//method to implement richardson extrapolation with error control
public void richardson_extrapolation_errorcontrol(funcref thefunc, double x, double h, int maxiter, double toler)
{
double deriv = 0;
double derivold = 0;
double err = 0;
double errold = 1000000000;
int itercount = 0;
int count = 0;
//firstly ensure that the "Richardsonoutput" list is empty
if (Richardsonoutput.Count() > 0)
Richardsonoutput.Clear();
//next run the algorithm and populate the l8ist
for (count = 1; count <= maxiter; count++)
{
//get the derivative estimate at iteration "count"
deriv = richardson_extrapolation(thefunc, x, h, count);
err = Math.Abs(deriv - derivold);
//is the error less than a specified tolerance
if (err < toler)
{
itercount = count;
Richardsonoutput.Add(deriv);
Richardsonoutput.Add(err);
Richardsonoutput.Add((double)count);
break;
}
//is the latest error greatest than the previous error
if (err > errold)
{
deriv = derivold;
err = errold;
itercount = count - 1;
Richardsonoutput.Add(deriv);
Richardsonoutput.Add(err);
Richardsonoutput.Add((double)count);
}
derivold = deriv;
errold = err;
Richardsonoutput.Add(deriv);
Richardsonoutput.Add(err);
Richardsonoutput.Add((double)count);
}
}
//recursive method to implement Richardsons extrapolation
public double richardson_extrapolation(funcref thefunc, double x, double h, double j)
{
double dval1 = 0;
double dval2 = 0;
if (j == 1)
return (1 / (2 * h)) * (thefunc(x + h) - thefunc(x - h));
else
{
dval1 = richardson_extrapolation(thefunc, x, h / 2.0, j - 1);
dval2 = richardson_extrapolation(thefunc, x, h, j - 1);
return dval1 + (1 / (Math.Pow(4, (double)j - 1) - 1)) * (dval1 - dval2);
}
//end of richardson_extrapolation() method
}
//NewtonRaphson_Richardson_ErrorControl() method
public void NewtonRaphson_Richardson_ErrorControl(double tol, funcref thefunc)
{
//clear the roots list
if (Roots.Count() > 0)
Roots.Clear();
if (NRiterations.Count() > 0)
NRiterations.Clear();
//ensure that there are intervals upon which to apply the Newton Raphson method)
if (Right.Count() == 0)
{
Console.WriteLine("No intervals are given, hence we cannot run the Newton Raphson method.\n");
}
else
{
int z = 0;
//running Newton Raphson on each interval
foreach (double d in Right)
{
double upper = d;
double lower = Left[z];
double rinit = (upper + lower) / 2;
double rold = rinit;
double numer = 0, denom = 0, dx = 0;
double rnew = 0;
int counter = 0;
//we will use "counter" to ensure Newton Raphson does not loop infinitely
while (counter < 1000)
{
numer = thefunc(rold);
richardson_extrapolation_errorcontrol(thefunc, rold, 0.025, 20, 0.0001);
int k = Richardsonoutput.Count();
denom = Richardsonoutput[k - 3];
if (Double.IsNaN(denom))
{
Console.WriteLine("Stopping Newton Raphson in interval ({0},{1}) as f'[x] at {2} is not defined", Left[z], d, rold);
break;
}
if (denom == 0.0)
{
Console.WriteLine("Stopping Newton Raphson in interval ({0},{1}) as f'[x] at {2} equals 0", Left[z], d, rold);
break;
}
dx = numer / denom;
rnew = rold - dx;
//if the change "dx" is less than the tolerance, then we have found an estimate of the root
if (Math.Abs(dx) < Math.Abs(tol))
{
Roots.Add(rnew);
NRiterations.Add(counter + 1);
break;
}
rold = rnew;
//prevent an infinite loop
counter++;
if (counter > 1000)
{
Console.WriteLine("The Newton Raphson method has done more than 1000 iterations and is still not within the specified tolerance.We will now assume that the root is given by the last iteration\n");
Roots.Add(rnew);
NRiterations.Add(counter);
}
}
double val = rnew;
z++;
//end of Newton Raphson on interval
}
}
//end of NewtonRaphson() method
}
//bracketing() method
public void Bracketing(double u, double l, double intervals, funcref thefunc)
{
double upper = 0;
double lower = 0;
double dx = 0;
//firstly we need to ensure that the Right and Left lists are empty
if (Right.Count() > 0)
{
Right.Clear();
Left.Clear();
}
dx = (u - l) / intervals;
lower = l;
upper = lower + dx;
while (upper <= u)
{
//test the bracket
if (thefunc(lower) * thefunc(upper) < 0)
{
//there is a root in this interval so add the values to the relevants list
Left.Add(lower);
Right.Add(upper);
}
else
{
//do nothing
}
//move to the next bracket
lower = upper;
upper = upper + dx;
}
//end of bracketing() method
}
static void Main(string[] args)
{
Solver s = new Solver();
funcref f1 = new funcref(val => Math.Exp(val) + Math.Pow(2, -val) + 2 * Math.Sin(val) - 6);
funcref f2 = new funcref(val => 2 * val * Math.Cos(2 * val) - (val - 2) * (val - 2));
funcref f3 = new funcref(val => Math.Exp(val) - 3 * Math.Pow(val, 2));
funcref f = new funcref(val => Math.Exp(val) + Math.Pow(2, -val) + 2 * Math.Sin(val) - 6);
double linterval = 0;
double uinterval = 0;
//Here we allow the user to specify which one of the three equations they would like to find the roots of
string str = "";
int choice = 0;
Console.Write("Three functions.\n\nf1(x) = e^x+2^-x+2Sin(x)-6\nf2(x) = 2xCos(2x) - (x-2)^2\nf3(x) = e^x -3x^2\n\nPlease enter an integer between 1 and 3:");
