int main()
{
int n;
FILE *fin=fopen("sprime.in","r");
fout=fopen("sprime.out","w");
fscanf(fin, "%d", &n);
sprime(2, n-1);
sprime(3, n-1);
sprime(5, n-1);
sprime(7, n-1);
return 0;
}
int main() {
freopen(INPUT, "r", stdin);
freopen(OUTPUT, "w", stdout);
int i, n;
scanf("%d", &n);
sprime(0, n, n);
exit(0);
}
void sprime(int n, int ndigit)
{
if(ndigit == 0) {
fprintf(fout, "%d\n", n);
return;
}
n *= 10;
if(isprime(n+1))
sprime(n+1, ndigit-1);
if(isprime(n+3))
sprime(n+3, ndigit-1);
if(isprime(n+7))
sprime(n+7, ndigit-1);
if(isprime(n+9))
sprime(n+9, ndigit-1);
}
int sprime(int n, int ndigit, int to) {
if (ndigit == 0) {
printf("%d\n", n);
return;
}
n *= 10;
int j = 1;
if (ndigit == to) {
j = 2;
}
for (; j <= 9; j++) {
if (isprime(n + j))
sprime(n + j, ndigit - 1, to);
}
}
/*!
* \brief Calculate natural spline interpolation of current data
*/
void Data2D::interpolate(bool constrained)
{
/* For a set of N points {x(i),y(i)} for i = 0 - (N-1), the i'th interval can be represented by a cubic spline
*
* S(i)(x) = a(i) + b(i)(x - x(i)) + c(i)(x - x(i))**2 + d(i)h(x - x(i))**3.
*
* As such, there are 4N unknown coefficients - {a(0),b(0),c(0),d(0),a(1),b(1),c(1),d(1),...,a(N-1),b(N-1),c(N-1),d(N-1)}
*
* Conditions:
* At the extreme values of x in each interval, the original y values are retained
* S(i)(x = x(i)) = y(i) and S(i)(x = x(i+1)) = y(i+1) == 2N equations
* First derivatives are continuous across intervals (i.e. match at extreme x values of intervals)
* S'(i)(x = x(i+1)) = S'(i+1)(x = x(i+1)) == (N-1) equations
* Second derivatives are continuous across intervals (i.e. match at extreme x values of intervals)
* S"(i)(x = x(i+1)) = S"(i+1)(x = x(i+1)) == (N-1) equations
*
* So, there are 4N-2 equations specified here - the remaining two come from specifying endpoint conditions of the spline fit.
* No derivative matching is (can) be performed at the extreme points i=0 and i=N-1, so specify exact derivatives at these points:
* Natural splines:
* S'(0) = 0 S'(N-1) = 0
*
* We need expressions for the derivatives of S:
* Let h(i) = x - x(i)
* S(i)(x) = a(i) + b(i)h(i) + c(i)h(i)**2 + d(i)h(i)**3
* S'(i)(x) = b(i) + 2 c(i)h(i) + 3 d(i)h(i)**2
* S"(i)(x) = 2 c(i) + 6 d(i)h(i)
*
* From the conditions outlined above:
* S(i)(x = x(i)) = y(i) ==> a(i) = y(i) since x = x(i) and so h(i) = x(i) - x(i) = 0
* S(i)(x = x(i+1)) = y(i+1) ==> a(i) + b(i)h(i) + c(i)h(i)**2 + d(i)h(i)**3 = y(i+1)
* S'(i)(x = x(i+1)) = S'(i+1)(x = x(i+1)) ==> b(i) + 2 c(i)h(i) + 3 d(i)h(i)**2 - b(i+1) = 0
* since at x = x(i+1), h(i+1) = 0 and so S'(i+1)(x = x(i+1)) = b(i+1)
