int cmp1(complex<double> a,complex<double> b)
{
if(a.real()==b.real())
return a.imag()<=b.imag();
else
return a.real()<b.real();
}
bool cmp(const complex <int> &a, const complex <int> &b)
{
if(a.imag()==b.imag())
return a.real()<b.real();
else
return a.imag()<b.imag();
}
complex<typename boost::units::subtract_typeof_helper<X,Y>::type>
operator-(const complex<X>& x,const complex<Y>& y)
{
typedef typename boost::units::subtract_typeof_helper<X,Y>::type type;
return complex<type>(x.real()-y.real(),x.imag()-y.imag());
}
void EMLTracker::integrate(complex<double> in)
{
localReplica.tick();
// First bring the signal level of the input to the same as the
// local replicas
in *= baseGain;
complex<double> carrier = localReplica.getCarrier();
complex<double> code = localReplica.getCode() ? plusOne : minusOne;
// mix in the carrier local replica
complex<double> m0 = in * conj(carrier);
// and sum them up.. (yea, the conj of the codes should be a NoOp)
code = conj(code);
early.process(m0, code);
prompt.process(m0, code);
late.process(m0, code);
// Update our sums for normalizing things...
complex<double> lr = conj(carrier) * code;
inSumSq += in.real()*in.real() + in.imag()*in.imag();
lrSumSq += lr.real()*lr.real() + lr.imag()*lr.imag();
}
/* https://github.com/AE9RB/fftbench/blob/master/cxlr.hpp
* Nice trick from AE9RB (David Turnbull) to get compiler to produce simpler
* fma (fused multiply-accumulate) instead of worrying about NaN handling
*/
inline complex<float>
operator*(const complex<float>& v1, const complex<float>& v2) {
return complex<float> {
v1.real() * v2.real() - v1.imag() * v2.imag(),
v1.real() * v2.imag() + v1.imag() * v2.real()
};
}
complex<typename add_typeof_helper<X,Y>::type>
operator+(const complex<X>& x,const complex<Y>& y)
{
typedef typename boost::units::add_typeof_helper<X,Y>::type type;
return complex<type>(x.real()+y.real(),x.imag()+y.imag());
}
bool operator()(const complex<double>& lhs, const complex<double>& rhs) {
if (norm(lhs) != norm(rhs)) {
return norm(lhs) < norm(rhs);
} else if (lhs.real() != rhs.real()) {
return lhs.real() < rhs.real();
} else {
return lhs.imag() < rhs.imag();
}
}
static type value(const complex<quantity<Unit,Y> >& x)
{
const complex<value_type> tmp =
root<static_rational<N,D> >(complex<Y>(x.real().value(),
x.imag().value()));
return type(quantity_type::from_value(tmp.real()),
quantity_type::from_value(tmp.imag()));
}
complex<double> complexAdd(complex<double> x1, complex<double> x2)
{
int x1_real = int(x1.real());
int x2_real = int(x2.real());
int x1_imag = int(x1.imag());
int x2_imag = int(x2.imag());
int r1 = int(bitset<16>(x1_real+x2_real).to_ulong());
int r2 = int(bitset<16>(x1_imag+x2_imag).to_ulong());
return complex<double>(double(r1), double(r2));
}
std::string fmtvalue(std::true_type, const complex<T>& x)
{
std::string restr = as_string(fmt<'g', number_width, number_precision>(x.real()));
if (restr.size() > number_width)
restr = as_string(fmt<'g', number_width, number_precision_short>(x.real()));
std::string imstr = as_string(fmt<'g', -1, number_precision>(std::abs(x.imag())));
if (imstr.size() > number_width)
imstr = as_string(fmt<'g', -1, number_precision_short>(std::abs(x.imag())));
return restr + (x.imag() < T(0) ? "-" : "+") + padleft(number_width, imstr + "j");
}
QTextStream& operator<<(QTextStream& ts, KCValue value)
{
ts << value.type();
switch (value.type()) {
case KCValue::Empty: break;
case KCValue::Boolean:
ts << ": ";
if (value.asBoolean()) ts << "TRUE";
else ts << "FALSE"; break;
case KCValue::Integer:
ts << ": " << value.asInteger(); break;
case KCValue::Float:
ts << ": " << (double) numToDouble(value.asFloat()); break;
case KCValue::Complex: {
const complex<KCNumber> complex(value.asComplex());
ts << ": " << (double) numToDouble(complex.real());
if (complex.imag() >= 0.0)
ts << '+';
ts << (double) numToDouble(complex.imag()) << 'i';
break;
}
case KCValue::String:
ts << ": " << value.asString(); break;
case KCValue::Array: {
ts << ": {" << value.asString();
const int cols = value.columns();
const int rows = value.rows();
for (int row = 0; row < rows; ++row) {
for (int col = 0; col < cols; ++col) {
ts << value.element(col, row);
if (col < cols - 1)
ts << ';';
}
if (row < rows - 1)
ts << '|';
}
ts << '}';
break;
}
case KCValue::Error:
ts << '(' << value.errorMessage() << ')'; break;
default: break;
}
return ts;
}
double recursiveColor(complex<double> complexNum, complex<double> constant, int iterations)
{
if(iterations == maxIterations)
return double(iterations);
if(sqrt(complexNum.real() * complexNum.real() + complexNum.imag() * complexNum.imag()) > 2)
return double(iterations);
else
{
complexNum = complex<double>(complexNum.real() * complexNum.real() - (complexNum.imag() * complexNum.imag()) + constant.real(), 2 * complexNum.real() * complexNum.imag() + constant.imag());
return recursiveColor(complexNum, constant, ++iterations);
}
}
/* Complex root forward deflation.
