CSharpNumerics

A numerical package that deals with scientific computing and mathematical analysis of discretizations and iterative processes https://www.nuget.org/packages/CSharpNumerics/

Numeric Extensions

Get the Factorial of int

Factorial(this int number)

E.g

5.Factorial()

Outputs 120

Findining roots using Newton–Raphson method

Func<double, double> func = (double x) => Math.Pow(x,2) - 4

func.NewtonRaphson()

Outputs 2

Derivative

To derivate a function use:

Derivate(this Func<double, double> func, double variablevalue, int order=1)

Calculate higher order derivative by setting the order parameter

E.g with Chain rule

double funcG(double x) => 4 * x - 3

Func<double, double> funcF=(double x) => Math.Pow(x, 2)

var result = funcF.Derivate(funcG,1)

By using Numerics.Enums.DerivateOperator the Chain rule, Product rule, or Quotient rule can be used

var result = funcF.Derivate(funcG,Numerics.Enums.DerivateOperator.Product)

var result = funcF.Derivate(funcG,Numerics.Enums.DerivateOperator.Quotient)

If several variables use:

Derivate(this Func<double[], double> func, double[] variables, int index, int order=1)

Or use the vector (x,y,z)

Derivate(this Func<Vector, double> func, Vector variables, Cartesian cartesian, int order=1)

Also possible derivate series

Func<double, double> displacement = (double time) => 9.81 * Math.Pow(time, 2) / 2

var velocity = displacement.GetSeries(0, 10, 1000).Derivate()

Integrals

To integrate a function with Trapezoidal rule use:

Integrate(this Func<double, double> func, double lowerLimit, double upperLimit)

To integrate a timeserie

Integrate(this List<Numerics.Models.TimeSerie> data)

TimeSerie is a model with the properties TimeStamp as DateTime and Value as double

To integrate a serie

Integrate(this List<Numerics.Models.Serie> data)

Serie model is a model with the properties Index as double and Value as double

Monte Carlo Integration

To solve double integrals with Monte Carlo method use:

Integrate(this Func<(double x, double y), double> func, (double lowerLimit, double upperLimit) xlimit, (double lowerLimit, double upperLimit) ylimit)

or triple integral

Integrate(this Func<Vector, double> func, Vector lowerLimit, Vector upperLimit)

The complex object

To work with Complex numbers use this struct:

ComplexNumber(double re, double im)

E.g Arithmetics

var a = new ComplexNumber(3, 2);

var b = new ComplexNumber(5, 3);

var sum= a + b;

var difference =a-b;

var product =a*b;

var quotient= a / b;

Power of complex number

var i = new ComplexNumber(3, 2); i.Pow(2);

output: 5+12*i

Calculate Imaginary exponents

var i = new ComplexNumber(0, Math.PI); i.Exponential()

output: -1

The vector object

To work with vectors use this struct:

Vector(double x, double y, double z)

or from two points

Vector((double,double, double) p1, (double, double, double) p2)

Following methods could be used:

1. Scalar product

Dot(Vector b)

2. Vector product

Cross(Vector b)

3. Projection between two vectors

Projection(Vector b)

4. Reflection between two vectors

Reflection(Vector b)

Using Sperical Coordinates

var v=Vector.FromSphericalCoordinates(radius, inclination, azimuth)

or covert to sperical from cartesian

v.ToSphericalCoordinates()

Metods to get radius, inclination, azimuth

GetMagnitude(), GetInclination(),GetAzimuth()

E.g

var a = new Vector(5, 3, 0);

var b = new Vector(2, 6, 0);

var skalar = a.Dot(b);

var vector = a.Cross(b);

E.g Arithmetics

var a = new Vector(2, 2, 0);

var b = new Vector(2, 2, 0);

var sum= a + b;

var difference =a-b;

var product =3*b;

The matrix object

To work with matrix use this struct:

Matrix(double[,] values)

E.g

var matrix = new Matrix(new double[,] { { 1, 3, 7 }, { 5, 2, 9 } });

Get Transpose

var transposematrix = matrix.Transpose();

Get Inverse

var inv = matrix.Inverse();

Get Pascal matrix

var pascalMatrix = new Matrix(new double[6, 6]).Pascal()

Get Adjugate

var adj = matrix.Adjugate();