* S"(i)(x = x(i+1)) = S"(i+1)(x = x(i+1)) ==> 2 c(i) + 6 d(i)h(i) - 2 c(i+1) = 0
* since at x = x(i+1), h(i+1) = 0 and so S"(i+1)(x = x(i+1)) = 2 c(i+1)
* To summarise:
* a(i) = y(i) for i = 0,N-1 (N eqs)
* a(i) + b(i)h(i) + c(i)h(i)**2 + d(i)h(i)**3 = y(i+1) for i = 0,N-1 (N eqs)
* b(i) + 2 c(i)h(i) + 3 d(i)h(i)**2 - b(i+1) = 0 for i = 0,N-2 (N-1 eqs)
* 2 c(i) + 6 d(i)h(i) - 2 c(i+1) = 0 for i = 0,N-2 (N-1 eqs)
*
* To simplify, consider value of second derivative in i'th interval at leftmost value:
* S"(i)(x = x(i)) = 2 c(i) since at x = x(i+1), h(i+1) = 0 and so S"(i+1)(x = x(i+1)) = 2 c(i+1)
* We will call these coefficients m(i):
* 2 c(i) + 6 d(i)h(i) - 2 c(i+1) = 0 ==> m(i) + 6 d(i)h(i) - m(i+1) = 0
* ==> d(i) = m(i+1) - m(i)
* -------------
* 6 h(i)
* m(i) m(i+1) - m(i)
* y(i) + b(i)h(i) + c(i)h(i)**2 + d(i)h(i)**3 = y(i+1) ==> y(i) + b(i)h(i) + --- h(i)**2 + ------------- h(i)**3 = y(i+1)
* 2 6 h(i)
*
* y(i+1) - y(i) m(i) m(i+1) - m(i)
* ==> b(i) = ------------- - --- h(i) - ------------- h(i)
* h(i) 2 6
*
*
* b(i) + 2 c(i)h(i) + 3 d(i)h(i)**2 = b(i+1) ==>
* ( y(i+1) - y(i) m(i) m(i+1) - m(i) ) m(i+1) - m(i) ( y(i+2) - y(i+1) m(i+1) m(i+2 - m(i+1) )
* ( ------------- - --- h(i) - ------------- h(i) ) + m(i)h(i) + 3 ------------- h(i)**2 = ( --------------- - ------ h(i+1) - ------------- h(i+1) )
* ( h(i) 2 6 ) 6 h(i) ( h(i+1) 2 6 )
*
* ( y(i+2) - y(i+1) y(i+1) - y(i) )
* ==> h(i)m(i) + 2 (h(i) + h(i+1))m(i+1) + h(i+1)m(i+2) = 6 ( --------------- - ------------- )
* ( h(i+1) h(i) )
*
* Written in matrix form, the above system of equations is:
*
* [ 1 0 0 0 ... 0 ] [ m(0) ] [ 0 ]
* [ h(0) 2(h(0) + h(1)) h(1) 0 ... 0 ] [ m(1) ] [ (y(2) - y(1)) / h(1) - (y(1) - y(0)) / h(0) ]
* [ 0 h(1) 2(h(1) + h(2)) h(2) ... 0 ] [ m(2) ] = 6 [ (y(3) - y(2)) / h(2) - (y(2) - y(1)) / h(1) ]
* [ ... ... ] [ ... ] [
* [ 0 0 0 0 ... 1 ] [ m(N-1) ] [ 0 ]
*
* Once solved, we have the set of m() and can thus determine a, b, c, and d as follows:
*
* y(i+1)-y(i) h(i) h(i) m(i) m(i+1)-m(i)
* a(i) = y(i) b(i) = ----------- - ---- m(i) - ---- (m(i+1)-m(i)) c(i) = ---- d(i) = -----------
* h(i) 2 6 2 6 h(i)
*
* The constrained spline fit removes the requirement that the second derivatives are equal at the boundaries of every interval, and instead specifies
* that an exact first derivative is obtained there instead. As such:
*
* S"(i)(x = x(i+1)) = S"(i+1)(x = x(i+1)) ==> S'(i)(x = x(i+1)) = S'(i+1)(x = x(i+1)) = f'(x = x(i+1))
*
* A suitable estimate of the slope comes from consideration of the slopes of the two adjacent intervals to x(i):
*
* ( x(i+1)-x(i) x(i)-x(i-1) )
* f'(i) = 0.5 * ( ----------- + ----------- ) for i=1, N-2
