/*! This method returns the zero of the function \f$ f \f$, determined with the given accuracy \f$ \epsilon \f$
* depending on the particular solver, this might mean that the returned \f$ x \f$ is such that \f$ |f(x)| < \epsilon
* \f$, or that \f$ |x-\xi| < \epsilon \f$ where \f$ \xi \f$ is the real zero.
*
* An initial guess must be supplied, as well as two values \f$ x_\mathrm{min} \f$ and \f$ x_\mathrm{max} \f$ which
* must bracket the zero (i.e., either \f$ f(x_\mathrm{min}) \leq 0 \leq f(x_\mathrm{max}) \f$, or \f$
* f(x_\mathrm{max}) \leq 0 \leq f(x_\mathrm{min}) \f$ must be true).
*/
public double solve(ISolver1d f, double accuracy, double guess, double xMin, double xMax)
{
Utils.QL_REQUIRE(accuracy > 0.0, () => "accuracy (" + accuracy + ") must be positive");
// check whether we really want to use epsilon
accuracy = Math.Max(accuracy, Const.QL_EPSILON);
xMin_ = xMin;
xMax_ = xMax;
Utils.QL_REQUIRE(xMin_ < xMax_, () => "invalid range: xMin_ (" + xMin_ + ") >= xMax_ (" + xMax_ + ")");
Utils.QL_REQUIRE(!lowerBoundEnforced_ || xMin_ >= lowerBound_, () =>
"xMin_ (" + xMin_ + ") < enforced low bound (" + lowerBound_ + ")");
Utils.QL_REQUIRE(!upperBoundEnforced_ || xMax_ <= upperBound_, () =>
"xMax_ (" + xMax_ + ") > enforced hi bound (" + upperBound_ + ")");
fxMin_ = f.value(xMin_);
if (Utils.close(fxMin_, 0.0))
{
return(xMin_);
}
fxMax_ = f.value(xMax_);
if (Utils.close(fxMax_, 0.0))
{
return(xMax_);
}
evaluationNumber_ = 2;
Utils.QL_REQUIRE(fxMin_ * fxMax_ < 0.0, () =>
"root not bracketed: f[" + xMin_ + "," + xMax_ + "] -> [" + fxMin_ + "," + fxMax_ + "]");
Utils.QL_REQUIRE(guess > xMin_, () => "guess (" + guess + ") < xMin_ (" + xMin_ + ")");
Utils.QL_REQUIRE(guess < xMax_, () => "guess (" + guess + ") > xMax_ (" + xMax_ + ")");
root_ = guess;
return(solveImpl(f, accuracy));
}
protected override double solveImpl(ISolver1d f, double xAccuracy)
{
/* The implementation of the algorithm was inspired by
* Press, Teukolsky, Vetterling, and Flannery,
* "Numerical Recipes in C", 2nd edition, Cambridge
* University Press
*/
double fl, froot, dx, xl;
// Pick the bound with the smaller function value
// as the most recent guess
if (Math.Abs(fxMin_) < Math.Abs(fxMax_))
{
root_ = xMin_;
froot = fxMin_;
xl = xMax_;
fl = fxMax_;
}
else
{
root_ = xMax_;
froot = fxMax_;
xl = xMin_;
fl = fxMin_;
}
while (evaluationNumber_ <= maxEvaluations_)
{
dx = (xl - root_) * froot / (froot - fl);
xl = root_;
fl = froot;
root_ += dx;
froot = f.value(root_);
++evaluationNumber_;
if (Math.Abs(dx) < xAccuracy || Utils.close(froot, 0.0))
{
return(root_);
}
}
Utils.QL_FAIL("maximum number of function evaluations (" + maxEvaluations_ + ") exceeded",
QLNetExceptionEnum.MaxNumberFuncEvalExceeded);
return(0);
}
protected override double solveImpl(ISolver1d f, double xAccuracy)
{
/* The implementation of the algorithm was inspired by
* Press, Teukolsky, Vetterling, and Flannery,
* "Numerical Recipes in C", 2nd edition, Cambridge
* University Press
*/
double dx, xMid, fMid;
