This class contains various mathematical functions and constants. More...

Static Public Member Functions
static double	Abs (this double number)
	Returns the absolute value of the double number. More...

static double	Cos (double d)
	Special cosine function as the Math.Cos function has problems with \(\cos\left(\frac{\pi}{2}\right)\) and with large values. More...

static double	DegreeToRadian (double degree)
	Convert Degree to Radian More...

static List< int[]>	DrawAllSubsetsOfGivenSizeFromSetOfIndices (int subsetSize, int populationSize)
	Method for creating all possible subsets of a given size out of a set of indices. More...

static double	Factorial (int m)
	Calculates the factorial \(n!\). More...

static int	GetQuantizationLevel (double value, double minimumQuantizedValue, double maximumQuantizedValue, double quantizationDifference)
	Gets the quantization level on which a certain value lies. More...

static double	GetQuantizedValue (double value, double minimumQuantizedValue, double maximumQuantizedValue, double quantizationDifference)
	Gets the quantized value. More...

static double	IndexOfTriangleNumber (int number)
	The i^th triangle number is i * (i+1) / 2 which we have solved for i to return the double index of the matching triangle number. For example this method returns 2.0 for 3 because 3 is the second triangle number. It returns 2.4 for 4 which means that 4 is above the second and below the third triangle number. More...

static void	InitQuantizationMinMaxRiser (double minimum, double maximum, int numberOfQuantizationLevels, out double minimumQuantizedValue, out double maximumQuantizedValue, out double quantizationDifference)
	Initializes a "Min-Max-Riser" quantization. More...

static void	InitQuantizationMinMaxTread (double minimum, double maximum, int numberOfQuantizationLevels, out double minimumQuantizedValue, out double maximumQuantizedValue, out double quantizationDifference)
	Initializes a "Min-Max-Tread" quantization. More...

static double	NumericDerivative (Func< double, double > function, double position, double epsilon=Globals.Epsilon)
	Calculates the numerical derivative of a function. More...

static double	Pow (double baseNumber, double exponent)
	Raises the double number to the given power. More...

static Complex	QuantizePoint (Complex point, int amplitudeLevels, int phaseLevels)
	Quantizes a value in the amplitude range \(0 \ldots 1\) and the phase range \(-\pi \ldots \pi\). The phase levels are calculated with the following formula: \(\frac{\pi}{2} - k \cdot \frac{\pi}{n}\). \(n\) is the number of phase levels, \(k = 0, \ldots, n-1\). More...

static Complex	QuantizePoint (Complex point, int amplitudeLevels, int phaseLevels, double factor)
	Quantizes a value in the amplitude range \(0 \ldots 1\) and the phase range \(-\pi \ldots \pi\). The phase levels are calculated with the following formula: \(\frac{\pi}{2} - k \cdot \frac{\pi}{n}\). \(n\) is the number of phase levels, \(k = 0, \ldots, n-1\). More...

static double	RadianToDegree (double radian)
	Convert Degree to Radian More...

static double	RMS (LargeList< double > values, bool relativeToAverage=false, LargeList< double > weights=null)
	Calculates the RMS (root mean square) of the given value. More...

static double	RMS (LargeList< double > values, double average)
	Calculates the RMS (root mean square) of the given value with respect to the given average. More...

static double	RombergIntegral (RealFunctionWithSingleArgument f, double xmin, double xmax, int n)
	Computes the definite integral of the real function \(f\) in the range from xmin to xmax. Romberg integration with \(2^{(n-1)+1}\) equidistant function evaluations is used to this end. More...

static double	Sin (double d)
	Special sine function as the Math.Sin function has problems with \(\sin\left(\pi\right)\) and with large values. More...

static int	SolveQuadraticEquation (double a2, double a1, double a0, out double x1, out double x2)
	Function solving the general quadratic equation \(0 = a_2 \cdot x^2 + a_1 \cdot x + a_0\) . More...

static double	SphericalPhase (VectorD position, double k, double radius, double signumZ, double signumKz)
	Calculates the phase of the spherical phase function on the position \(\boldsymbol{r}\). The spherical phase function is defined as \(f(\boldsymbol{r}) = \exp{(\text{signumZ} \cdot \text{signumKz} \cdot k \cdot a)}\) with \(a = \sqrt{(x^2 + y^2 + z^2)}\). More...