str = Console.ReadLine();
bool b = int.TryParse(str, out choice);
while (!b || ((choice < 1) || (choice > 3)))
{
Console.Write("\nInvalid Input. Please renter an integer between 1 and 3: ");
str = Console.ReadLine();
b = int.TryParse(str, out choice);
}
//user chooses which function to examine
switch (choice)
{
case 1:
Console.Write("\nWe will work with function f1(x). Enter number of intervals:");
f = f1;
linterval = -4;
uinterval = 4;
break;
case 2:
Console.Write("\nWe will work with function f2(x).Enter number of intervals:");
f = f2;
linterval = 0;
uinterval = 4;
break;
case 3:
Console.Write("\nWe will work with function f3(x).Enter number of intervals:");
linterval = -2;
uinterval = 5;
f = f3;
break;
}
//user chooses the number of intervals
int intervals = 0;
str = Console.ReadLine();
b = int.TryParse(str, out intervals);
while (!b || ((intervals < 1) || (intervals > 1000)))
{
Console.Write("\nInvalid Input. Please enter an integer between 1 and 1000: ");
str = Console.ReadLine();
b = int.TryParse(str, out intervals);
}
//user chooses a tolerance level
Console.Write("\nNow enter the tolerance level:");
double tolerance = 0;
str = Console.ReadLine();
b = double.TryParse(str, out tolerance);
while (!b || (tolerance <= 0))
{
Console.Write("\nInvalid Input. Please enter a double >=0: ");
str = Console.ReadLine();
b = double.TryParse(str, out tolerance);
}
//bracketing
Console.Write("\nBracketing Output\n");
Console.Write("*****************\n");
s.Bracketing(uinterval, linterval, intervals, f);
s.print_bracketing_roots();
//Bisection
Console.Write("\nBisection Method\n");
Console.Write("*****************\n");
s.Bisection(tolerance, f);
s.print_roots();
//Newton Raphson
Console.Write("\nNewton Raphson Method\n");
Console.Write("*****************\n");
s.NewtonRaphson(tolerance, f);
s.print_roots_iterations();
int milliseconds = 3000;
Thread.Sleep(milliseconds);
//Richardson extrapolation with error control
//user chooses a tolerance level
Console.Write("\nRichardson Extrapolation with error Control\n");
Console.Write("*********************************************\n");
Console.Write("Provide a double at which you wish to evaluate the derivative:");
double x = 0;
str = Console.ReadLine();
b = double.TryParse(str, out x);
while (!b)
{
Console.Write("\nInvalid Input. Please enter a double: ");
str = Console.ReadLine();
b = double.TryParse(str, out x);
}
s.richardson_extrapolation_errorcontrol(f, x, 0.5, 40, tolerance);
s.print_richardson_output();
milliseconds = 5000;
Thread.Sleep(milliseconds);
//Advanced test
Console.Write("\n\n**********************************************************************");
Console.Write("\n**********************OPTICAL WAGEGUIDE*******************************\n");
Console.Write("\nBracketing Output\n");
Console.Write("*****************\n");
Waveguide w = new Waveguide();
funcref func = new funcref(w.Calc);
double lambda = 1.3, nc = 3.44, ns = 3.34;
double k0 = 2.0 * Math.PI / lambda;
linterval = k0 * ns;
uinterval = k0 * nc;
//bracketing
intervals = 20;
s.Bracketing(uinterval, linterval, intervals, func);
s.print_bracketing_roots();
milliseconds = 3000;
Thread.Sleep(milliseconds);
//Bisection
Console.Write("\nBiscection Output\n");
Console.Write("*****************\n");
s.Bisection(tolerance, func);
s.print_roots();
milliseconds = 3000;
Thread.Sleep(milliseconds);
//Newton Raphson Richardson Error Control
Console.Write("\nNewton Raphson Error Control Output\n");
Console.Write("*************************************\n");
s.NewtonRaphson_Richardson_ErrorControl(tolerance, func);
s.print_roots_iterations();
milliseconds = 3000;
Thread.Sleep(milliseconds);
Console.Write("\n\n*************************************\n");
Console.Write("End of Program\n");
Console.Write("*************************************\n");
Console.ReadLine();
}
//bisection() method
public void Bisection(double tol, funcref thefunc)
{
//clear the roots list
if (Roots.Count() > 0)
Roots.Clear();
//ensure that there are intervals upon which to apply the bisection method)
if (Right.Count() == 0)
{
Console.WriteLine("No intervals are given, hence we cannot run the bisection method.\n");
}
else
{
int z = 0;
foreach (double d in Right)
{
double upper = d;
double lower = Left[z];
int counter = 0;
//check upper and lower intervals are closer than the tolerance level
while ((upper - lower > tol) & (counter < 1000))
{
double m = (upper + lower) / 2;
double s = thefunc(lower) * thefunc(m);
if (s < 0)
{
upper = m;
}
else
{
lower = m;
}
//prevent an infinite loop
counter++;
if (counter > 1000)
Console.WriteLine("The Bisection method has done more than 1000 iterations and is still not within the specified tolerance.We will now assume that the root is given by the last iteration\n");
}
double val = (upper + lower) / 2;
Roots.Add(val);
z++;
}
}
//end of Bisection() method
}