private static double IntegrateSimpsonsRule(FunctionOneVar f, double low, double high, int steps)
{
double retval = 0;
double stepSize = (high - low) / (double)steps;
double leftVal = 0;
double midVal = 0;
double rightVal = 0;
leftVal = f(low);
if (steps % 2 == 1)
{
steps++;
}
for (int i = 2; i <= steps; i += 2)
{
double xmid = low + (i - 1) * stepSize;
double xright = low + i * stepSize;
midVal = f(xmid);
rightVal = f(xright);
retval += (leftVal + 4 * midVal + rightVal) * stepSize / 3.0;
leftVal = rightVal;
}
return(retval);
}
public static double Derivative(FunctionOneVar f, double x, double step)
{
double val = f(x);
if (double.IsNaN(val) || double.IsInfinity(val))
return double.NaN;
double valRight = f(x + step);
double valLeft = f(x - step);
// if they're both NaN, we can't do anything.
if (double.IsNaN(valRight) && double.IsNaN(valLeft))
return double.NaN;
// if only one is NaN, we can still do a first order derivative.
else if (double.IsNaN(valLeft) || double.IsInfinity(valLeft))
{
valLeft = val;
return (valRight - valLeft) / step;
}
else if (double.IsNaN(valRight) || double.IsInfinity(valRight))
{
valRight = val;
return (valRight - valLeft) / step;
}
// wel, everything seems good here:
else
return (valRight - valLeft) / (2 * step);
}
/// <summary>
/// Finds the point at which given function is at the given value, using the bisection algorithm.
/// </summary>
/// <param name="f">The function whose root we are supposed to find.</param>
/// <param name="low">The lower end of the range within to look for a root.</param>
/// <param name="high">The upper end of the range within to look for a root.</param>
/// <param name="value">The value to look for.</param>
/// <returns></returns>
public double ValueByBisection(FunctionOneVar f, double low, double high, double value)
{
mFunctionToCall = f;
mFunctionValue = value;
double retval = s_RootByBisection(pValueFunction, low, high, mTolerance, mMaxIterations);
mFunctionToCall = null;
mFunctionValue = 0;
return(retval);
}
/// <summary>
/// Finds the root of a passed function. Returns the value of x for which f(x) = 0.
/// </summary>
/// <param name="f">The function whose root we are supposed to find.</param>
/// <param name="low">The lower end of the range within to look for a root.</param>
/// <param name="high">The upper end of the range within to look for a root.</param>
/// <param name="maxIterations">The maximum number of iterations in which to look.</param>
/// <param name="tolerance">The level of tolerance at which the result should differ from the
/// correct value. The actual value used is (tolerance * (f(high) - f(low) / 2)). </param>
/// <returns></returns>
public static double s_RootByBisection(FunctionOneVar f, double low, double high, double tolerance, int maxIterations)
{
double x = (low + high) / 2.0;
double f_val = f(x);
double f_low = f(low);
while (double.IsNaN(f_low))
{
low += (high - low) / 1000.0;
f_low = f(low);
}
double f_high = f(high);
while (double.IsNaN(f_high))
{
high -= (high - low) / 1000.0;
f_high = f(high);
}
double tol = Math.Abs(tolerance * (f_high - f_low) / 2);
if (f_low * f_high >= 0)
{
throw new Exception("Error: Range given is not guaranteed to bracket a zero.");
}
int nIterations = 0;
do
{
f_val = f(x);
if (f_val * f_low < 0)
{
high = x;
f_high = f_val;
}
else if (f_val * f_high < 0)
{
low = x;
f_low = f_val;
}
x = (low + high) / 2.0;
nIterations++;
} while (nIterations < maxIterations && Math.Abs(f_val) > tol);
return(x);
}
private static double IntegrateTrapezoidRule(FunctionOneVar f, double low, double high, int steps)
{
double retval = 0;
double stepSize = (high - low) / (double)steps;
double leftVal = 0;
double rightVal = 0;
leftVal = f(low);
for (int i = 1; i <= steps; i++)
{
double x = low + i * stepSize;
rightVal = f(x);
retval += 0.5 * (rightVal + leftVal) * stepSize;
leftVal = rightVal;
}
return(retval);
}
public static double Derivative(FunctionOneVar f, double x, double step)
{
double val = f(x);
if (double.IsNaN(val) || double.IsInfinity(val))
{
return(double.NaN);
}
double valRight = f(x + step);
double valLeft = f(x - step);
// if they're both NaN, we can't do anything.