static VectorD	SphericalPhaseDerivative (VectorD position, double k, double radius, double signumZ, double signumKz)
	Calculates the first derivative of spherical phase function (not a complex number but just the phase value). The spherical phase function is defined as \(f(\boldsymbol{r}) = \exp{(signumZ * signumKz * k * a)}\) with \(a = \sqrt{(x^2 + y^2 + z^2)}\). More...

static Complex	SphericalPhaseFunction (VectorD position, double k, double radius, double signumZ, double signumKz)
	Calculates the value of a spherical phase function on the position \(\boldsymbol{r}\). The spherical phase function is defined as \(f(\boldsymbol{r}) = \exp{(\text{signumZ} \cdot \text{signumKz} \cdot k \cdot a)}\) with \(a = \sqrt{(x^2 + y^2 + z^2)}\). More...

static void	SymDefSolve (double[,] matr, ref double[] vect)
	Solves the linear system matr * x = vect by using the Cholesky decomposition of matr. More...

static double	Tan (double d)
	Special tangent function as the Math.Tan function has problems with \(\tan\left(\frac{\pi}{2}\right)\) and \(\tan\left(\pi\right)\). More...

static double	ToThePowerOf (this double baseNumber, double exponent)
	Raises the double number to the given power. More...

static int	TriangleNumber (int index)
	Calculates the i^th triangle number which is i * (i+1) / 2 More...

Static Public Attributes
const double	HalfPi
	Constant value of \(\frac{\pi}{2}\).

const double	TwoPi
	Constant value of \(2 \cdot \pi\). More...

Detailed Description

This class contains various mathematical functions and constants.

Member Function Documentation

◆ Abs()

static double Abs ( this double number )

static

Returns the absolute value of the double number.

Parameters

number The double number.

Returns: The absolute value of the double number.

◆ Cos()

static double Cos ( double d )

static

Special cosine function as the Math.Cos function has problems with \(\cos\left(\frac{\pi}{2}\right)\) and with large values.

Parameters

d	The argument of the cosine.

Returns: The cosine of d.

◆ DegreeToRadian()

static double DegreeToRadian ( double degree )

static

Convert Degree to Radian

Parameters

degree The angle in degree.

Returns: The angle in radian.

◆ DrawAllSubsetsOfGivenSizeFromSetOfIndices()

static List< int[]> DrawAllSubsetsOfGivenSizeFromSetOfIndices	(	int	subsetSize,
		int	populationSize
	)

static

Method for creating all possible subsets of a given size out of a set of indices.

Parameters

subsetSize	Size of the subses to draw.
populationSize	Size of the (zero based) index set. So the highest index is populationSize - 1.

Returns: List of all subsets of given size.

◆ Factorial()

static double Factorial ( int m )

static

Calculates the factorial \(n!\).

Parameters

m	Order of factorial to calculate: \(n\).

Returns: \(n!\)

◆ GetQuantizationLevel()

static int GetQuantizationLevel	(	double	value,
		double	minimumQuantizedValue,
		double	maximumQuantizedValue,
		double	quantizationDifference
	)

static

Gets the quantization level on which a certain value lies.

Parameters

value	The value to quantize.
minimumQuantizedValue	The value corresponding to the lowest quantization level.
maximumQuantizedValue	The value corresponding to the highest quantization level.
quantizationDifference	The difference between two consecutive quantization levels.

Returns: Index of the quantization level closest to the given value.

◆ GetQuantizedValue()

static double GetQuantizedValue	(	double	value,
		double	minimumQuantizedValue,
		double	maximumQuantizedValue,
		double	quantizationDifference
	)

static

Gets the quantized value.

Parameters

value	The value to quantize.
minimumQuantizedValue	The value corresponding to the lowest quantization level.
maximumQuantizedValue	The value corresponding to the highest quantization level.
quantizationDifference	The difference between two consecutive quantization levels.

Returns: The quantized value.

◆ IndexOfTriangleNumber()

static double IndexOfTriangleNumber ( int number )

static

The i^th triangle number is i * (i+1) / 2 which we have solved for i to return the double index of the matching triangle number. For example this method returns 2.0 for 3 because 3 is the second triangle number. It returns 2.4 for 4 which means that 4 is above the second and below the third triangle number.