// Orient the search so that f>0 lies at root_+dx
if (fxMin_ < 0.0)
{
dx = xMax_ - xMin_;
root_ = xMin_;
}
else
{
dx = xMin_ - xMax_;
root_ = xMax_;
}
while (evaluationNumber_ <= maxEvaluations_)
{
dx /= 2.0;
xMid = root_ + dx;
fMid = f.value(xMid);
evaluationNumber_++;
if (fMid <= 0.0)
{
root_ = xMid;
}
if (Math.Abs(dx) < xAccuracy || Utils.close(fMid, 0.0))
{
return(root_);
}
}
Utils.QL_FAIL("maximum number of function evaluations (" + maxEvaluations_ + ") exceeded",
QLNetExceptionEnum.MaxNumberFuncEvalExceeded);
return(0);
}
protected override double solveImpl(ISolver1d f, double xAccuracy)
{
/* The implementation of the algorithm was inspired by Press, Teukolsky, Vetterling, and Flannery,
* "Numerical Recipes in C", 2nd edition, Cambridge University Press */
double min1, min2;
double froot, p, q, r, s, xAcc1, xMid;
// dummy assignements to avoid compiler warning
double d = 0.0, e = 0.0;
root_ = xMax_;
froot = fxMax_;
while (evaluationNumber_ <= maxEvaluations_)
{
if ((froot > 0.0 && fxMax_ > 0.0) ||
(froot < 0.0 && fxMax_ < 0.0))
{
// Rename xMin_, root_, xMax_ and adjust bounds
xMax_ = xMin_;
fxMax_ = fxMin_;
e = d = root_ - xMin_;
}
if (Math.Abs(fxMax_) < Math.Abs(froot))
{
xMin_ = root_;
root_ = xMax_;
xMax_ = xMin_;
fxMin_ = froot;
froot = fxMax_;
fxMax_ = fxMin_;
}
// Convergence check
xAcc1 = 2.0 * Const.QL_EPSILON * Math.Abs(root_) + 0.5 * xAccuracy;
xMid = (xMax_ - root_) / 2.0;
if (Math.Abs(xMid) <= xAcc1 || Utils.close(froot, 0.0))
{
return(root_);
}
if (Math.Abs(e) >= xAcc1 &&
Math.Abs(fxMin_) > Math.Abs(froot))
{
// Attempt inverse quadratic interpolation
s = froot / fxMin_;
if (Utils.close(xMin_, xMax_))
{
p = 2.0 * xMid * s;
q = 1.0 - s;
}
else
{
q = fxMin_ / fxMax_;
r = froot / fxMax_;
p = s * (2.0 * xMid * q * (q - r) - (root_ - xMin_) * (r - 1.0));
q = (q - 1.0) * (r - 1.0) * (s - 1.0);
}
if (p > 0.0)
{
q = -q; // Check whether in bounds
}
p = Math.Abs(p);
min1 = 3.0 * xMid * q - Math.Abs(xAcc1 * q);
min2 = Math.Abs(e * q);
if (2.0 * p < (min1 < min2 ? min1 : min2))
{
e = d; // Accept interpolation
d = p / q;
}
else
{
d = xMid; // Interpolation failed, use bisection
e = d;
}
}
else
{
// Bounds decreasing too slowly, use bisection
d = xMid;
e = d;
}
xMin_ = root_;
fxMin_ = froot;
if (Math.Abs(d) > xAcc1)
{
root_ += d;
}
else
{
root_ += Math.Abs(xAcc1) * Math.Sign(xMid);
}
froot = f.value(root_);
evaluationNumber_++;
}
Utils.QL_FAIL("maximum number of function evaluations (" + maxEvaluations_ + ") exceeded",
QLNetExceptionEnum.MaxNumberFuncEvalExceeded);
return(0);
}
/*! This method returns the zero of the function \f$ f \f$, determined with the given accuracy \f$ \epsilon \f$
* depending on the particular solver, this might mean that the returned \f$ x \f$ is such that \f$ |f(x)| < \epsilon
* \f$, or that \f$ |x-\xi| < \epsilon \f$ where \f$ \xi \f$ is the real zero.
*
* This method contains a bracketing routine to which an initial guess must be supplied as well as a step used to
* scan the range of the possible bracketing values.