*/
static int complexdeflation( const register int n, double a[], const complex<double> z )
{
register int i;
double r, u;
r = -2.0 * z.real();
u = z.real() * z.real() + z.imag() * z.imag();
a[ 1 ] -= r * a[ 0 ];
for( i = 2; i < n - 1; i++ )
a[ i ] = a[ i ] - r * a[ i - 1 ] - u * a[ i - 2 ];
return n - 2;
}
complex<typename boost::units::divide_typeof_helper<X,Y>::type>
operator/(const complex<X>& x,const complex<Y>& y)
{
// naive implementation of complex division
typedef typename boost::units::divide_typeof_helper<X,Y>::type type;
return complex<type>((x.real()*y.real()+x.imag()*y.imag())/
(y.real()*y.real()+y.imag()*y.imag()),
(x.imag()*y.real()-x.real()*y.imag())/
(y.real()*y.real()+y.imag()*y.imag()));
// fully correct implementation has more complex return type
//
// typedef typename boost::units::multiply_typeof_helper<X,Y>::type xy_type;
// typedef typename boost::units::multiply_typeof_helper<Y,Y>::type yy_type;
//
// typedef typename boost::units::add_typeof_helper<xy_type, xy_type>::type
// xy_plus_xy_type;
// typedef typename boost::units::subtract_typeof_helper<
// xy_type,xy_type>::type xy_minus_xy_type;
//
// typedef typename boost::units::divide_typeof_helper<
// xy_plus_xy_type,yy_type>::type xy_plus_xy_over_yy_type;
// typedef typename boost::units::divide_typeof_helper<
// xy_minus_xy_type,yy_type>::type xy_minus_xy_over_yy_type;
//
// BOOST_STATIC_ASSERT((boost::is_same<xy_plus_xy_over_yy_type,
// xy_minus_xy_over_yy_type>::value == true));
//
// return complex<xy_plus_xy_over_yy_type>(
// (x.real()*y.real()+x.imag()*y.imag())/
// (y.real()*y.real()+y.imag()*y.imag()),
// (x.imag()*y.real()-x.real()*y.imag())/
// (y.real()*y.real()+y.imag()*y.imag()));
}
complex<double> complexMultiply(complex<double> x1, complex<double> x2)
{
int x1_real = int(x1.real());
int x2_real = int(x2.real());
int x1_imag = int(x1.imag());
int x2_imag = int(x2.imag());
int r1 = int(bitset<16>(x1_real*x2_real).to_ulong());
int r2 = int(bitset<16>(x1_imag*x2_imag).to_ulong());
int r3 = int(bitset<16>(x1_real*x2_imag).to_ulong());
int r4 = int(bitset<16>(x1_imag*x2_real).to_ulong());
int m_real = int(bitset<16>(r1-r2).to_ulong());
int m_imag = int(bitset<16>(r3+r4).to_ulong());
return complex<double>(double(m_real), double(m_imag));
}
complex<typename unary_plus_typeof_helper<X>::type>
operator+(const complex<X>& x)
{
typedef typename unary_plus_typeof_helper<X>::type type;
return complex<type>(x.real(),x.imag());
}
string num_to_string(const complex<T>& num, int pad=0){
string ret;
// both components non-zero
if(num.real()!=0 && num.imag()!=0){
// don't add "+" in the middle if imag is negative and will start with "-"
string ret=num_to_string(num.real(),/*pad*/0)+(num.imag()>0?"+":"")+num_to_string(num.imag(),/*pad*/0)+"j";
if(pad==0 || (int)ret.size()>=pad) return ret;
return string(pad-ret.size(),' ')+ret; // left-pad with spaces
}
// only imaginary is non-zero: skip the real part, and decrease padding to accomoadate the trailing "j"
if(num.imag()!=0){
return num_to_string(num.imag(),/*pad*/pad>0?pad-1:0)+"j";
}
// non-complex (zero or not)
return num_to_string(num.real(),pad);
}
complex<Float> Comms::GlobalSum(complex<Float> z)
{
if (!initialized) { Initialize(); }
#ifdef USE_MPI
Float sendbuf[2];
Float recvbuf[2];
sendbuf[0] = z.real();
sendbuf[1] = z.imag();
MPI_Barrier(MPI_COMM_WORLD);