Parameters

number The number for which the index of the matching triangle number is determined.

Returns: The index of the matching triangle number. If you want to obtain the largest triangle number less than or equal the given number use a floor rounding on the result. If you want to obtain the smallest triangle number greater than or equal the given number use a ceiling rounding on the result.

Exceptions

System.ArgumentException Number must be larger or equal than zero.

◆ InitQuantizationMinMaxRiser()

static void InitQuantizationMinMaxRiser	(	double	minimum,
		double	maximum,
		int	numberOfQuantizationLevels,
		out double	minimumQuantizedValue,
		out double	maximumQuantizedValue,
		out double	quantizationDifference
	)

static

Initializes a "Min-Max-Riser" quantization.

Parameters

minimum	The minimum of the value range to quantize.
maximum	The maximum of the value range to quantize.
numberOfQuantizationLevels	The number of quantization levels.
minimumQuantizedValue	Out parameter to return the minimum quantized value.
maximumQuantizedValue	Out parameter to return the maximum quantized value.
quantizationDifference	Out parameter to return the difference between two consecutive quantization levels.

◆ InitQuantizationMinMaxTread()

static void InitQuantizationMinMaxTread	(	double	minimum,
		double	maximum,
		int	numberOfQuantizationLevels,
		out double	minimumQuantizedValue,
		out double	maximumQuantizedValue,
		out double	quantizationDifference
	)

static

Initializes a "Min-Max-Tread" quantization.

Parameters

minimum	The minimum of the value range to quantize.
maximum	The maximum of the value range to quantize.
numberOfQuantizationLevels	The number of quantization levels.
minimumQuantizedValue	Out parameter to return the minimum quantized value.
maximumQuantizedValue	Out parameter to return the maximum quantized value.
quantizationDifference	Out parameter to return the difference between two consecutive quantization levels.

◆ NumericDerivative()

static double NumericDerivative	(	Func< double, double >	function,
		double	position,
		double	epsilon = `Globals.Epsilon`
	)

static

Calculates the numerical derivative of a function.

Parameters

function	The function 𝑦 = f(𝑥).
position	The position 𝑥 at which the derivative is calculated.
epsilon	The numerical derivative is calculated by calculating the function values at two positions 𝑥 + ε and 𝑥 - ε.

Returns: The numerical derivative of the function at the given position.

◆ Pow()

static double Pow	(	double	baseNumber,
		double	exponent
	)

static

Raises the double number to the given power.

Parameters

baseNumber	The base.
exponent	The exponent.

Returns: The double number to the power of the exponent.

◆ QuantizePoint() [1/2]

static Complex QuantizePoint	(	Complex	point,
		int	amplitudeLevels,
		int	phaseLevels
	)

static

Quantizes a value in the amplitude range \(0 \ldots 1\) and the phase range \(-\pi \ldots \pi\). The phase levels are calculated with the following formula: \(\frac{\pi}{2} - k \cdot \frac{\pi}{n}\). \(n\) is the number of phase levels, \(k = 0, \ldots, n-1\).

Parameters

point	The complex value to quantize.
amplitudeLevels	Number of amplitude levels. Zero means no quantization.
phaseLevels	Number of phase levels. Zero means no quantization.

Returns: The quantized complex value.

◆ QuantizePoint() [2/2]

static Complex QuantizePoint	(	Complex	point,
		int	amplitudeLevels,
		int	phaseLevels,
		double	factor
	)

static

Quantizes a value in the amplitude range \(0 \ldots 1\) and the phase range \(-\pi \ldots \pi\). The phase levels are calculated with the following formula: \(\frac{\pi}{2} - k \cdot \frac{\pi}{n}\). \(n\) is the number of phase levels, \(k = 0, \ldots, n-1\).

Parameters

point	The complex value to quantize.
amplitudeLevels	Number of amplitude levels. Zero means no quantization.
phaseLevels	Number of phase levels. Zero means no quantization.
factor	The projection strength \(\lambda\). \(\lambda = 1\) means hard quantization and \(\lambda = 0\) means no quantization. Values of \(\lambda\) between 0 and 1 lead to soft quantization. The used formulas can be found in the quantization chapter of the manual.

Returns: The quantized complex value.

◆ RadianToDegree()

static double RadianToDegree ( double radian )

static

Convert Degree to Radian

Parameters

radian The angle in radian.