*/
public double solve(ISolver1d f, double accuracy, double guess, double step)
{
Utils.QL_REQUIRE(accuracy > 0.0, () => "accuracy (" + accuracy + ") must be positive");
// check whether we really want to use epsilon
accuracy = Math.Max(accuracy, Const.QL_EPSILON);
const double growthFactor = 1.6;
int flipflop = -1;
root_ = guess;
fxMax_ = f.value(root_);
// monotonically crescent bias, as in optionValue(volatility)
if (Utils.close(fxMax_, 0.0))
{
return(root_);
}
else if (fxMax_ > 0.0)
{
xMin_ = enforceBounds_(root_ - step);
fxMin_ = f.value(xMin_);
xMax_ = root_;
}
else
{
xMin_ = root_;
fxMin_ = fxMax_;
xMax_ = enforceBounds_(root_ + step);
fxMax_ = f.value(xMax_);
}
evaluationNumber_ = 2;
while (evaluationNumber_ <= maxEvaluations_)
{
if (fxMin_ * fxMax_ <= 0.0)
{
if (Utils.close(fxMin_, 0.0))
{
return(xMin_);
}
if (Utils.close(fxMax_, 0.0))
{
return(xMax_);
}
root_ = (xMax_ + xMin_) / 2.0;
return(solveImpl(f, accuracy));
}
if (Math.Abs(fxMin_) < Math.Abs(fxMax_))
{
xMin_ = enforceBounds_(xMin_ + growthFactor * (xMin_ - xMax_));
fxMin_ = f.value(xMin_);
}
else if (Math.Abs(fxMin_) > Math.Abs(fxMax_))
{
xMax_ = enforceBounds_(xMax_ + growthFactor * (xMax_ - xMin_));
fxMax_ = f.value(xMax_);
}
else if (flipflop == -1)
{
xMin_ = enforceBounds_(xMin_ + growthFactor * (xMin_ - xMax_));
fxMin_ = f.value(xMin_);
evaluationNumber_++;
flipflop = 1;
}
else if (flipflop == 1)
{
xMax_ = enforceBounds_(xMax_ + growthFactor * (xMax_ - xMin_));
fxMax_ = f.value(xMax_);
flipflop = -1;
}
evaluationNumber_++;
}
Utils.QL_FAIL("unable to bracket root in " + maxEvaluations_
+ " function evaluations (last bracket attempt: " + "f[" + xMin_ + "," + xMax_ +
"] "
+ "-> [" + fxMin_ + "," + fxMax_ + "])",
QLNetExceptionEnum.RootNotBracketException);
return(0);
}
protected override double solveImpl(ISolver1d f, double xAcc)
{
/* The implementation of the algorithm was inspired by
* Press, Teukolsky, Vetterling, and Flannery,
* "Numerical Recipes in C", 2nd edition, Cambridge
* University Press
*/
double fxMid, froot, s, xMid, nextRoot;
// test on Black-Scholes implied volatility show that
// Ridder solver algorithm actually provides an
// accuracy 100 times below promised
double xAccuracy = xAcc / 100.0;
// Any highly unlikely value, to simplify logic below
root_ = double.MinValue;
while (evaluationNumber_ <= maxEvaluations_)
{
xMid = 0.5 * (xMin_ + xMax_);
// First of two function evaluations per iteraton
fxMid = f.value(xMid);
++evaluationNumber_;
s = Math.Sqrt(fxMid * fxMid - fxMin_ * fxMax_);
if (Utils.close(s, 0.0))
{
return(root_);
}
// Updating formula
nextRoot = xMid + (xMid - xMin_) *
((fxMin_ >= fxMax_ ? 1.0 : -1.0) * fxMid / s);
if (Math.Abs(nextRoot - root_) <= xAccuracy)
{
return(root_);
}
root_ = nextRoot;
// Second of two function evaluations per iteration
froot = f.value(root_);
++evaluationNumber_;
if (Utils.close(froot, 0.0))
{
return(root_);
}
// Bookkeeping to keep the root bracketed on next iteration
if (sign(fxMid, froot).IsNotEqual(fxMid))
{
xMin_ = xMid;
fxMin_ = fxMid;
xMax_ = root_;
fxMax_ = froot;
}
else if (sign(fxMin_, froot).IsNotEqual(fxMin_))
{
xMax_ = root_;
fxMax_ = froot;
}
else if (sign(fxMax_, froot).IsNotEqual(fxMax_))
{
xMin_ = root_;
fxMin_ = froot;
}
else
{
Utils.QL_FAIL("never get here.");
}
if (Math.Abs(xMax_ - xMin_) <= xAccuracy)
{
return(root_);
}
}
Utils.QL_FAIL("maximum number of function evaluations (" + maxEvaluations_ + ") exceeded",
QLNetExceptionEnum.MaxNumberFuncEvalExceeded);
return(0);
}
protected override double solveImpl(ISolver1d f, double xAccuracy)
{
// Orient the search so that f(xl) < 0
double xh, xl;
if (fxMin_ < 0.0)
{
xl = xMin_;
xh = xMax_;
}
else
{
xh = xMin_;
xl = xMax_;
}
double froot = f.value(root_);
++evaluationNumber_;
// first order finite difference derivative
double dfroot = xMax_ - root_ < root_ - xMin_ ?
