public static double Solve()
{
a = 2;
b = 4;
Func <double, double> d = new Func <double, double>(ArcLength);
OneVariableFunction f = new OneVariableFunction(d);
double x = f.Integrate(0, 2 * Math.PI);
return(x);
}
public double Get_Equilibrium_Chemical_Potential()
{
OneVariableFunction charge_func = new OneVariableFunction(new Func <double, double>(Get_ChargeDensity));
// initialise root finder and search for a root for the chemical potential
RiddersRootFinder finder = new RiddersRootFinder();
double result = finder.Find(charge_func, -1.0 * band_gap, band_gap);
return(result);
}
public static double Solve()
{
Func <double, double> d = new Func <double, double>(function);
OneVariableFunction f = new OneVariableFunction(d);
double x = f.Integrate(0, Math.PI / 2);
double outVolume = -2 * Math.PI * x;
double inVolume = 4 * Math.PI * b * b * a / 3;
return(outVolume - inVolume);
}
double Interpolate_Fermi_Function_Derivative(DoubleHermitianEigDecomp eig_decomp_old, DoubleHermitianEigDecomp eig_decomp_new, int wavefunction)
{
double E0 = eig_decomp_old.EigenValue(wavefunction);
double dE_dk = (eig_decomp_new.EigenValue(wavefunction) - eig_decomp_old.EigenValue(wavefunction)) / delta_k;
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func <double, double>)((double k) => - 1.0 * beta * Math.Exp(beta * (E0 + dE_dk * k)) * Math.Pow(Math.Exp(beta * (E0 + dE_dk * k)) + 1, -2.0)));
dos_integrand.Integrator = new GaussKronrodIntegrator();
return(dos_integrand.Integrate(0, delta_k));
}
/// <summary>
/// calculates the density of electrons in the conduction band for a given chemical potential
/// </summary>
double Perform_Integral(Func <double, double> function, double lower_limit, double upper_limit)
{
if (chem_pot != double.MaxValue)
{
throw new Exception("Error - Something isn't reseting the chemical potential... this is very unsafe");
}
chem_pot = mu;
// integrates the density for a given energy up to a given number of kB * T above the conduction band edge
OneVariableFunction integrand = new OneVariableFunction(function);
GaussKronrodIntegrator tmp = new GaussKronrodIntegrator();
double result = tmp.Integrate(integrand, lower_limit, upper_limit);
// reset chem_pot
chem_pot = double.MaxValue;
return(result);
}
protected double Get_OneD_DoS_Deriv(double band_edge, double no_kb_T)
{
if (band_edge >= no_kb_T * Physics_Base.kB * temperature)
{
return(0.0);
}
else if (temperature == 0.0)
{
return(-1.0 * Math.Sqrt(-2.0 * mass / band_edge) / (Math.PI * Physics_Base.hbar));
}
else
{
// calculate the density of states integral directly
double alpha = 2.0 * Math.Sqrt(2.0 * mass) / (Math.PI * Physics_Base.hbar * Physics_Base.kB * temperature);
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func <double, double>)((double E) => Math.Sqrt(E - band_edge) * Math.Exp(beta * E) * Math.Pow(Math.Exp(beta * E) + 1, -3.0) * (beta * (Math.Exp(beta * E) + 1) - 2.0 * beta * Math.Exp(beta * E))));
return(Perform_DoS_Deriv_Integral(band_edge, no_kb_T, alpha, dos_integrand));
}
}
double Perform_DoS_Integral(double band_edge, double no_kb_T, double alpha, OneVariableFunction dos_integrand)
{
dos_integrand.Integrator = new GaussKronrodIntegrator();
// check to see if the band edge is less than a large tolerance... it should be
// physically, it is difficult for a band edge to dip far below the fermi surface as the charge flowing into the system should pin it
