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A numerical package that deals with scientific computing and mathematical analysis of discretizations and iterative processes https://www.nuget.org/packages/CSharpNumerics/
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
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()
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
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)
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
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:
- Scalar product
Dot(Vector b)
- Vector product
Cross(Vector b)
- Projection between two vectors
Projection(Vector b)
- 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;
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;
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(@"${path}\${file}.csv");
Calculate for one point
double fx(Vector p) => Math.Sin(p.x * p.y);
double fy(Vector p) => Math.Cos(p.x * p.y);
double fz(Vector p) => Math.Pow(Math.E, p.z);
var field = new VectorField(fx, fy, fz);
var div = field.Divergence((1, 2, 2))
Calculate for one point
double fx(Vector p) => 4*p.z;
double fy(Vector p) => p.y *Math.Pow(p.x,3);
double fz(Vector p) => p.z * Math.Pow(p.y,2);
var field = new VectorField(fx, fy, fz);
var v = field.Curl((1, 4, 2));
Calculate for range. It is done in same way as for gradient E.g save both the vector field and curl to file:
double fx(Vector p) => p.y;
double fy(Vector p) => -p.x;
double fz(Vector p) => 0;
var w = new VectorField(fx, fy, fz);
var data= w.EvaluateRange(-4,-4,4,1,8);
var curl = w.Curl(-4, -4, 4, 1, 8);
data.Save(@"${path}\${file}.csv");
curl.Save(@"${path}\${file}.csv");
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))
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)
To test if analytic fuction in a point using Cauchy–Riemann equations:
fz.IsAnalytical((x0,y0))
To get the Jacobian as a Matrix
fz.Jacobian((x0,y0))
E.g use a lowpass filter to remove noise from a signal
var result =input.LowPassFilter(output).ToList()
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
Use in the same way as a fast fourier transform
DiscreteFourierTransform(this List<ComplexNumber> numbers)
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)
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);
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)
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)
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)
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)