MathFunctions Class Reference

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.
static double	Cos (double d)
	Special cosine function as the Math.Cos function has problems with with large values (in .Net Framework 4.8) and with \(\cos\left(\frac{\pi}{2}\right)\) in general.
static double	DegreeToRadian (double degree)
	Convert Degree to Radian.
static List< int[]>	DrawAllSubsetsOfGivenSizeFromSetOfIndices (int subsetSize, int populationSize)
	Method for creating all possible subsets of a given size out of a set of indices.
static double	Factorial (int m)
	Calculates the factorial \(n!\).
static int	GetQuantizationLevel (double value, double minimumQuantizedValue, double maximumQuantizedValue, double quantizationDifference)
	Gets the quantization level on which a certain value lies.
static double	GetQuantizedValue (double value, double minimumQuantizedValue, double maximumQuantizedValue, double quantizationDifference)
	Gets the quantized value.
static double	IndexOfTriangleNumber (int number)
	The ith 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.
static void	InitQuantizationMinMaxRiser (double minimum, double maximum, int numberOfQuantizationLevels, out double minimumQuantizedValue, out double maximumQuantizedValue, out double quantizationDifference)
	Initializes a "Min-Max-Riser" quantization.
static void	InitQuantizationMinMaxTread (double minimum, double maximum, int numberOfQuantizationLevels, out double minimumQuantizedValue, out double maximumQuantizedValue, out double quantizationDifference)
	Initializes a "Min-Max-Tread" quantization.
static double	NumericDerivative (Func< double, double > function, double position, double epsilon=Globals.Epsilon)
	Calculates the numerical derivative of a function.
static double	Pow (double baseNumber, double exponent)
	Raises the double number to the given power.
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, \(k = 0, \ldots, n-1\).
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, \(k = 0, \ldots, n-1\).
static double	RadianToDegree (double radian)
	Convert Degree to Radian.
static double	RMS (LargeList< double > values, bool relativeToAverage=false, LargeList< double > weights=null)
	Calculates the RMS (root mean square) of the given value.
static double	RMS (LargeList< double > values, double average)
	Calculates the RMS (root mean square) of the given value with respect to the given average.
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.
static double	Sin (double d)
	Special sine function as the Math.Sin function has problems with large values (in .Net Framework 4.8) and with \(\sin\left(\pi\right)\) in general.
static int	SolveQuadraticEquation (double a2, double a1, double a0, out double x1, out double x2)
	Function solving the general quadratic equation \(0 = a_2 \cdot x^2 + a_1 \cdot x + a_0\) .
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 \(a = \sqrt{(x^2 + y^2 + z^2)}\).
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 \(a = \sqrt{(x^2 + y^2 + z^2)}\).
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 \(a = \sqrt{(x^2 + y^2 + z^2)}\).
static void	SymDefSolve (double[,] matr, ref double[] vect)
	Solves the linear system matr * x = vect by using the Cholesky decomposition of matr.
static double	Tan (double d)
	Special tangent function as the Math.Tan function has problems with \(\tan\left(\frac{\pi}{2}\right)\) and \(\tan\left(\pi\right)\).
static double	ToThePowerOf (this double baseNumber, double exponent)
	Raises the double number to the given power.
static int	TriangleNumber (int index)
	Calculates the ith triangle number which is i * (i+1) / 2.

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

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 with large values (in .Net Framework 4.8) and with \(\cos\left(\frac{\pi}{2}\right)\) in general.

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 large values (in .Net Framework 4.8) and with \(\sin\left(\pi\right)\) in general.

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