// here, we use a value of 0.1eV... integral_tol should never be greater than 1.0eV as this is roughly the band gap and holes/electrons
// should be induced
double integral_lower = -100.0;
if (band_edge > integral_lower)
{
integral_lower = band_edge;
}
double result = alpha * dos_integrand.Integrate(integral_lower, no_kb_T * Physics_Base.kB * temperature);
return(result);
}
protected double Get_TwoD_DoS(double band_edge, double no_kb_T)
{
if (band_edge >= no_kb_T * Physics_Base.kB * temperature)
{
return(0.0);
}
else if (temperature == 0.0)
{
return(-1.0 * band_edge * mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI));
}
else
{
// calculate the density of states integral directly
double alpha = mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI);
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func <double, double>)((double E) => 1.0 / (Math.Exp(beta * E) + 1)));
return(Perform_DoS_Integral(band_edge, no_kb_T, alpha, dos_integrand));
}
}
protected double Get_TwoD_DoS_Deriv(double band_edge, double no_kb_T)
{
if (band_edge >= no_kb_T * Physics_Base.kB * temperature)
{
return(0.0);
}
else if (temperature == 0.0)
{
return(mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI));
}
else
{
// calculate the derivative of the density of states integral directly
// NOTE: The 2D DoS is constant so this is just the integral of the derivative of the Fermi function
double alpha = mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI);
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func <double, double>)((double E) => beta * Math.Exp(beta * E) * Math.Pow(Math.Exp(beta * E) + 1, -2.0)));
return(Perform_DoS_Deriv_Integral(band_edge, no_kb_T, alpha, dos_integrand));
}
}
/// <summary>
/// calculates the density of electrons in the conduction band for a given chemical potential
/// </summary>
double Perform_Integral(Func<double, double> function, double lower_limit, double upper_limit)
{
if (chem_pot != double.MaxValue)
throw new Exception("Error - Something isn't reseting the chemical potential... this is very unsafe");
chem_pot = mu;
// integrates the density for a given energy up to a given number of kB * T above the conduction band edge
OneVariableFunction integrand = new OneVariableFunction(function);
GaussKronrodIntegrator tmp = new GaussKronrodIntegrator();
double result = tmp.Integrate(integrand, lower_limit, upper_limit);
// reset chem_pot
chem_pot = double.MaxValue;
return result;
}
public double Get_Equilibrium_Chemical_Potential()
{
OneVariableFunction charge_func = new OneVariableFunction(new Func<double, double>(Get_ChargeDensity));
// initialise root finder and search for a root for the chemical potential
RiddersRootFinder finder = new RiddersRootFinder();
double result = finder.Find(charge_func, -1.0 * band_gap, band_gap);
return result;
}
protected double Get_TwoD_DoS_Deriv(double band_edge, double no_kb_T)
{
if (band_edge >= no_kb_T * Physics_Base.kB * temperature)
return 0.0;
else if (temperature == 0.0)
return mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI);
else
{
// calculate the derivative of the density of states integral directly
// NOTE: The 2D DoS is constant so this is just the integral of the derivative of the Fermi function
double alpha = mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI);
