Description

Various mathematical functions.

Namespace for new Math classes and functions. See the Math Libraries page for a detailed description.

Namespaces
	Bessel
	Real and complex Bessel functions.

Functions
double	cot (double x)
	cotangent function: $cot(x)\equiv1/tan(x)$ More...

double	erf (double arg)
	Error function of real-valued argument. More...

double	Gaussian (double x, double average, double std_dev)

double	GeneratePoissonRandom (double average)

double	IntegratedGaussian (double x, double average, double std_dev)

double	Laue (double x, size_t N)
	Real Laue function: $Laue(x,N)\equiv\sin(Nx)/sin(x)$ . More...

complex_t	sinc (complex_t z)
	Complex sinc function: $sinc(x)\equiv\sin(x)/x$ . More...

double	sinc (double x)
	sinc function: $sinc(x)\equiv\sin(x)/x$ More...

double	StandardNormal (double x)

complex_t	tanhc (complex_t z)
	Complex tanhc function: $tanhc(x)\equiv\tanh(x)/x$ . More...

Function Documentation

◆ cot()

double Math::cot ( double x )

cotangent function: $cot(x)\equiv1/tan(x)$

Definition at line 47 of file Functions.cpp.

 {
     return tan(M_PI_2 - x);
 }

References M_PI_2.

Referenced by Bipyramid4::Bipyramid4(), Cone::Cone(), Pyramid2::Pyramid2(), Pyramid3::Pyramid3(), Pyramid4::Pyramid4(), and Pyramid6::Pyramid6().

◆ erf()

double Math::erf ( double arg )

Error function of real-valued argument.

Definition at line 88 of file Functions.cpp.

 {
     if (arg < 0.0)
         throw std::runtime_error("Error in Math::erf: negative argument is not allowed");
     if (std::isinf(arg))
         return 1.0;
     return gsl_sf_erf(arg);
 }

Referenced by FootprintGauss::calculate().

◆ Gaussian()

double Math::Gaussian	(	double	x,
		double	average,
		double	std_dev
	)

Definition at line 35 of file Functions.cpp.

 {
     return StandardNormal((x - average) / std_dev) / std_dev;
 }

References StandardNormal().

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◆ GeneratePoissonRandom()

double Math::GeneratePoissonRandom ( double average )

Definition at line 124 of file Functions.cpp.

 {
     unsigned seed =
         static_cast<unsigned>(std::chrono::system_clock::now().time_since_epoch().count());
     std::default_random_engine generator(seed);
     if (average <= 0.0)
         return 0.0;
     if (average < 1000.0) { // Use std::poisson_distribution
         std::poisson_distribution<int> distribution(average);
         int sample = distribution(generator);
         return (double)sample;
     }
     // Use normal approximation
     std::normal_distribution<double> distribution(average, std::sqrt(average));
     double sample = distribution(generator);
     return std::max(0.0, sample);
 }

Referenced by PoissonBackground::addBackground().

◆ IntegratedGaussian()

double Math::IntegratedGaussian	(	double	x,
		double	average,
		double	std_dev
	)

Definition at line 40 of file Functions.cpp.

 {
     double normalized_x = (x - average) / std_dev;
     static double root2 = std::sqrt(2.0);
     return (gsl_sf_erf(normalized_x / root2) + 1.0) / 2.0;
 }

Referenced by ResolutionFunction2DGaussian::evaluateCDF().

◆ Laue()

double Math::Laue	(	double	x,
		size_t	N
	)

Real Laue function: $Laue(x,N)\equiv\sin(Nx)/sin(x)$ .

Definition at line 77 of file Functions.cpp.

 {
     static const double SQRT6DOUBLE_EPS = std::sqrt(6.0 * std::numeric_limits<double>::epsilon());
     auto nd = static_cast<double>(N);
     if (std::abs(nd * x) < SQRT6DOUBLE_EPS)
         return nd;
     double num = std::sin(nd * x);
     double den = std::sin(x);
     return num / den;
 }

References N.

Referenced by InterferenceFinite3DLattice::iff_without_dw(), Interference2DSuperLattice::iff_without_dw(), and InterferenceFinite2DLattice::interferenceForXi().

◆ sinc() [1/2]

complex_t Math::sinc ( complex_t z )

Complex sinc function: $sinc(x)\equiv\sin(x)/x$ .

Definition at line 59 of file Functions.cpp.

 {
     // This is an exception from the rule that we must not test floating-point numbers for equality.
     // For small non-zero arguments, sin(z) returns quite accurately z or z-z^3/6.
     // There is no loss of precision in computing sin(z)/z.
     // Therefore there is no need for an expensive test like abs(z)<eps.
     if (z == complex_t(0., 0.))
         return 1.0;
     return std::sin(z) / z;
 }

◆ sinc() [2/2]

double Math::sinc ( double x )

sinc function: $sinc(x)\equiv\sin(x)/x$

Definition at line 52 of file Functions.cpp.

 {
     if (x == 0)
         return 1;
     return std::sin(x) / x;
 }

Referenced by Profile1DTriangle::decayFT(), ripples::factor_x_box(), Box::formfactor_at_bottom(), Cylinder::formfactor_at_bottom(), EllipsoidalCylinder::formfactor_at_bottom(), HorizontalCylinder::formfactor_at_bottom(), LongBoxGauss::formfactor_at_bottom(), LongBoxLorentz::formfactor_at_bottom(), ripples::profile_yz_bar(), ripples::profile_yz_cosine(), ripples::profile_yz_triangular(), Profile1DGate::standardizedFT(), Profile1DTriangle::standardizedFT(), and Profile1DCosine::standardizedFT().

◆ StandardNormal()

double Math::StandardNormal ( double x )

Definition at line 30 of file Functions.cpp.

 {
     return std::exp(-x * x / 2.0) / std::sqrt(M_TWOPI);
 }

References M_TWOPI.

Referenced by Gaussian().

◆ tanhc()

complex_t Math::tanhc ( complex_t z )

Complex tanhc function: $tanhc(x)\equiv\tanh(x)/x$ .

Definition at line 70 of file Functions.cpp.

 {
     if (std::abs(z) < std::numeric_limits<double>::epsilon())
         return 1.0;
     return std::tanh(z) / z;
 }