utl Namespace Reference

Classes
class	AsymptoticPolynomialA
class	AsymptoticPolynomialB
class	Branch
class	BranchPoint
struct	Horner
struct	Horner< Float, Tag, 0 >
struct	Iterator
struct	LogRecursionImpl
struct	LogRecursionImpl< Float, branch, 0 >
struct	Pade
struct	Pade< Float, -1, 4 >
struct	Pade< Float, -1, 5 >
struct	Pade< Float, -1, 6 >
struct	Pade< Float, -1, 7 >
struct	Pade< Float, 0, 1 >
struct	Pade< Float, 0, 2 >
struct	Polynomial

Functions
	HORNER_COEFF (BranchPoint, 0, -1)
	HORNER_COEFF (BranchPoint, 1, 1)
	HORNER_COEFF (BranchPoint, 2, -0.333333333333333333e0)
	HORNER_COEFF (BranchPoint, 3, 0.152777777777777777e0)
	HORNER_COEFF (BranchPoint, 4, -0.79629629629629630e-1)
	HORNER_COEFF (BranchPoint, 5, 0.44502314814814814e-1)
	HORNER_COEFF (BranchPoint, 6, -0.25984714873603760e-1)
	HORNER_COEFF (BranchPoint, 7, 0.15635632532333920e-1)
	HORNER_COEFF (BranchPoint, 8, -0.96168920242994320e-2)
	HORNER_COEFF (BranchPoint, 9, 0.60145432529561180e-2)
	HORNER_COEFF (BranchPoint, 10, -0.38112980348919993e-2)
	HORNER_COEFF (BranchPoint, 11, 0.24408779911439826e-2)
	HORNER_COEFF (BranchPoint, 12, -0.15769303446867841e-2)
	HORNER_COEFF (BranchPoint, 13, 0.10262633205076071e-2)
	HORNER_COEFF (BranchPoint, 14, -0.67206163115613620e-3)
	HORNER_COEFF (BranchPoint, 15, 0.44247306181462090e-3)
	HORNER_COEFF (BranchPoint, 16, -0.29267722472962746e-3)
	HORNER_COEFF (BranchPoint, 17, 0.19438727605453930e-3)
	HORNER_COEFF (BranchPoint, 18, -0.12957426685274883e-3)
	HORNER_COEFF (BranchPoint, 19, 0.86650358052081260e-4)
	HORNER_COEFF (AsymptoticPolynomialB< 2 >, 0, 0)
	HORNER_COEFF (AsymptoticPolynomialB< 2 >, 1, -1)
	HORNER_COEFF (AsymptoticPolynomialB< 2 >, 2, 1./2)
	HORNER_COEFF (AsymptoticPolynomialB< 3 >, 0, 0)
	HORNER_COEFF (AsymptoticPolynomialB< 3 >, 1, 1)
	HORNER_COEFF (AsymptoticPolynomialB< 3 >, 2, -3./2)
	HORNER_COEFF (AsymptoticPolynomialB< 3 >, 3, 1./3)
	HORNER_COEFF (AsymptoticPolynomialB< 4 >, 0, 0)
	HORNER_COEFF (AsymptoticPolynomialB< 4 >, 1, -1)
	HORNER_COEFF (AsymptoticPolynomialB< 4 >, 2, 3)
	HORNER_COEFF (AsymptoticPolynomialB< 4 >, 3, -11./6)
	HORNER_COEFF (AsymptoticPolynomialB< 4 >, 4, 1./4)
	HORNER_COEFF (AsymptoticPolynomialB< 5 >, 0, 0)
	HORNER_COEFF (AsymptoticPolynomialB< 5 >, 1, 1)
	HORNER_COEFF (AsymptoticPolynomialB< 5 >, 2, -5)
	HORNER_COEFF (AsymptoticPolynomialB< 5 >, 3, 35./6)
	HORNER_COEFF (AsymptoticPolynomialB< 5 >, 4, -25./12)
	HORNER_COEFF (AsymptoticPolynomialB< 5 >, 5, 1./5)
	HORNER_COEFF2 (AsymptoticPolynomialA, 0, -y)
	HORNER_COEFF2 (AsymptoticPolynomialA, 1, y)
	HORNER_COEFF2 (AsymptoticPolynomialA, 2,(Horner< Float, AsymptoticPolynomialB< 2 >, 2 >::Eval(y)))
	HORNER_COEFF2 (AsymptoticPolynomialA, 3,(Horner< Float, AsymptoticPolynomialB< 3 >, 3 >::Eval(y)))
	HORNER_COEFF2 (AsymptoticPolynomialA, 4,(Horner< Float, AsymptoticPolynomialB< 4 >, 4 >::Eval(y)))
	HORNER_COEFF2 (AsymptoticPolynomialA, 5,(Horner< Float, AsymptoticPolynomialB< 5 >, 5 >::Eval(y)))
template<typename Float , int order>
Float	AsymptoticExpansionImpl (const Float a, const Float b)
template<typename Float >
Float	HalleyStep (const Float x, const Float w)
template<typename Float >
Float	FritschStep (const Float x, const Float w)
template<typename Float >
Float	SchroederStep (const Float x, const Float w)
template<int branch>
double	LambertW (const double x)
template<>
double	LambertW< 0 > (const double x)
template<>
double	LambertW<-1 > (const double x)
double	LambertW (const int branch, const double x)