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func<double, double>)((double E) => beta * Math.Exp(beta * E) * Math.Pow(Math.Exp(beta * E) + 1, -2.0)));
return Perform_DoS_Deriv_Integral(band_edge, no_kb_T, alpha, dos_integrand);
}
}
protected double Get_TwoD_DoS(double band_edge, double no_kb_T)
{
if (band_edge >= no_kb_T * Physics_Base.kB * temperature)
return 0.0;
else if (temperature == 0.0)
return -1.0 * band_edge * mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI);
else
{
// calculate the density of states integral directly
double alpha = mass / (Physics_Base.hbar * Physics_Base.hbar * 2.0 * Math.PI);
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func<double, double>)((double E) => 1.0 / (Math.Exp(beta * E) + 1)));
return Perform_DoS_Integral(band_edge, no_kb_T, alpha, dos_integrand);
}
}
protected double Get_OneD_DoS_Deriv(double band_edge, double no_kb_T)
{
if (band_edge >= no_kb_T * Physics_Base.kB * temperature)
return 0.0;
else if (temperature == 0.0)
return -1.0 * Math.Sqrt(-2.0 * mass / band_edge) / (Math.PI * Physics_Base.hbar);
else
{
// calculate the density of states integral directly
double alpha = 2.0 * Math.Sqrt(2.0 * mass) / (Math.PI * Physics_Base.hbar * Physics_Base.kB * temperature);
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func<double, double>)((double E) => Math.Sqrt(E - band_edge) * Math.Exp(beta * E) * Math.Pow(Math.Exp(beta * E) + 1, -3.0) * (beta * (Math.Exp(beta * E) + 1) - 2.0 * beta * Math.Exp(beta * E))));
return Perform_DoS_Deriv_Integral(band_edge, no_kb_T, alpha, dos_integrand);
}
}
double Perform_DoS_Deriv_Integral(double band_edge, double no_kb_T, double alpha, OneVariableFunction dos_integrand)
{
dos_integrand.Integrator = new GaussKronrodIntegrator();
if (band_edge < -1.0 * no_kb_T * Physics_Base.kB * temperature)
{
return(alpha * dos_integrand.Integrate(-1.0 * no_kb_T * Physics_Base.kB * temperature, no_kb_T * Physics_Base.kB * temperature));
}
else
{
return(alpha * dos_integrand.Integrate(band_edge, no_kb_T * Physics_Base.kB * temperature));
}
}
/// <summary></summary>
protected static void UpdateContinuousDistribution( ref ChartControl chart, ProbabilityDistribution dist, List<string> titles, DistributionFunction function, int numInterpolatedValues )
{
string xTitle = "x";
string yTitle;
double xmin = dist.InverseCDF( 0.0001 );
double xmax = dist.InverseCDF( 0.9999 );
OneVariableFunction f;
if( function == DistributionFunction.PDF )
{
yTitle = "Probability Density Function";
f = new OneVariableFunction( new Func<double, double> ( delegate( double x ) { return dist.PDF( x ); } ) );
}
else
{
yTitle = "Cumulative Distribution Function";
f = new OneVariableFunction( new Func<double, double> ( delegate( double x ) { return dist.CDF( x ); } ) );
}
ChartSeries series = BindXY( f, xmin, xmax, numInterpolatedValues, ChartSeriesType.Line, ChartSymbolShape.None );
Update( ref chart, series, titles, xTitle, yTitle );
}
double Interpolate_Fermi_Function_Derivative(DoubleHermitianEigDecomp eig_decomp_old, DoubleHermitianEigDecomp eig_decomp_new, int wavefunction)
{
double E0 = eig_decomp_old.EigenValue(wavefunction);
double dE_dk = (eig_decomp_new.EigenValue(wavefunction) - eig_decomp_old.EigenValue(wavefunction)) / delta_k;
double beta = 1.0 / (Physics_Base.kB * temperature);