if (double.IsNaN(valRight) && double.IsNaN(valLeft))
{
return(double.NaN);
}
// if only one is NaN, we can still do a first order derivative.
else if (double.IsNaN(valLeft) || double.IsInfinity(valLeft))
{
valLeft = val;
return((valRight - valLeft) / step);
}
else if (double.IsNaN(valRight) || double.IsInfinity(valRight))
{
valRight = val;
return((valRight - valLeft) / step);
}
// wel, everything seems good here:
else
{
return((valRight - valLeft) / (2 * step));
}
}
/// <summary>
/// Finds the root of the given function using the bisection algorithm.
/// </summary>
/// <param name="f">The function whose root we are supposed to find.</param>
/// <param name="low">The lower end of the range within to look for a root.</param>
/// <param name="high">The upper end of the range within to look for a root.</param>
/// <returns></returns>
public double RootByBisection(FunctionOneVar f, double low, double high)
{
return(s_RootByBisection(f, low, high, mTolerance, mMaxIterations));
}
public static double Integrate(FunctionOneVar f, double low, double high, int steps)
{
return(IntegrateSimpsonsRule(f, low, high, steps));
//return IntegrateTrapezoidRule(f, low, high, steps);
}
public static double s_MinimumByBisection(FunctionOneVar f, double x_low, double x_high, double tolerance, int maxIterations)
{
double x_center = (x_low + x_high) / 2.0;
double f_center = f(x_center);
double f_low = f(x_low);
double f_high = f(x_high);
double tol = tolerance * (f_high - f_low) / 2;
int nIterations = 0;
// check to see if we've failed to bracket a minimum.
while (nIterations < maxIterations && f_center > f_low)
{
x_high = x_center;
f_high = f_center;
x_center = (x_low + x_high) / 2.0;
f_center = f(x_center);
nIterations++;
}
while (nIterations < maxIterations && f_center > f_high)
{
x_low = x_center;
f_low = f_center;
x_center = (x_low + x_high) / 2.0;
f_center = f(x_center);
nIterations++;
}
do
{
// examine right bracket
double x = (x_high + x_center) / 2;
double f_val = f(x);
if (f_center < f_val)
{
x_high = x;
f_high = f_val;
}
else
{
x_low = x_center;
x_center = x;
f_low = f_center;
f_center = f_val;
}
// examine left bracket
x = (x_low + x_center) / 2;
f_val = f(x);
if (f_center < f_val)
{
x_low = x;
f_low = f_val;
}
else
{
x_high = x_center;
x_center = x;
f_high = f_center;
f_center = f_val;
}
nIterations++;
} while (nIterations < maxIterations && Math.Abs(f_high - f_low) > tol);
return(x_center);
}
/// <summary>
/// Finds the root of a passed function. Returns the value of x for which f(x) = 0.
/// </summary>
/// <param name="f">The function whose root we are supposed to find.</param>
/// <param name="low">The lower end of the range within to look for a root.</param>
/// <param name="high">The upper end of the range within to look for a root.</param>
/// <param name="maxIterations">The maximum number of iterations in which to look.</param>
/// <param name="tolerance">The level of tolerance at which the result should differ from the
/// correct value. The actual value used is (tolerance * (f(high) - f(low) / 2)). </param>
/// <returns></returns>
public static double s_RootByBisection(FunctionOneVar f, double low, double high, double tolerance, int maxIterations)
{
double x = (low + high) / 2.0;
double f_val = f(x);
double f_low = f(low);
while (double.IsNaN(f_low))
{
low += (high - low) / 1000.0;
f_low = f(low);
}
double f_high = f(high);
while (double.IsNaN(f_high))
{
high -= (high - low) / 1000.0;
f_high = f(high);
}
double tol = Math.Abs(tolerance * (f_high - f_low) / 2);
if (f_low * f_high >= 0)
throw new Exception("Error: Range given is not guaranteed to bracket a zero.");
int nIterations = 0;
do
{
f_val = f(x);
if (f_val * f_low < 0)
{
high = x;
f_high = f_val;
}
else if (f_val * f_high < 0)
{
low = x;
f_low = f_val;
}
x = (low + high) / 2.0;
nIterations++;
} while (nIterations < maxIterations && Math.Abs(f_val) > tol) ;
return x;
}
