std::vector<std::complex<double> >
PolyBase::calcRoots(const double epsilon)
/**
Calculate all the roots of the polynominal.
Uses the GSL which uses a Hessian reduction of the
characteristic compainion matrix.
\f[ A= \left( -a_{m-1}/a_m -a_{m-2}/a_m ... -a_0/a_m \right) \f]
where the matrix component below A is the Indenty.
However, GSL requires that the input coefficient is a_m == 1,
hence the call to this->compress().
@param epsilon :: tolerance factor (-ve to use default)
@return roots (not sorted/uniqued)
*/
{
compress(epsilon);
std::vector<std::complex<double> > Out(iDegree);
// Zero State:
if (iDegree==0)
return Out;
// x+a_0 =0
if (iDegree==1)
{
Out[0]=std::complex<double>(-afCoeff[0]);
return Out;
}
// x^2+a_1 x+c = 0
if (iDegree==2)
{
solveQuadratic(Out[0],Out[1]);
return Out;
}
// x^3+a_2 x^2+ a_1 x+c=0
if (iDegree==2)
{
solveCubic(Out[0],Out[1],Out[2]);
return Out;
}
// THERE IS A QUARTIC / QUINTIC Solution availiable but...
// WS contains the the hessian matrix if required (eigenvalues/vectors)
//
gsl_poly_complex_workspace* WS
(gsl_poly_complex_workspace_alloc(iDegree+1));
double* RZ=new double[2*(iDegree+1)];
gsl_poly_complex_solve(&afCoeff.front(),iDegree+1, WS, RZ);
for(int i=0;i<iDegree;i++)
Out[i]=std::complex<double>(RZ[2*i],RZ[2*i+1]);
gsl_poly_complex_workspace_free (WS);
delete [] RZ;
return Out;
}
void
redlichKwong( const CriticalParameters& inCP,
double inT, // Temperature (K)
double inP, // Pressure (bar)
double* outZ, // Compressibility Factor
//double* outState,
double* outVol, // Molar volume (m^3/mol)
double* outPhi, // Fugacity coefficient
double* outHdep, // Enthalpy departure function
double* outSdep) // Entropy departure function
{
// Calculate a and b parameters (depend only on critical parameters)…
double a = 0.42748 * kR * kR * std::pow(inCP.mTc, 2.5) / (inCP.mPc * 1.0e5);
double b = 0.08664 * kR * inCP.mTc / (inCP.mPc * 1.0e5);
double kappa = 0.0;
// Calculate coefficients in the cubic equation of state…
//
// coeffs: (C0,C1,C2,A,B);
double A = a * inP * 1.0e5/ (std::sqrt(inT) * std::pow(kR * inT, 2));
double B = b * inP * 1e5 / (kR * inT);
double C2 = -1.0;
double C1 = A - B - B * B;
double C0 = -A * B;
// Solve the cubic equation for Z0 - Z2, D…
double Z0;
double Z1;
double Z2;
double D;
solveCubic(C0, C1, C2, &Z0, &Z1, &Z2, &D);
// Determine the fugacity coefficient of first root and departure functions…
//
// calcdepfns(coeffs[3], coeffs[4], paramsab[0], Z[0]);
// calcdepfns(A, B, kappa, Z)
calcDepartureFunctions(inT, A, B, kappa, Z0, outPhi, outHdep, outSdep);
}
tuple solveCubicWrapper( double a, double b, double c, double d )
{
double x[3];
int s = solveCubic( a, b, c, d, x );
switch( s )
{
case 0 :
return tuple();
case 1 :
return make_tuple( x[0] );
case 2 :
return make_tuple( x[0], x[1] );
case 3 :
return make_tuple( x[0], x[1], x[2] );
default :
PyErr_SetString( PyExc_ArithmeticError, "Infinite solutions." );
throw_error_already_set();
return tuple(); // should never get here
}
}
bool CubicSpline::intersects(const CubicSpline &rt, VectorF ignoreAxis) const
{
#if 1
vector<VectorF> path1 = getSplinePoints(*this);
vector<VectorF> path2 = getSplinePoints(rt);
for(size_t i = 1; i < path1.size(); i++)
{
for(size_t j = 1; j < path2.size(); j++)
{
if(linesIntersect(path1[i - 1], path1[i], path2[j - 1], path2[j], 0))
return true;
}
}
return false;
#else
const int splitCount = 50; // number of line segments to split spline into
ignoreAxis = normalize(ignoreAxis);
for(int segmentNumber = 0; segmentNumber < splitCount; segmentNumber++)
{
float startT = (float)(segmentNumber) / splitCount;
float endT = (float)(segmentNumber + 1) / splitCount;
VectorF startP = rt.evaluate(startT);
startP -= ignoreAxis * dot(ignoreAxis, startP); // move to plane normal to ignoreAxis