OneVariableFunction dos_integrand = new OneVariableFunction((Func<double, double>)((double k) => -1.0 * beta * Math.Exp(beta * (E0 + dE_dk * k)) * Math.Pow(Math.Exp(beta * (E0 + dE_dk * k)) + 1, -2.0)));
dos_integrand.Integrator = new GaussKronrodIntegrator();
return dos_integrand.Integrate(0, delta_k);
}
double Perform_DoS_Deriv_Integral(double band_edge, double no_kb_T, double alpha, OneVariableFunction dos_integrand)
{
dos_integrand.Integrator = new GaussKronrodIntegrator();
if (band_edge < -1.0 * no_kb_T * Physics_Base.kB * temperature)
return alpha * dos_integrand.Integrate(-1.0 * no_kb_T * Physics_Base.kB * temperature, no_kb_T * Physics_Base.kB * temperature);
else
return alpha * dos_integrand.Integrate(band_edge, no_kb_T * Physics_Base.kB * temperature);
}
static void Main(string[] args)
{
//начало и конец отрезка
double a = 0, b = 1;
//число промежутков деления [a,b]
int m = 100;
double h = (b - a) / m;
//N для пп. 3)+4)
int N1 = 2;
//N для п. 5)
int N2 = 6;
//для пп. 3)+4)
double f1(double x)
{
//многочлен нулевой степени
//return 1;
//многочлен первой степени
//return x;
//многочлен третьей степени
//return Math.Pow(x, 3);
return(Math.Sin(x));
}
//для п. 5)
double f2(double x)
{
return(Math.Cos(x));
}
//вес
double r(double x)
{
return(Math.Pow(x, 0.25));
}
//первообразная
double Rk(double x, int k)
{
return(Math.Pow(x, k + 1.25) / (k + 1.25));
}
Console.WriteLine("Приближённое вычисление интегралов при помощи квадратурных формул\n" +
"Наивысшей Алгебраической Степени Точности");
Console.WriteLine("Вариант №2\n");
Console.WriteLine("[a, b] = [{0}, {1}], m = {2}", a, b, m);
Console.WriteLine("Хотите изменить параметры задачи?");
string answ = Console.ReadLine();
if (answ == "y")
{
Console.WriteLine("Введите параметр a:");
a = Convert.ToDouble(Console.ReadLine());
Console.WriteLine("Введите параметр b:");
b = Convert.ToDouble(Console.ReadLine());
Console.WriteLine("Введите параметр m:");
m = Convert.ToInt32(Console.ReadLine());
Console.WriteLine("Успешно изменёны!\n");
}
h = (b - a) / m;
OneVariableFunction F1 = new OneVariableFunction(x => f1(x) * r(x));
double J1 = F1.Integrate(a, b);
double J1_real = 0.406538;
OneVariableFunction F2 = new OneVariableFunction(x => f2(x) * (1 / Math.Sqrt(1 - x * x)));
double J2 = F2.Integrate(-1, 1);
double J2_real = 2.4039394306344129982733248925915112237;
Console.WriteLine("________________________________________________________________________________________________");
Console.WriteLine("\nf(x) = sin(x)");
Console.WriteLine("r(x) = x^(1/4)");
Console.WriteLine("N = {0}", N1);
double JcompoundQFGauss = compoundQFGauss(N1, m, a, b, h);
Console.WriteLine("\nСоставная КФ Гаусса с {0} узлами: {1}", N1, JcompoundQFGauss);
if (!Double.IsNaN(J1))
{
Console.WriteLine("Библиотека NMath {0}", Math.Abs(JcompoundQFGauss - J1));
}
if (a == 0 && b == 1)
{
Console.WriteLine("Wolfram Alpha {0}", Math.Abs(JcompoundQFGauss - J1_real));
}
Console.WriteLine("________________________________________________________________________________________________");
double JQFtypeGauss = QFtypeGauss(N1, a, b);
Console.WriteLine("\nКФ типа Гаусса с {0} узлами: {1}", N1, JQFtypeGauss);
if (!Double.IsNaN(J1))
{
Console.WriteLine("Библиотека NMath {0}", Math.Abs(JQFtypeGauss - J1));
}
if (a == 0 && b == 1)
{
Console.WriteLine("Wolfram Alpha {0}", Math.Abs(JQFtypeGauss - J1_real));
}