Function Documentation

HORNER_COEFF() [1/38]

utl::HORNER_COEFF	(	BranchPoint	,
		0	,
		-	1 )

HORNER_COEFF() [2/38]

utl::HORNER_COEFF	(	BranchPoint	,
		1	,
		1	)

HORNER_COEFF() [3/38]

utl::HORNER_COEFF	(	BranchPoint	,
		2	,
		-0.	333333333333333333e0 )

HORNER_COEFF() [4/38]

utl::HORNER_COEFF	(	BranchPoint	,
		3	,
		0.	152777777777777777e0 )

HORNER_COEFF() [5/38]

utl::HORNER_COEFF	(	BranchPoint	,
		4	,
		-0.79629629629629630e-	1 )

HORNER_COEFF() [6/38]

utl::HORNER_COEFF	(	BranchPoint	,
		5	,
		0.44502314814814814e-	1 )

HORNER_COEFF() [7/38]

utl::HORNER_COEFF	(	BranchPoint	,
		6	,
		-0.25984714873603760e-	1 )

HORNER_COEFF() [8/38]

utl::HORNER_COEFF	(	BranchPoint	,
		7	,
		0.15635632532333920e-	1 )

HORNER_COEFF() [9/38]

utl::HORNER_COEFF	(	BranchPoint	,
		8	,
		-0.96168920242994320e-	2 )

HORNER_COEFF() [10/38]

utl::HORNER_COEFF	(	BranchPoint	,
		9	,
		0.60145432529561180e-	2 )

HORNER_COEFF() [11/38]

utl::HORNER_COEFF	(	BranchPoint	,
		10	,
		-0.38112980348919993e-	2 )

HORNER_COEFF() [12/38]

utl::HORNER_COEFF	(	BranchPoint	,
		11	,
		0.24408779911439826e-	2 )

HORNER_COEFF() [13/38]

utl::HORNER_COEFF	(	BranchPoint	,
		12	,
		-0.15769303446867841e-	2 )

HORNER_COEFF() [14/38]

utl::HORNER_COEFF	(	BranchPoint	,
		13	,
		0.10262633205076071e-	2 )

HORNER_COEFF() [15/38]

utl::HORNER_COEFF	(	BranchPoint	,
		14	,
		-0.67206163115613620e-	3 )

HORNER_COEFF() [16/38]

utl::HORNER_COEFF	(	BranchPoint	,
		15	,
		0.44247306181462090e-	3 )

HORNER_COEFF() [17/38]

utl::HORNER_COEFF	(	BranchPoint	,
		16	,
		-0.29267722472962746e-	3 )

HORNER_COEFF() [18/38]

utl::HORNER_COEFF	(	BranchPoint	,
		17	,
		0.19438727605453930e-	3 )

HORNER_COEFF() [19/38]

utl::HORNER_COEFF	(	BranchPoint	,
		18	,
		-0.12957426685274883e-	3 )

HORNER_COEFF() [20/38]

utl::HORNER_COEFF	(	BranchPoint	,
		19	,
		0.86650358052081260e-	4 )

HORNER_COEFF() [21/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 2 >	,
		0	,
		0	)

HORNER_COEFF() [22/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 2 >	,
		1	,
		-	1 )

HORNER_COEFF() [23/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 2 >	,
		2	,
		1./	2 )

HORNER_COEFF() [24/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 3 >	,
		0	,
		0	)

HORNER_COEFF() [25/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 3 >	,
		1	,
		1	)

HORNER_COEFF() [26/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 3 >	,
		2	,
		-3./	2 )

HORNER_COEFF() [27/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 3 >	,
		3	,
		1./	3 )

HORNER_COEFF() [28/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 4 >	,
		0	,
		0	)

HORNER_COEFF() [29/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 4 >	,
		1	,
		-	1 )

HORNER_COEFF() [30/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 4 >	,
		2	,
		3	)

HORNER_COEFF() [31/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 4 >	,
		3	,
		-11./	6 )

HORNER_COEFF() [32/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 4 >	,
		4	,
		1./	4 )

HORNER_COEFF() [33/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 5 >	,
		0	,
		0	)

HORNER_COEFF() [34/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 5 >	,
		1	,
		1	)

HORNER_COEFF() [35/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 5 >	,
		2	,
		-	5 )

HORNER_COEFF() [36/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 5 >	,
		3	,
		35./	6 )

HORNER_COEFF() [37/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 5 >	,
		4	,
		-25./	12 )