VectorF endP = rt.evaluate(endT);
endP -= ignoreAxis * dot(ignoreAxis, endP); // move to plane normal to ignoreAxis
VectorF delta = endP - startP;
if(absSquared(delta) < eps * eps) // if delta is too small
{
continue;
}
// solve dot(evaluate(t), cross(ignoreAxis, delta)) == 0 and it intersects if dot(evaluate(t) - startP, delta) / dot(delta, delta) is in [0, 1] and t is in [0, 1]
VectorF normal = cross(ignoreAxis, delta);
float cubic = dot(getCubic(), normal);
float quadratic = dot(getQuadratic(), normal);
float linear = dot(getLinear(), normal);
float constant = dot(getConstant(), normal);
float intersections[3];
int intersectionCount = solveCubic(constant, linear, quadratic, cubic, intersections);
for(int i = 0; i < intersectionCount; i++)
{
float t = intersections[i];
if(t < 0 || t > 1)
{
continue;
}
float v = dot(evaluate(t) - startP, delta) / dot(delta, delta);
if(v < 0 || v > 1)
{
continue;
}
return true;
}
}
return false;
#endif
}
int PolynomialSolver::solveQuartic(float c[5], float s[4])
{
float coeffs[4],
z, u, v, sub,
A, B, C, D,
sq_A, p, q, r;
int i, num;
// normalize the equation:x ^ 4 + Ax ^ 3 + Bx ^ 2 + Cx + D = 0
A = c[3] / c[4];
B = c[2] / c[4];
C = c[1] / c[4];
D = c[0] / c[4];
// subsitute x = y - A / 4 to eliminate the cubic term: x^4 + px^2 + qx + r = 0
sq_A = A * A;
p = -3.0f / 8.0f * sq_A + B;
q = 1.0f / 8.0f * sq_A * A - 1.0f / 2.0f * A * B + C;
r = -3.0f / 256.0f * sq_A * sq_A + 1.0f / 16.0f * sq_A * B - 1.0f / 4.0f * A * C + D;
if(isZero(r))
{
// no absolute term:y(y ^ 3 + py + q) = 0
coeffs[0] = q;
coeffs[1] = p;
coeffs[2] = 0.0;
coeffs[3] = 1.0;
num = solveCubic(coeffs, s);
s[num++] = 0;
}
else
{
// solve the resolvent cubic...
coeffs[0] = 1.0f / 2.0f * r * p - 1.0f / 8.0f * q * q;
coeffs[1] = -r;
coeffs[2] = -1.0f / 2.0f * p;
coeffs[3] = 1.0f;
(void) solveCubic(coeffs, s);
// ...and take the one real solution...
z = s[0];
// ...to build two quadratic equations
u = z * z - r;
v = 2.0f * z - p;
if(isZero(u))
u = 0.0;
else if(u > 0.0f)
u = std::sqrt(u);
else
return 0;
if(isZero(v))
v = 0;
else if(v > 0.0f)
v = std::sqrt(v);
else
return 0;
coeffs[0] = z - u;
coeffs[1] = q < 0 ? -v : v;
coeffs[2] = 1.0f;
num = solveQuadric(coeffs, s);
coeffs[0] = z + u;
coeffs[1] = q < 0 ? v : -v;
coeffs[2] = 1.0f;
num += solveQuadric(coeffs, s + num);
}
// resubstitute
sub = 1.0f / 4 * A;
for(i = 0; i < num; i++)
s[i] -= sub;
return num;
}
int solveQuartic(float c[5], float s[4])
{
float coeffs[4];
float z, u, v, sub;
float A, B, C, D;
float sq_A, p, q, r;
int i, num;
/* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
A = c[ 3 ] / c[ 4 ];
B = c[ 2 ] / c[ 4 ];
C = c[ 1 ] / c[ 4 ];
D = c[ 0 ] / c[ 4 ];
/* substitute x = y - A/4 to eliminate cubic term:
x^4 + px^2 + qx + r = 0 */
sq_A = A * A;
p = - 3.0/8 * sq_A + B;
q = 1.0/8 * sq_A * A - 1.0/2 * A * B + C;
r = - 3.0/256*sq_A*sq_A + 1.0/16*sq_A*B - 1.0/4*A*C + D;
if (IS_ZERO(r))
{
/* no absolute term: y(y^3 + py + q) = 0 */
coeffs[ 0 ] = q;
coeffs[ 1 ] = p;
coeffs[ 2 ] = 0;
coeffs[ 3 ] = 1;
num = solveCubic(coeffs, s);
s[ num++ ] = 0;
}
else {
/* solve the resolvent cubic ... */
coeffs[ 0 ] = 1.0/2 * r * p - 1.0/8 * q * q;
coeffs[ 1 ] = - r;
coeffs[ 2 ] = - 1.0/2 * p;
coeffs[ 3 ] = 1;
(void) solveCubic(coeffs, s);
/* ... and take the one real solution ... */
z = s[ 0 ];
/* ... to build two quadric equations */
u = z * z - r;
v = 2 * z - p;
if (IS_ZERO(u))
u = 0;
else if (u > 0)
u = sqrt(u);
else
return 0;
if (IS_ZERO(v))
v = 0;
else if (v > 0)
v = sqrt(v);
else
return 0;
coeffs[ 0 ] = z - u;
coeffs[ 1 ] = q < 0 ? -v : v;
coeffs[ 2 ] = 1;
num = solveQuadric(coeffs, s);
coeffs[ 0 ]= z + u;
coeffs[ 1 ] = q < 0 ? v : -v;
coeffs[ 2 ] = 1;
num += solveQuadric(coeffs, s + num);
}
/* resubstitute */
sub = 1.0/4 * A;
for (i = 0; i < num; ++i)
s[ i ] -= sub;
return num;
}