* ( y(i+1)-y(i) y(i)-y(i-1) )
*
* At the endpoints:
*
* 3( y(1)-y(0) ) f'(1) 3( y(i-1) - y(n-2) ) f'(N-1)
* f'(0) = -------------- - ----- f'(N-1) = -------------------- - -------
* 2( x(1)-x(0) ) 2 2( x(i-1) - x(n-2) ) 2
*
* Since the slope at each point is known, there is no longer a need to solve a system of equations, and each coefficient may be calculated as follows:
*
* 2( f'(i)+2f'(i-1) ) 6(y(i) - y(i-1)
* S"(i) = ------------------- - ---------------- and S"(i-1) = -S"(i)
* x(i) - x(i-1) (x(i)-x(i-1))**2
*
* S"(i) - S"(i-1) x(i)S"(i-1)-x(i-1)S"(i)
* d(i) = --------------- c(i) = -----------------------
* 6(x(i) - x(i-1)) 2(x(i) - x(i-1))
*
* (y(i) - y(i-1)) - c(i)(x(i)**2 - x(i-1)**2) - d(i)(x(i)**3 - x(i-1)**3)
* b(i) =------------------------------------------------------------------------
* x(i) - x(i-1)
*/
int i, nPoints = x_.nItems();
// Do a quick test on splineH_ and constrainedSpline_ to see if we need to recalculate the interpolation
// if ((splineH_.nItems() != 0) && (constrainedSpline_ == constrained)) return;
// Calculate interval array 'h'
splineH_.createEmpty(nPoints);
for (i=0; i<nPoints-1; ++i) splineH_[i] = x_[i+1] - x_[i];
// Initialise parameter arrays and working array
splineA_.createEmpty(nPoints);
splineB_.createEmpty(nPoints);
splineC_.createEmpty(nPoints);
splineD_.createEmpty(nPoints);
constrainedSpline_ = constrained;
if (!constrainedSpline_)
{
/*
* Unconstrained Spline Fit
*/
// Determine 'm' values (== C) now by solving the tridiagonal matrix above. Use Thomas algorithm...
// -- First stage
double p, q, r, s;
Array<double> rprime(nPoints), sprime(nPoints);
rprime[0] = 0.0; // Would be 0.0 / 1.0 in the case of Natural spline
sprime[0] = 0.0;
for (i=1; i<nPoints-1; ++i)
{
// For a given i, p(i) = h(i-1), q(i) = 2(h(i-1)+h(i)), r(i) = h(i), s(i) = 6 ((y(n+1) - y(n)) / h(n) - (y(n) - y(n-1)) / h(n-1)
p = splineH_[i-1];
q = 2.0*(splineH_[i-1]+splineH_[i]);
r = splineH_[i];
s = 6.0 * ((y_[i+1] - y_[i]) / splineH_[i] - (y_[i] - y_[i-1]) / splineH_[i-1]);
// -- Calculate r'(i) = r(i) / ( q(i) - r'(i-1)p(i) )
rprime[i] = r / (q - rprime[i-1]*p);
// -- Calculate s'(i) = (s(i) - s'(i-1)p(i) ) / ( q(i) - r'(i-1)p(i))
sprime[i] = (s - sprime[i-1]*p) / (q - rprime[i-1]*p);
}
rprime[nPoints-1] = 0.0;
sprime[nPoints-1] = (0.0 - sprime[nPoints-2]/splineH_[i-1]) / (2.0*splineH_[i-1]);
// -- Second stage - backsubstitution
splineC_[nPoints-1] = 0.0;
for (i=nPoints-2; i>=0; --i)
{
// For a given i, m(i) = s'(i) - r'(i)m(i+1)
splineC_[i] = sprime[i] - rprime[i]*splineC_[i+1];
}
// splineC_ array now contains m(i)...