Get Determinant

var det = matrix.Determinant()

output values

matrix.values

or Identity matrix

matrix.identity

E.g Arithmetics

var a = new Matrix(new double[,] { { 5, 7, 2 }, { -2, 9, 4 } });

var b = new Matrix(new double[,] {{ 1, 3, 7 }, { 5, 2, 9} });

var sum= a + b;

var difference =a-b;

var product =a*b;

Or

var b = 3

var product =b*a;

var quotient= a / b;

Or

var a = new Matrix(new double[,] { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 }});

var b = new Vector(2, 1, 3);

var product =a*b;

The vectorfield object

Gradient

Calculate for one point

Func<Vector, double> func = (Vector p) => Math.Pow(p.x, 2) * Math.Pow(p.y, 3);

var v=func.Gradient((1, -2, 0))

Calculate for range

var grad = func.Gradient(-4, -4, -4, 1, 8);

Where the parameters are minimum value of x,y,z the step size, and the length. The range in this example is -4<=x<=4,-4<=y<=4,-4<=z<=4. This method will return new Dictionary<Vector, Vector> where the key is the point and value is the calculated function value for that point. Save the data to csv

grad.Save(@"${path}\${file}.csv");

Divergence

Calculate for one point

double fx(Vector p) => Math.Sin(p.x * p.y);

double fy(Vector p) => Math.Cos(p.x * p.y);

double fz(Vector p) => Math.Pow(Math.E, p.z);

var field = new VectorField(fx, fy, fz);

var div = field.Divergence((1, 2, 2))

Curl

Calculate for one point

double fx(Vector p) => 4*p.z;

double fy(Vector p) => p.y *Math.Pow(p.x,3);

double fz(Vector p) => p.z * Math.Pow(p.y,2);

var field = new VectorField(fx, fy, fz);

var v = field.Curl((1, 4, 2));

Calculate for range. It is done in same way as for gradient E.g save both the vector field and curl to file:

double fx(Vector p) => p.y;

double fy(Vector p) => -p.x;

double fz(Vector p) => 0;

var w = new VectorField(fx, fy, fz);

var data= w.EvaluateRange(-4,-4,4,1,8);

var curl = w.Curl(-4, -4, 4, 1, 8);

data.Save(@"${path}\${file}.csv");

curl.Save(@"${path}\${file}.csv");

Laplacian

Calculate for one point

Func<Vector, double> func = (Vector p) => Math.Pow(p.x, 2) * Math.Pow(p.y, 3);

var v=func.Laplacian((1, -2, 0))

The complex function object

To work with Complex functions use this struct:

ComplexFunction(Func<(double x, double y), double> re, Func<(double x, double y), double> im)

E.g:

double fx((double x, double y) p) => Math.Pow(Math.E, p.x) * Math.Cos(p.y);

double fy((double x, double y) p) => Math.Pow(Math.E, p.x) * Math.Sin(p.y);

var fz = new ComplexFunction(fx, fy);

or

ComplexNumber fz(ComplexNumber z) => new ComplexNumber(Math.Pow(Math.E, z.realPart) * Math.Cos(z.imaginaryPart), Math.Pow(Math.E, z.realPart) * Math.Sin(z.imaginaryPart));

To derivate a complex function use:

Derivate(this ComplexFunction func, ComplexNumber variables, int order = 1)

Cauchy–Riemann equations

To test if analytic fuction in a point using Cauchy–Riemann equations:

fz.IsAnalytical((x0,y0))

Jacobian

To get the Jacobian as a Matrix

fz.Jacobian((x0,y0))

Transform

E.g use a lowpass filter to remove noise from a signal

var result =input.LowPassFilter(output).ToList()

Fast Fourier transform

To use a fast fourier transform use extentionsmethods from a list of complexnumber

FastFouriertransform(this List<ComplexNumber> numbers)

to calculate the inverse fast fourier transform

List<ComplexNumber> InverseFastFouriertransform(this List<ComplexNumber> numbers)

E.g Convert a Gaussian pulse from the time domain to the frequency domain and save result.