#ifdef USE_SINGLE
MPI_Reduce( sendbuf, recvbuf, 2, MPI_FLOAT, MPI_SUM, 0, MPI_COMM_WORLD );
#endif
#ifdef USE_DOUBLE
MPI_Reduce( sendbuf, recvbuf, 2, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD );
#endif
#ifdef USE_LONG_DOUBLE
MPI_Reduce( sendbuf, recvbuf, 2, MPI_LONG_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD );
#endif
return complex<Float>(recvbuf[0], recvbuf[1]);
#else
return z;
#endif
}
complex<typename unary_minus_typeof_helper<X>::type>
operator-(const complex<X>& x)
{
typedef typename unary_minus_typeof_helper<X>::type type;
return complex<type>(-x.real(),-x.imag());
}
complex<double> cos(complex<double> x) {
double cosh(double x);
double cos(double x);
double r = cosh(x.imag())*cos(x.real());
double i = sinh(x.imag())*sin(x.real());
return(complex<double>(r,i));
}
void write_mandel(std::string name, int x, int y)
{
GIF_image gif(x, y);
for(int i=0; i<256;i++)
{
gif.setColor(i, i, i, (char)255);
}
for (int i = 0; i < y; i++)
{
for (int j = 0; j < x; j++)
{
complex<double> c(c1.real()+j*(c2-c1).real()/X, c1.imag()+i*(c2-c1).imag()/Y);
complex<double> z = 0;
int iter = 0;
while (abs(z) < 2 && iter < MAX_ITER)
{
z = z*z+c;
iter++;
}
gif.setPixel(j, i, ((MAX_ITER-iter)*255)/40);
}
}
gif.writeToFile(name.c_str());
}
/** Implement Exponential Integral function, E1(z), based on formulaes in
*Abramowitz and Stegun (A&S)
* In fact this implementation returns exp(z)*E1(z) where z is a complex number
*
* @param z :: input
* @return exp(z)*E1(z)
*/
complex<double> exponentialIntegral(const complex<double> &z) {
double z_abs = abs(z);
if (z_abs == 0.0) {
// Basically function is infinite in along the real axis for this case
return complex<double>(std::numeric_limits<double>::max(), 0.0);
} else if (z_abs < 10.0) // 10.0 is a guess based on formula 5.1.55 (no idea
// how good it really is)
{
// use formula 5.1.11 in A&S. rewrite last term in 5.1.11 as
// x*sum_n=0 (-1)^n x^n / (n+1)*(n+1)! and then calculate the
// terms in the sum recursively
complex<double> z1(1.0, 0.0);
complex<double> z2(1.0, 0.0);
for (int i = 1; i <= 100; i++) // is max of 100 here best?
{
z2 = -static_cast<double>(i) * (z2 * z) / ((i + 1.0) * (i + 1.0));
z1 += z2;
if (abs(z2) < 1.0E-10 * abs(z1))
break; // i.e. if break loop if little change to term added
}
return exp(z) * (-log(z) - M_EULER + z * z1);
} else {
// use formula 5.1.22 in A&S. See discussion page 231
complex<double> z1(0.0, 0.0);
for (int i = 20; i >= 1; i--) // not sure if 20 is always sensible?
z1 = static_cast<double>(i) / (1.0 + static_cast<double>(i) / (z + z1));
complex<double> retVal = 1.0 / (z + z1);
if (z.real() <= 0.0 && z.imag() == 0.0)
retVal -= exp(z) * complex<double>(0.0, 1.0) * M_PI; // see formula 5.1.7
return retVal;
}
}
void put(index_type i, complex<T> val)
{
assert(this->format_ != no_user_format);
if (this->format_ == array_format)
this->u_.data_[i] = val;
else if (this->format_ == interleaved_format)
{
this->u_.split_.real_[2*i+0] = val.real();
this->u_.split_.real_[2*i+1] = val.imag();
}
else // if (format_ == split_format)
{
this->u_.split_.real_[i] = val.real();
this->u_.split_.imag_[i] = val.imag();
}
}
complex<Y>
pow(const complex<Y>& x,const Y& y)
{
std::complex<Y> tmp(x.real(),x.imag());
tmp = std::pow(tmp,y);
return complex<Y>(tmp.real(),tmp.imag());
}
/* Performed function evaluation. Horners algorithm.