Console.WriteLine("________________________________________________________________________________________________");
Console.WriteLine("\nКвадратурная формула Мелера");
Console.WriteLine("\nf(x) = cos(x)");
Console.WriteLine("r(x) = 1/sqrt(1-x^2)");
Console.WriteLine("[a, b] = [-1, 1], N = {0}", N2);
Console.WriteLine("Хотите изменить параметр N для КФ Мелера?");
answ = Console.ReadLine();
if (answ == "y")
{
Console.WriteLine("Введите параметр N:");
N2 = Convert.ToInt32(Console.ReadLine());
Console.WriteLine("Успешно изменён\n");
}
double JMehler = Mehler(N2);
Console.WriteLine("\nКвадратурная формула Мелера с {0} узлами по [a, b] = [-1; 1]: {1}", N2, JMehler);
if (!Double.IsNaN(J2))
{
Console.WriteLine("Библиотека NMath {0}", Math.Abs(JMehler - J2));
}
Console.WriteLine("Wolfram Alpha {0}", Math.Abs(JMehler - J2_real));
double compoundQFGauss(int N, int M, double A, double B, double H)
{
double res = 0;
double z = A;
double t1 = 1 / Math.Sqrt(3);
double t2 = -1 / Math.Sqrt(3);
double x1 = 0;
double x2 = 0;
for (int i = 0; i < M; i++)
{
x1 = (t1 + 1) * H / 2 + z;
x2 = (t2 + 1) * H / 2 + z;
res += f1(x1) * r(x1) + f1(x2) * r(x2);
z += H;
}
res *= H / 2;
return(res);
}
double QFtypeGauss(int N, double A, double B)
{
Console.WriteLine("\nКФ типа Гаусса с {0} узлами", N1);
double [] mu = new double[2 * N];
Console.WriteLine("\nМоменты весовой функции");
for (int i = 0; i < mu.Length; i++)
{
mu[i] = Rk(B, i) - Rk(A, i);
Console.WriteLine("μ{0} = {1}", i, mu[i]);
}
double a1 = (mu[0] * mu[3] - mu[2] * mu[1]) / (mu[1] * mu[1] - mu[2] * mu[0]);
double a2 = (mu[2] * mu[2] - mu[3] * mu[1]) / (mu[1] * mu[1] - mu[2] * mu[0]);
Console.WriteLine("\nОртогональный многочлен x^2+{0}x+{1}=0\n", a1, a2);
//корни уравнения
double x1 = (-a1 + Math.Sqrt(a1 * a1 - 4 * a2)) / 2;
double x2 = (-a1 - Math.Sqrt(a1 * a1 - 4 * a2)) / 2;
Console.WriteLine("Узлы КФ: x1 = {0}, x2 = {1}", x1, x2);
if (x1 != x2 && x1 > A && x1 < B && x2 > A && x2 < B)
{
Console.WriteLine("Различны и принадлежат (a, b).");
}
else
{
Console.WriteLine("Узлаы не различны и/или не принадлежат (a, b).");
}
double A1 = (mu[1] - x2 * mu[0]) / (x1 - x2);
double A2 = (mu[1] - x1 * mu[0]) / (x2 - x1);
Console.WriteLine("\nКоэффициенты КФ: A1 = {0}, A2 = {1}", A1, A2);
if (A1 + A2 - mu[0] < 1E-10 && A1 * A2 > 0)
{
Console.WriteLine("Коэффициенты одного знака и A1 + A2 = μ0.");
}
double JGauss = A1 * f1(x1) + A2 * f1(x2);
return(JGauss);
}
double Mehler(int N)
{
double res = 0;
for (int k = 1; k <= N; k++)
{
res += f2(Math.Cos((2 * k - 1) * Math.PI / (2 * N)));
Console.WriteLine(Math.Cos((2 * k - 1) * Math.PI / (2 * N)));
}
res *= Math.PI / N;
return(res);
}
}
double Perform_DoS_Integral(double band_edge, double no_kb_T, double alpha, OneVariableFunction dos_integrand)
{
dos_integrand.Integrator = new GaussKronrodIntegrator();
// check to see if the band edge is less than a large tolerance... it should be
// physically, it is difficult for a band edge to dip far below the fermi surface as the charge flowing into the system should pin it
// here, we use a value of 0.1eV... integral_tol should never be greater than 1.0eV as this is roughly the band gap and holes/electrons
// should be induced
double integral_lower = -100.0;
if (band_edge > integral_lower)
integral_lower = band_edge;
double result = alpha * dos_integrand.Integrate(integral_lower, no_kb_T * Physics_Base.kB * temperature);
return result;
}