Func<double, double> func = (double t) => 1 / (4 * Math.Sqrt(2 * Math.PI * 0.01)) * (Math.Exp(-t * t / (2 * 0.01)));

var timeseries = func.GetSeries(-0.5, 0.5, 100);

timeseries.Save(@"\timeserie.csv");

GetSeries takes the interval and how many values to return

var frequency = func.FastFouriertransform(-0.5, 0.5, 100).ToFrequencyResolution(100);

frequency.Save(@"\frequency.csv");

ToFrequencyResolution takes the sample rate and will return the frequency as index and the magnitude of the complex number as value

Discrete Fourier transform

Use in the same way as a fast fourier transform

DiscreteFourierTransform(this List<ComplexNumber> numbers)

LaplaceTransform

Calculate the laplace transform for s value

LaplaceTransform(this Func<double, double> func, double s)

or it's invers from t value

InverseLaplaceTransform(this Func<double, double> func, double t)

Differential Equations

Runge–Kutta

The Runge–Kutta (R4) method uses this extension method

RungeKutta(this Func<(double t, double y), double> func, double min, double max, double stepSize, double yInitial)

E.g yprim =tan(y)+1 with the initial-value problem y0=1 and 1<= t <= 1.1 and step size 0.025

Func<(double y, double t), double> func = ((double t, double y) v) => Math.Tan(v.y) +1

var result = func.RungeKutta(1,1.1,0.025,1)

It is also possible using Explicit Runge–Kutta methods by defining the Runge–Kutta matrix, weights and nodes

var result = func.RungeKutta(1,1.1,0.025,1,new Matrix(new double[,] { { 0, 0, 0 }, { 0.5, 0, 0 }, { 0, 0.5, 0 }, { 0, 0, 1 } }), new double[] { 1.0 / 6.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 6.0 }, new double[] { 0.0, 0.5, 0.5, 1 })

Or solve ode by using the Trapezoidal rule

var result = func.TrapezoidalRule(1, 1.1, 0.00025, 1);

Matrix differential equation

Extension methods to solve linear equation system

LinearSystemSolver(this Matrix matrix, Vector vector)

or gauss elimination

GaussElimination(this Matrix matrix, Vector vector)

for N values

GaussElimination(this Matrix matrix, List<double> vector)

Find eigen values of matrix

var result = matrix.EigenValues()

Find Eigenvector if knowing a eigenvalue of a matrix (in this example 1)

var result =matrix.EigenVector(1)

To solve a linear system of differential equations use OdeSolver with initial value y(0)=x(0)=tZero when t=0

List<Func<double,double>> OdeSolver(this Matrix matrix, double tZero)

Statistics

Generate zero-mean white noise with a variance of 4 using Random

var rnd = new Random()

rnd.GenerateNoise(4)

To calculate the median use linq in the same way as calculating avarerage, sum, max or min

timeseries.Median(p => p.Value)

To calculate the standard deviation

timeseries.StandardDeviation(p => p.Value)

To calculate the variance

timeseries.Variance(p => p.Value)

To calculate the covariance if model has X,Y properties

series.Covariance(p => (p.X,p.Y))

There is also a Statistics class containing static methods

E.g get normal distribution curve

Numerics.Methods.Statistics.NormalDistribution(variance, mean)

Extension method for calculating cumulative sum of list

CumulativeSum<T>(this IEnumerable<T> enumerable, Func<T, double> func)

Interpolation

Linear interpolation of a timeserie

LinearInterpolationTimeSerie(this IEnumerable<TimeSerie> ts, DateTime timeStamp)

Or serie

LinearInterpolation<T>(this IEnumerable<T> ts, Func<T, (double x, double y)> func, double index)

E.g

var value = serie.LinearInterpolation(p=>(p.Index, p.Value),1)

Regression

Linear regression that will return intercept correlation and slope

LinearRegression<T>(this IEnumerable<T> enumerable, Func<T, (double x, double y)> func)

E.g

var serie = new List<Serie>() { new Serie() { Index = 3.0, Value = 0.62}, new Serie() { Index = 3.4, Value = 0.93 }, new Serie() { Index = 3.8, Value = 1.08 }};

var (slope, intercept, correlation) = serie.LinearRegression(p=>(p.Index, p.Value))

Exponetial regression that will return a exponetial function

var func = serie.ExponentialRegression(p => (p.Index, p.Value));

Logistic regression using slope and intercept

LogisticRegression<T>(this IEnumerable<T> enumerable, Func<T, (double x, double y)> func, double slope, double intercept)

Calculate Confidence Intervals

var (lower,upper) = timeserie.ConfidenceIntervals(p => p.Value, 0.95)

Get K nearest neighbors

var timeserie = new List<(double x, double y, int classification)>() { (7, 7, 0), (7, 4, 0), (3, 4, 1), (1, 4, 1) }

var classification = timeserie.KnearestNeighbors(p=> (p.x, p.y, p.classification),(3,7),3)