HORNER_COEFF() [38/38]

utl::HORNER_COEFF	(	AsymptoticPolynomialB< 5 >	,
		5	,
		1./	5 )

HORNER_COEFF2() [1/6]

utl::HORNER_COEFF2	(	AsymptoticPolynomialA	,
		0	,
		-	y )

HORNER_COEFF2() [2/6]

utl::HORNER_COEFF2	(	AsymptoticPolynomialA	,
		1	,
		y	)

HORNER_COEFF2() [3/6]

utl::HORNER_COEFF2	(	AsymptoticPolynomialA	,
		2	,
		(Horner< Float, AsymptoticPolynomialB< 2 >, 2 >::Eval(y))	)

HORNER_COEFF2() [4/6]

utl::HORNER_COEFF2	(	AsymptoticPolynomialA	,
		3	,
		(Horner< Float, AsymptoticPolynomialB< 3 >, 3 >::Eval(y))	)

HORNER_COEFF2() [5/6]

utl::HORNER_COEFF2	(	AsymptoticPolynomialA	,
		4	,
		(Horner< Float, AsymptoticPolynomialB< 4 >, 4 >::Eval(y))	)

HORNER_COEFF2() [6/6]

utl::HORNER_COEFF2	(	AsymptoticPolynomialA	,
		5	,
		(Horner< Float, AsymptoticPolynomialB< 5 >, 5 >::Eval(y))	)

AsymptoticExpansionImpl()

template<typename Float , int order>

Float utl::AsymptoticExpansionImpl ( const Float a, const Float b )

inline

Definition at line 158 of file LambertW.hh.

  {
    return a + Horner<Float, AsymptoticPolynomialA, order>::Eval(1/a, b);
  }

HalleyStep()

template<typename Float >

Float utl::HalleyStep ( const Float x, const Float w )

inline

Definition at line 210 of file LambertW.hh.

  {
    const Float ew = exp(w);
    const Float wew = w * ew;
    const Float wewx = wew - x;
    const Float w1 = w + 1;
    return w - wewx / (ew * w1 - (w + 2) * wewx/(2*w1));
  }

FritschStep()

template<typename Float >

Float utl::FritschStep ( const Float x, const Float w )

inline

Definition at line 223 of file LambertW.hh.

  {
    const Float z = log(x/w) - w;
    const Float w1 = w + 1;
    const Float q = 2 * w1 * (w1 + Float(2./3)*z);
    const Float eps = z / w1 * (q - z) / (q - 2*z);
    return w * (1 + eps);
  }

SchroederStep()

template<typename Float >

Float utl::SchroederStep ( const Float x, const Float w )

inline

Definition at line 236 of file LambertW.hh.

  {
    const Float y = x * exp(-w);
    const Float f0 = w - y;
    const Float f1 = 1 + y;
    const Float f00 = f0*f0;
    const Float f11 = f1*f1;
    const Float f0y = f0*y;
    return w - 4*f0*(6*f1*(f11 + f0y) + f00*y) / (f11*(24*f11 + 36*f0y) + f00*(6*y*y + 8*f1*y + f0y));
  }

LambertW() [1/2]

template<int branch>

double utl::LambertW

(

const double

x

)

LambertW< 0 >()

template<>

template double utl::LambertW< 0 >

(

const double

x

)

Definition at line 345 of file LambertW.hh.

  {
    typedef double d;
    return Y5(
      (Branch<d, 0>::BranchPointExpansion<8>(x)),
      x < -0.367679,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Branch<d, 0>::BranchPointExpansion<10>(x))),
      x < -0.311,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Pade<d, 0, 1>::Approximation(x))),
      x < 1.38,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Pade<d, 0, 2>::Approximation(x))),
      x < 236,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Branch<d, 0>::AsymptoticExpansion<6-1>(x)))
    );
  }

LambertW<-1 >()

template<>

template double utl::LambertW<-1 >

(

const double

x

)

Definition at line 364 of file LambertW.hh.

  {
    typedef double d;
    return Y7(
      (Branch<d, -1>::BranchPointExpansion<8>(x)),
      x < -0.367579,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Branch<d, -1>::BranchPointExpansion<4>(x))),
      x < -0.366079,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Pade<d, -1, 7>::Approximation(x))),
      x < -0.289379,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Pade<d, -1, 4>::Approximation(x))),
      x < -0.0509,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Pade<d, -1, 5>::Approximation(x))),
      x < -0.000131826,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Pade<d, -1, 6>::Approximation(x))),
      x < -6.30957e-31,
      (Iterator<d, HalleyStep<d> >::Depth<1>::Recurse(x, Branch<d, -1>::LogRecursion<3>(x)))
    );
  }

LambertW() [2/2]

double utl::LambertW ( const int branch, const double x )

inline

Definition at line 397 of file LambertW.hh.

  {
    switch (branch) {
    case -1: return LambertW<-1>(x);
    case  0: return LambertW<0>(x);
    default: return std::numeric_limits<double>::quiet_NaN();
    }
  }