction1;
player_body.dynamic_friction = friction1;
}
plane( const point_type& a, const vector_type& u, const vector_type& v )
: m_a( a )
, m_u( u )
, m_v( v )
, m_n( normalize( cross_product( m_u, m_v ) ) )
, m_d( dot_product(m_n, as_vector(m_a)) )
{}
struct circle *new_contact_circle (struct surface *this_srf, struct torus *torus_ptr, int which_side)
{
int k;
double signed_distance, circle_radius;
double circle_center[3], circle_atom_vector[3], circle_axis[3];
struct circle *circle_ptr;
struct sphere *atm_ptr;
atm_ptr = torus_ptr -> atm[which_side];
/* computations for circle */
circle_radius =
torus_ptr -> radius * atm_ptr -> radius /
(atm_ptr -> radius + this_srf -> probe_radius);
for (k = 0; k < 3; k++) {
circle_center[k] =
(atm_ptr -> radius * torus_ptr -> center[k] +
this_srf -> probe_radius * atm_ptr -> center[k])
/ (atm_ptr -> radius + this_srf -> probe_radius);
circle_atom_vector[k] =
circle_center[k] - atm_ptr -> center[k];
circle_axis[k] = (2 * which_side - 1) * torus_ptr -> axis[k];
}
/* allocate memory, setup fields */
circle_ptr = new_circle (circle_center, circle_radius, circle_axis);
if (error()) return(NULL);
link_circle (this_srf, circle_ptr);
circle_ptr -> subtype = CONTACT_SUBTYPE;
signed_distance = (-dot_product (circle_axis, circle_atom_vector));
circle_ptr -> theta = atan2 (signed_distance, circle_radius);
circle_ptr -> atm = atm_ptr;
torus_ptr -> cir[which_side] = circle_ptr;
return (circle_ptr);
}
inline double operator()(const TPoint& x, const TPoint& y) const {
VectorType xv = x.GetVnlVector();
VectorType yv = y.GetVnlVector();
VectorType r = yv - xv;
return exp(-dot_product(r, r) / m_sigma2);
}
vgl_vector_3d<double> ProjectAonB(const vgl_vector_3d<double> &A, const vgl_vector_3d<double> &B)
{
vgl_vector_3d<double> a = A;
vgl_vector_3d<double> b = normalize(B);
return (dot_product(a,b) * b );
}
void PointLight::ApplyLight( ID3DXEffect* _effect )
{
D3DXHANDLE handle;
handle = _effect->GetParameterByName(m_szLightName.c_str(), "Position");
float3 pos = GetFrame().GetWorldMat().axis_w;
D3DXVECTOR4 position(pos.x, pos.y, pos.z, 1);
_effect->SetVector(handle, &position);
handle = _effect->GetParameterByName(m_szLightName.c_str(), "Attenuation");
_effect->SetVector(handle, &D3DXVECTOR4(m_v3fAttenuation, 1));
handle = _effect->GetParameterByName(m_szLightName.c_str(), "Range" );
_effect->SetFloat(handle, m_fRadius );
D3DXMATRIX cam = RenderEngine::GetCamera()->GetWorldMatrix();
float3 diff = float3( GetFrame().GetWorldMat().wx, GetFrame().GetWorldMat().wy, GetFrame().GetWorldMat().wz ) - float3( cam._41, cam._42, cam._43 );
if( dot_product( diff, diff ) + 0.001f < ( m_fRadius * m_fRadius ) )
{
_effect->SetTechnique( "RenderPointLightInside" );
}
else
{
_effect->SetTechnique( "RenderPointLightOutside" );
}
Light::ApplyLight( _effect );
}
plane( const point_type& a, const point_type& b, const point_type& c )
: m_a( a )
, m_u( b-a )
, m_v( c-a )
, m_n( normalize( cross_product( m_u, m_v ) ) )
, m_d( dot_product( m_n, as_vector( m_a ) ) )
{}
int main(int argc, char **argv) {
struct timeval start;
struct timeval stop;
unsigned long elapsed;
double result;
double *B = malloc(size*sizeof(double));
tareador_ON ();
int i;
tareador_start_task("init_A");
for (i=0; i< size; i++) A[i]=i;
tareador_end_task();
tareador_start_task("init_B");
for (i=0; i< size; i++) B[i]=2*i;
tareador_end_task();
gettimeofday(&start,NULL);
dot_product (size, A, B, &result);
tareador_OFF ();
gettimeofday(&stop,NULL);
elapsed = 1000000 * (stop.tv_sec - start.tv_sec);
elapsed += stop.tv_usec - start.tv_usec;
printf("Result of Dot product i= %le\n", result);
printf("Execution time (us): %lu \n", elapsed);
return 0;
}