(fxMax_ - froot) / (xMax_ - root_) :
(fxMin_ - froot) / (xMin_ - root_);
// xMax_-xMin_>0 is verified in the constructor
double dx = xMax_ - xMin_;
while (evaluationNumber_ <= maxEvaluations_)
{
double frootold = froot;
double rootold = root_;
double dxold = dx;
// Bisect if (out of range || not decreasing fast enough)
if ((((root_ - xh) * dfroot - froot) *
((root_ - xl) * dfroot - froot) > 0.0) ||
(Math.Abs(2.0 * froot) > Math.Abs(dxold * dfroot)))
{
dx = (xh - xl) / 2.0;
root_ = xl + dx;
}
else // Newton
{
dx = froot / dfroot;
root_ -= dx;
}
// Convergence criterion
if (Math.Abs(dx) < xAccuracy)
{
return(root_);
}
froot = f.value(root_);
++evaluationNumber_;
dfroot = (frootold - froot) / (rootold - root_);
if (froot < 0.0)
{
xl = root_;
}
else
{
xh = root_;
}
}
Utils.QL_FAIL("maximum number of function evaluations (" + maxEvaluations_ + ") exceeded",
QLNetExceptionEnum.MaxNumberFuncEvalExceeded);
return(0);
}
//! safe %Newton 1-D solver
/*! \note This solver requires that the passed function object
* implement a method <tt>Real derivative(Real)</tt>.
*/
protected override double solveImpl(ISolver1d f, double xAccuracy)
{
/* The implementation of the algorithm was inspired by Press, Teukolsky, Vetterling, and Flannery,
* "Numerical Recipes in C", 2nd edition, Cambridge University Press */
double froot, dfroot, dx, dxold;
double xh, xl;
// Orient the search so that f(xl) < 0
if (fxMin_ < 0.0)
{
xl = xMin_;
xh = xMax_;
}
else
{
xh = xMin_;
xl = xMax_;
}
// the "stepsize before last"
dxold = xMax_ - xMin_;
// it was dxold=std::fabs(xMax_-xMin_); in Numerical Recipes
// here (xMax_-xMin_ > 0) is verified in the constructor
// and the last step
dx = dxold;
froot = f.value(root_);
dfroot = f.derivative(root_);
if (dfroot.IsEqual(default(double)))
{
throw new ArgumentException("Newton requires function's derivative");
}
++evaluationNumber_;
while (evaluationNumber_ <= maxEvaluations_)
{
// Bisect if (out of range || not decreasing fast enough)
if ((((root_ - xh) * dfroot - froot) *
((root_ - xl) * dfroot - froot) > 0.0) ||
(Math.Abs(2.0 * froot) > Math.Abs(dxold * dfroot)))
{
dxold = dx;
dx = (xh - xl) / 2.0;
root_ = xl + dx;
}
else
{
dxold = dx;
dx = froot / dfroot;
root_ -= dx;
}
// Convergence criterion
if (Math.Abs(dx) < xAccuracy)
{
return(root_);
}
froot = f.value(root_);
dfroot = f.derivative(root_);
evaluationNumber_++;
if (froot < 0.0)
{
xl = root_;
}
else
{
xh = root_;
}
}
Utils.QL_FAIL("maximum number of function evaluations (" + maxEvaluations_ + ") exceeded",
QLNetExceptionEnum.MaxNumberFuncEvalExceeded);
return(0);
}