Returns: The angle in degree.

◆ RMS() [1/2]

static double RMS	(	LargeList< double >	values,
		bool	relativeToAverage = `false`,
		LargeList< double >	weights = `null`
	)

static

Calculates the RMS (root mean square) of the given value.

Parameters

values	The list of values.
relativeToAverage	If this optional parameter is set to `true` the RMS refers to the average value. Otherwise it refers to zero.
weights	Optional list containing a weight per value.

Returns: The RMS. NaN if zero values are given.

Exceptions

ArgumentException RMS calculation: For every value there must be exactly one weight.

◆ RMS() [2/2]

static double RMS	(	LargeList< double >	values,
		double	average
	)

static

Calculates the RMS (root mean square) of the given value with respect to the given average.

Parameters

values	The list of values.
average	Optional average. Zero by default.

Returns: The RMS. NaN if zero values are given.

◆ RombergIntegral()

static double RombergIntegral	(	RealFunctionWithSingleArgument	f,
		double	xmin,
		double	xmax,
		int	n
	)

static

Computes the definite integral of the real function \(f\) in the range from xmin to xmax. Romberg integration with \(2^{(n-1)+1}\) equidistant function evaluations is used to this end.

Parameters

f	Delegate of the real function \(f\) to integrate.
xmin	Lower border of integration interval.
xmax	Upper border of integration interval.
n	Order of the method. 1 means trapezoidal, 2 means Simpson rule.

Returns: Approximate integral value.

◆ Sin()

static double Sin ( double d )

static

Special sine function as the Math.Sin function has problems with \(\sin\left(\pi\right)\) and with large values.

Parameters

d	The argument of the sine.

Returns: The sine of d.

◆ SolveQuadraticEquation()

static int SolveQuadraticEquation	(	double	a2,
		double	a1,
		double	a0,
		out double	x1,
		out double	x2
	)

static

Function solving the general quadratic equation \(0 = a_2 \cdot x^2 + a_1 \cdot x + a_0\) .

Parameters

a2	Coefficient of quadratic term.
a1	Coefficient of linear term.
a0	Coefficient of constant term.
x1	Out parameter giving the solution if only one solution exists or the smaller of the two solutions.
x2	Out parameter giving the larger solution in case of two solutions.

Returns: The number of solutions. Either -1, 0, 1, or 2. Two equal values in case of zero discriminant. -1 indicates that any x is a solution.

◆ SphericalPhase()

static double SphericalPhase	(	VectorD	position,
		double	k,
		double	radius,
		double	signumZ,
		double	signumKz
	)

static

Calculates the phase of the spherical phase function on the position \(\boldsymbol{r}\). The spherical phase function is defined as \(f(\boldsymbol{r}) = \exp{(\text{signumZ} \cdot \text{signumKz} \cdot k \cdot a)}\) with \(a = \sqrt{(x^2 + y^2 + z^2)}\).

Parameters

position	Position \(\boldsymbol{r}\) spherical phase is to be calculated for.
k	Absolute value of the wave vector \(k\).
radius	Distance \(z\) from the source plane of the spherical phase function.
signumZ	Sign of the spherical phase radius, i.e. is the definition plane before or behind the source. The product of signumZ and signumKz ( \(\operatorname{sgn}(z) \cdot \operatorname{sgn}(k_z)\)) determines if the phase is diverging or converging.
signumKz	Sign of the z-component of the k-vector of the harmonic field, i.e. whether propagation in positive or negative z-direction is considered. The product of signumZ and signumKz ( \(\operatorname{sgn}(radius) \cdot \operatorname{sgn}(k_z)\)) determines if the phase is diverging or converging.

Returns: The phase as real value with double precision.

◆ SphericalPhaseDerivative()

static VectorD SphericalPhaseDerivative	(	VectorD	position,
		double	k,
		double	radius,
		double	signumZ,
		double	signumKz
	)

static

Calculates the first derivative of spherical phase function (not a complex number but just the phase value). The spherical phase function is defined as \(f(\boldsymbol{r}) = \exp{(signumZ * signumKz * k * a)}\) with \(a = \sqrt{(x^2 + y^2 + z^2)}\).