public static double Integrate(FunctionOneVar f, double low, double high, int steps)
{
return IntegrateSimpsonsRule(f, low, high, steps);
//return IntegrateTrapezoidRule(f, low, high, steps);
}
private static double IntegrateTrapezoidRule(FunctionOneVar f, double low, double high, int steps)
{
double retval = 0;
double stepSize = (high - low) / (double)steps;
double leftVal = 0;
double rightVal = 0;
leftVal = f(low);
for (int i = 1; i <= steps; i++)
{
double x = low + i * stepSize;
rightVal = f(x);
retval += 0.5 * (rightVal + leftVal) * stepSize;
leftVal = rightVal;
}
return retval;
}
private static double IntegrateSimpsonsRule(FunctionOneVar f, double low, double high, int steps)
{
double retval = 0;
double stepSize = (high - low) / (double)steps;
double leftVal = 0;
double midVal = 0;
double rightVal = 0;
leftVal = f(low);
if (steps % 2 == 1)
steps++;
for (int i = 2; i <= steps; i += 2)
{
double xmid = low + (i - 1) * stepSize;
double xright = low + i * stepSize;
midVal = f(xmid);
rightVal = f(xright);
retval += (leftVal + 4 * midVal + rightVal) * stepSize / 3.0;
leftVal = rightVal;
}
return retval;
}
/// <summary>
/// Finds the point at which given function is at the given value, using the bisection algorithm.
/// </summary>
/// <param name="f">The function whose root we are supposed to find.</param>
/// <param name="low">The lower end of the range within to look for a root.</param>
/// <param name="high">The upper end of the range within to look for a root.</param>
/// <param name="value">The value to look for.</param>
/// <returns></returns>
public double ValueByBisection(FunctionOneVar f, double low, double high, double value)
{
mFunctionToCall = f;
mFunctionValue = value;
double retval = s_RootByBisection(pValueFunction, low, high, mTolerance, mMaxIterations);
mFunctionToCall = null;
mFunctionValue = 0;
return retval;
}
/// <summary>
/// Finds the root of the given function using the bisection algorithm.
/// </summary>
/// <param name="f">The function whose root we are supposed to find.</param>
/// <param name="low">The lower end of the range within to look for a root.</param>
/// <param name="high">The upper end of the range within to look for a root.</param>
/// <returns></returns>
public double RootByBisection(FunctionOneVar f, double low, double high)
{
return s_RootByBisection(f, low, high, mTolerance, mMaxIterations);
}
public double MinimumByBisection(FunctionOneVar f, double low, double high)
{
return s_MinimumByBisection(f, low, high, Tolerance, MaxIterations);
}
public double MinimumByBisection(FunctionOneVar f, double low, double high)
{
return(s_MinimumByBisection(f, low, high, Tolerance, MaxIterations));
}
public static double s_MinimumByBisection(FunctionOneVar f, double x_low, double x_high, double tolerance, int maxIterations)
{
double x_center = (x_low + x_high) / 2.0;
double f_center = f(x_center);
double f_low = f(x_low);
double f_high = f(x_high);
double tol = tolerance * (f_high - f_low) / 2;
int nIterations = 0;
// check to see if we've failed to bracket a minimum.
while (nIterations < maxIterations && f_center > f_low)
{
x_high = x_center;
f_high = f_center;
x_center = (x_low + x_high) / 2.0;
f_center = f(x_center);
nIterations++;
}
while (nIterations < maxIterations && f_center > f_high)
{
x_low = x_center;
f_low = f_center;
x_center = (x_low + x_high) / 2.0;
f_center = f(x_center);
nIterations++;
}
do
{
// examine right bracket
double x = (x_high + x_center) / 2;
double f_val = f(x);
if (f_center < f_val)
{
x_high = x;
f_high = f_val;
}
else
{
x_low = x_center;
x_center = x;
f_low = f_center;
f_center = f_val;
}
// examine left bracket
x = (x_low + x_center) / 2;
f_val = f(x);
if (f_center < f_val)
{
x_low = x;
f_low = f_val;
}
else
{
x_high = x_center;
x_center = x;
f_high = f_center;
f_center = f_val;
}
nIterations++;
} while (nIterations < maxIterations && Math.Abs(f_high - f_low) > tol);
return x_center;
}