*/
static double feval( const int n, const double a[], const complex<double> z, complex<double> *fz )
{
register int i;
double p, q, r, s, t;
p = -2.0 * z.real();
q = z.real() * z.real() + z.imag() * z.imag();
s = 0; r = a[ 0 ];
for( i = 1; i < n; i++ )
{
t = a[ i ] - p * r - q * s;
s = r;
r = t;
}
*fz = complex<double>( a[ n ] + z.real() * r - q * s, z.imag() * r );
return fz->real() * fz->real() + fz->imag() * fz->imag();
}
complex<long double> sum_to_all(complex<long double> in) {
complex<long double> out = in;
#ifdef HAVE_MPI
if (MPI_LONG_DOUBLE == MPI_DATATYPE_NULL) {
complex<double> dout;
dout = sum_to_all(complex<double>(double(in.real()), double(in.imag())));
out = complex<long double>(dout.real(), dout.imag());
}
else
MPI_Allreduce(&in,&out,2,MPI_LONG_DOUBLE,MPI_SUM,mycomm);
#endif
return out;
}
complex ApproximateDoSFunctor::generateGreensResult(const arma::cx_vec &alphas, const arma::vec &betas, const complex &energy, long currentindex, complex &lastGreen)
{
if(currentindex == betas.size() -1)
{
lastGreen = (energy - alphas(currentindex) - sqrt(pow(energy-alphas(currentindex),2) - 4. * pow(betas(currentindex-1),2) ) )/( 2. * pow(betas(currentindex-1),2) );
if(lastGreen.imag() > 0)
return lastGreen;
return (lastGreen = (energy - alphas(currentindex) + sqrt(pow(energy-alphas(currentindex),2) - 4. * pow(betas(currentindex-1),2) ) )/( 2. * pow(betas(currentindex-1),2) ));
}
else
return (lastGreen = 1./(energy - alphas(currentindex) - pow(betas(currentindex),2)*lastGreen));
}
/* Calculate a upper bound for the rounding errors performed in a
polynomial at a complex point.
( Adam's test )
*/
static double upperbound( const int n, const double a[], const complex<double> z )
{
register int i;
double p, q, r, s, t, u, e;
p = - 2.0 * z.real();
q = z.real() * z.real() + z.imag() * z.imag();
u = sqrt( q );
s = 0.0; r = a[ 0 ]; e = fabs( r ) * ( 3.5 / 4.5 );
for( i = 1; i < n; i++ )
{
t = a[ i ] - p * r - q * s;
s = r;
r = t;
e = u * e + fabs( t );
}
t = a[ n ] + z.real() * r - q * s;
e = u * e + fabs( t );
e = ( 9.0 * e - 7.0 * ( fabs( t ) + fabs( r ) * u ) +
fabs( z.real() ) * fabs( r ) * 2.0 ) * pow( (double)FLT_RADIX, -DBL_MANT_DIG+1);
return e * e;
}
complex<typename boost::units::multiply_typeof_helper<X,Y>::type>
operator*(const complex<X>& x,const complex<Y>& y)
{
typedef typename boost::units::multiply_typeof_helper<X,Y>::type type;
return complex<type>(x.real()*y.real() - x.imag()*y.imag(),
x.real()*y.imag() + x.imag()*y.real());
// fully correct implementation has more complex return type
//
// typedef typename boost::units::multiply_typeof_helper<X,Y>::type xy_type;
//
// typedef typename boost::units::add_typeof_helper<
// xy_type,xy_type>::type xy_plus_xy_type;
// typedef typename
// boost::units::subtract_typeof_helper<xy_type,xy_type>::type
// xy_minus_xy_type;
//
// BOOST_STATIC_ASSERT((boost::is_same<xy_plus_xy_type,
// xy_minus_xy_type>::value == true));
//
// return complex<xy_plus_xy_type>(x.real()*y.real()-x.imag()*y.imag(),
// x.real()*y.imag()+x.imag()*y.real());
}
void ElectronicStructure::rot(complex<double> Hij,double dt,int i,int j){
/***********************************************************************
Action of operator: exp(iL_ij*dt) = exp(-(i/hbar)*dt*(H_ij*c_j*d/dc_i + H_ji*c_i*d/dc_j)):
exp(iL_ij*dt) = A * B * A, where
A = exp(iL_ij^-), phi = (dt/2) * Im(H_ij)/hbar
B = exp(iL_ij^+), phi = dt * -Re(H_ij)/hbar
composite rotation in plane i,j due to Hamiltonian matrix elements H_ij
note Hamiltonian is Hermitian: H_ij^* = H_ji
***********************************************************************/
double phi1 = 0.5*dt*Hij.imag()/hbar;
double phi2 = -dt*Hij.real()/hbar;
rot1(phi1,i,j);
rot2(phi2,i,j);
rot1(phi1,i,j);
}