Parameters

position	Position \(\boldsymbol{r}\) spherical phase is to be calculated for.
k	Absolute value of the wave vector \(k\).
radius	Distance \(z\) from the source plane of the spherical phase function.
signumZ	Sign of the spherical phase radius, i.e. is the definition plane before or behind the source. The product of signumZ and signumKz ( \(\operatorname{sgn}(z) \cdot \operatorname{sgn}(k_z)\)) determines if the phase is diverging or converging.
signumKz	Sign of the z-component of the k-vector of the harmonic field, i.e. whether propagation in positive or negative z-direction is considered. The product of signumZ and signumKz ( \(\operatorname{sgn}(radius) \cdot \operatorname{sgn}(k_z)\)) determines if the phase is diverging or converging.

Returns: Result of operation.

◆ SphericalPhaseFunction()

static Complex SphericalPhaseFunction	(	VectorD	position,
		double	k,
		double	radius,
		double	signumZ,
		double	signumKz
	)

static

Calculates the value of a spherical phase function on the position \(\boldsymbol{r}\). The spherical phase function is defined as \(f(\boldsymbol{r}) = \exp{(\text{signumZ} \cdot \text{signumKz} \cdot k \cdot a)}\) with \(a = \sqrt{(x^2 + y^2 + z^2)}\).

Parameters

position	Position \(\boldsymbol{r}\) spherical phase is to be calculated for.
k	Absolute value of the wave vector \(k\).
radius	Distance \(z\) from the source plane of the spherical phase function.
signumZ	Sign of the spherical phase radius, i.e. is the definition plane before or behind the source. The product of signumZ and signumKz ( \(\operatorname{sgn}(z) \cdot \operatorname{sgn}(k_z)\)) determines if the phase is diverging or converging.
signumKz	Sign of the z-component of the k-vector of the harmonic field, i.e. whether propagation in positive or negative z-direction is considered. The product of signumZ and signumKz ( \(\operatorname{sgn}(radius) \cdot \operatorname{sgn}(k_z)\)) determines if the phase is diverging or converging.

Returns: Result of operation.

◆ SymDefSolve()

static void SymDefSolve	(	double	matr[,],
		ref double[]	vect
	)

static

Solves the linear system matr * x = vect by using the Cholesky decomposition of matr.

Parameters

matr	A two dimensional coefficients matrix. The matrix must be symmetric and positive definite.
vect	In the function call this parameter must contain the function values vector of the equation system. After the evaluation of the function this parameter contains the solution vector x.

◆ Tan()

static double Tan ( double d )

static

Special tangent function as the Math.Tan function has problems with \(\tan\left(\frac{\pi}{2}\right)\) and \(\tan\left(\pi\right)\).

Parameters

d	The argument of the tangent.

Returns: The tangent of d.

◆ ToThePowerOf()

static double ToThePowerOf	(	this double	baseNumber,
		double	exponent
	)

static

Raises the double number to the given power.

Parameters

baseNumber	The base.
exponent	The exponent.

Returns: The double number to the power of the exponent.

◆ TriangleNumber()

static int TriangleNumber ( int index )

static

Calculates the i^th triangle number which is i * (i+1) / 2

Parameters

index The index of the triangle number to calculate.

Returns: The i^th triangle number

Member Data Documentation

◆ TwoPi

const double TwoPi

static

Constant value of \(2 \cdot \pi\).

Static Public Member Functions

Static Public Attributes

Detailed Description

Member Function Documentation

◆ Abs()

◆ Cos()

◆ DegreeToRadian()

◆ DrawAllSubsetsOfGivenSizeFromSetOfIndices()

◆ Factorial()

◆ GetQuantizationLevel()

◆ GetQuantizedValue()

◆ IndexOfTriangleNumber()

◆ InitQuantizationMinMaxRiser()

◆ InitQuantizationMinMaxTread()

◆ NumericDerivative()

◆ Pow()

◆ QuantizePoint() [1/2]

◆ QuantizePoint() [2/2]

◆ RadianToDegree()

◆ RMS() [1/2]

◆ RMS() [2/2]

◆ RombergIntegral()

◆ Sin()

◆ SolveQuadraticEquation()

◆ SphericalPhase()

◆ SphericalPhaseDerivative()

◆ SphericalPhaseFunction()

◆ SymDefSolve()

◆ Tan()

◆ ToThePowerOf()

◆ TriangleNumber()

Member Data Documentation

◆ TwoPi