for (i=0; i<nPoints-1; ++i)
{
splineB_[i] = (y_[i+1] - y_[i]) / splineH_[i] - 0.5 * splineH_[i] * splineC_[i] - (splineH_[i] * (splineC_[i+1]-splineC_[i]))/6.0;
splineD_[i] = (splineC_[i+1] - splineC_[i]) / (6.0 * splineH_[i]);
splineC_[i] *= 0.5;
splineA_[i] = y_[i];
}
}
else
{
/*
* Constrained Spline fit
*/
// Calculate first derivatives at each point
Array<double> fp(nPoints);
double gradA, gradB;
for (i=1; i<nPoints-1; ++i)
{
gradA = (x_[i+1] - x_[i])/(y_[i+1] - y_[i]);
gradB = (x_[i] - x_[i-1])/(y_[i] - y_[i-1]);
if (UChromaMath::sgn(gradA) != UChromaMath::sgn(gradB)) fp[i] = 0.0;
else fp[i] = 2.0 / (gradA + gradB);
}
// fp[0] = (3.0*(y_[1] - y_[0])) / (2.0*x_[1]-x_[0]) - 0.5*fp[1];
// fp[nPoints-1] = (3.0*(y_[nPoints-1] - y_[nPoints-2])) / (2.0*x_[nPoints-1]-x_[nPoints-2]) - 0.5*fp[nPoints-2];
fp[0] = 0.0;
fp[nPoints-1] = 0.0;
// Calculate coefficients
double fppi, fppim1, dx, dy;
for (i=1; i<nPoints; ++i)
{
dx = x_[i] - x_[i-1];
dy = y_[i] - y_[i-1];
fppim1 = -2.0*(fp[i]+2.0*fp[i-1]) / dx + 6.0*dy/(dx*dx);
fppi = 2.0*(2.0*fp[i]+fp[i-1]) / dx - 6.0*dy/(dx*dx);
splineD_[i-1] = (fppi - fppim1) / (6.0*dx);
splineC_[i-1] = (x_[i]*fppim1 - x_[i-1]*fppi) / (2.0*dx);
splineB_[i-1] = (dy - splineC_[i-1]*(x_[i]*x_[i] - x_[i-1]*x_[i-1]) - splineD_[i-1]*(x_[i]*x_[i]*x_[i] - x_[i-1]*x_[i-1]*x_[i-1])) / dx;
splineA_[i-1] = y_[i-1] - splineB_[i-1]*x_[i-1] - splineC_[i-1]*x_[i-1]*x_[i-1] - splineD_[i-1]*x_[i-1]*x_[i-1]*x_[i-1];
}
}
splineInterval_ = 0;
}
void QpGenData::datarandom( OoqpVector & x, OoqpVector & y,
OoqpVector & z, OoqpVector & s )
{
double drand( double * );
double ix = 3074.20374;
OoqpVectorHandle xdual(la->newVector( nx ));
this->randomlyChooseBoundedVariables( x, *xdual,
*blx, *ixlow, *bux, *ixupp,
&ix, .25, .25, .25 );
{
// ofstream x_vec( "x" );
// x->writeToStream( x_vec );
// ofstream eta_vec( "xdual" );
// eta_vec.precision(16);
// xdual->writeToStream( eta_vec );
// ofstream blx_vec( "blx" );
// blx->writeToStream( blx_vec );
// ofstream bux_vec( "bux" );
// bux->writeToStream( bux_vec );
}
OoqpVectorHandle sprime(la->newVector( mz ));
this->randomlyChooseBoundedVariables( *sprime, z,
*bl, *iclow, *bu, *icupp,
&ix, .25, .25, .5 );
{
// ofstream z_vec( "z" );
// z_vec << z << endl;
}
{
Q->randomizePSD( &ix );
// ofstream Q_mat( "Q" );
// Q_mat << Q << endl;
}
{
A->randomize( -10.0, 10.0, &ix );
// ofstream A_mat( "A" );
// A_mat << A << endl;
}
{
C->randomize( -10.0, 10.0, &ix );
// ofstream C_mat( "C" );
// C_mat << C << endl;
}
y.randomize( -10.0, 10.0, &ix );
{
// ofstream y_vec( "y" );
// y_vec << y << endl;
}
// g = - Q x + A\T y + C\T z + xdual
g->copyFrom( *xdual );
Q->mult( 1.0, *g, -1.0, x );
A->transMult( 1.0, *g, 1.0, y );
C->transMult( 1.0, *g, 1.0, z );
// bA = A x
A->mult( 0.0, *bA, 1.0, x );
{
// ofstream bA_vec( "bA" );
// bA_vec << bA << endl;
}
// Have a randomly generated sprime.
// C x - s = 0. Let q + sprime = s, i.e. q = s - sprime.
// Compute s and temporarily store in q
C->mult( 0.0, s, 1.0, x );
// Now compute the real q = s - sprime
OoqpVectorHandle q(la->newVector( mz ));
q->copyFrom( s );
q->axpy( -1.0, *sprime );
// Adjust bl and bu appropriately
bl->axpy( 1.0, *q );
bu->axpy( 1.0, *q );
bl->selectNonZeros( *iclow );
bu->selectNonZeros( *icupp );
{
// ofstream bl_vec( "bl" );
// bl_vec << bl << endl;
// ofstream bu_vec( "bu" );
// bu_vec << bu << endl;
}
}
