/** {@inheritDoc} */
public new ArrayRealVector mapToSelf(UnivariateFunction function)
{
for (int i = 0; i < data.Length; i++)
{
data[i] = function.value(data[i]);
}
return(this);
}
/// <summary>
/// Convenience method to find a zero of a univariate real function. A default
/// solver is used.
/// </summary>
/// <param name="function">Function.</param>
/// <param name="x0">Lower bound for the interval.</param>
/// <param name="x1">Upper bound for the interval.</param>
/// <param name="absoluteAccuracy">Accuracy to be used by the solver.</param>
/// <returns>a value where the function is zero.</returns>
/// <exception cref="NoBracketingException"> if the function has the same sign at the
/// endpoints.</exception>
/// <exception cref="NullArgumentException"> if <c>function</c> is <c>null</c>.</exception>
public static double solve(UnivariateFunction function, double x0, double x1, double absoluteAccuracy)
{
if (function == null)
{
throw new NullArgumentException(new LocalizedFormats("FUNCTION"));
}
UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
return(solver.solve(Int32.MaxValue, function, x0, x1));
}
/// <summary>
/// Convenience method to find a zero of a univariate real function. A default
/// solver is used.
/// </summary>
/// <param name="function"> Function. </param>
/// <param name="x0"> Lower bound for the interval. </param>
/// <param name="x1"> Upper bound for the interval. </param>
/// <param name="absoluteAccuracy"> Accuracy to be used by the solver. </param>
/// <returns> a value where the function is zero. </returns>
/// <exception cref="NoBracketingException"> if the function has the same sign at the
/// endpoints. </exception>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
public static double Solve(UnivariateFunction function, double x0, double x1, double absoluteAccuracy)
{
if (function == null)
{
throw new ArgumentNullException("LocalizedFormats.FUNCTION");
}
UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
return(solver.Solve(int.MaxValue, function, x0, x1));
}
/// <summary>
/// Check whether the interval bounds bracket a root. That is, if the
/// values at the endpoints are not equal to zero, then the function takes
/// opposite signs at the endpoints.
/// </summary>
/// <param name="function"> Function. </param>
/// <param name="lower"> Lower endpoint. </param>
/// <param name="upper"> Upper endpoint. </param>
/// <returns> {@code true} if the function values have opposite signs at the
/// given points. </returns>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
public static bool IsBracketing(UnivariateFunction function, double lower, double upper)
{
if (function == null)
{
throw new ArgumentNullException("LocalizedFormats.FUNCTION");
}
double fLo = function.Value(lower);
double fHi = function.Value(upper);
return((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0));
}
/// <summary>
/// Check whether the interval bounds bracket a root. That is, if the
/// values at the endpoints are not equal to zero, then the function takes
/// opposite signs at the endpoints.
/// </summary>
/// <param name="function">Function.</param>
/// <param name="lower">Lower endpoint.</param>
/// <param name="upper">Upper endpoint.</param>
/// <returns><c>true</c> if the function values have opposite signs at the
/// given points.</returns>
/// <exception cref="NullArgumentException"> if <c>function</c> is <c>null</c>.</exception>
public static Boolean isBracketing(UnivariateFunction function, double lower, double upper)
{
if (function == null)
{
throw new NullArgumentException(new LocalizedFormats("FUNCTION"));
}
double fLo = function.value(lower);
double fHi = function.value(upper);
return((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0));
}
/**
* Acts as if it is implemented as:
* <pre>
* Entry e = null;
* for(Iterator<Entry> it = iterator(); it.hasNext(); e = it.next()) {
* e.setValue(function.value(e.getValue()));
* }
* </pre>
* Entries of this vector are modified in-place by this method.
*
* @param function Function to apply to each entry.
* @return a reference to this vector.
*/
public virtual RealVector mapToSelf(UnivariateFunction function)
{
var it = iterator();
while (it.MoveNext())
{
Entry e = it.Current;
e.setValue(function.value(e.getValue()));
}
return(this);
}
/// <summary>
/// Check that the endpoints specify an interval and the end points
/// bracket a root.
/// </summary>
/// <param name="function"> Function. </param>
/// <param name="lower"> Lower endpoint. </param>
/// <param name="upper"> Upper endpoint. </param>
/// <exception cref="NoBracketingException"> if the function has the same sign at the
/// endpoints. </exception>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
public static void VerifyBracketing(UnivariateFunction function, double lower, double upper)
{
if (function == null)
{
throw new ArgumentNullException("LocalizedFormats.FUNCTION");
}
VerifyInterval(lower, upper);
if (!IsBracketing(function, lower, upper))
{
throw new NoBracketingException(lower, upper, function.Value(lower), function.Value(upper));
}
}
/// <summary>
/// Check that the endpoints specify an interval and the end points
/// bracket a root.
/// </summary>
/// <param name="function">Function.</param>
/// <param name="lower">Lower endpoint.</param>
/// <param name="upper">Upper endpoint.</param>
/// <exception cref="NoBracketingException"> if the function has the same sign at the
/// endpoints.</exception>
/// <exception cref="NullArgumentException"> if <c>function</c> is <c>null</c>.</exception>
public static void verifyBracketing(UnivariateFunction function, double lower, double upper)
{
if (function == null)
{
throw new NullArgumentException(new LocalizedFormats("FUNCTION"));
}
verifyInterval(lower, upper);
if (!isBracketing(function, lower, upper))
{
throw new NoBracketingException(lower, upper, function.value(lower), function.value(upper));
}
}
/// <summary>
/// Convenience method to find a zero of a univariate real function. A default
/// solver is used.
/// </summary>
/// <param name="function"> Function. </param>
/// <param name="x0"> Lower bound for the interval. </param>
/// <param name="x1"> Upper bound for the interval. </param>
/// <param name="absoluteAccuracy"> Accuracy to be used by the solver. </param>
/// <returns> a value where the function is zero. </returns>
/// <exception cref="NoBracketingException"> if the function has the same sign at the
/// endpoints. </exception>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
public static double Solve(UnivariateFunction function, double x0, double x1, double absoluteAccuracy)
{
if (function == null)
{
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
return(solver.solve(int.MaxValue, function, x0, x1));
}
/// <summary>
/// Prepare for computation.
/// Subclasses must call this method if they override any of the
/// {@code solve} methods.
/// </summary>
/// <param name="maxEval"> Maximum number of evaluations. </param>
/// <param name="f"> the integrand function </param>
/// <param name="lower"> the min bound for the interval </param>
/// <param name="upper"> the upper bound for the interval </param>
/// <exception cref="NullArgumentException"> if {@code f} is {@code null}. </exception>
/// <exception cref="MathIllegalArgumentException"> if {@code min >= max}. </exception>
//JAVA TO C# CONVERTER WARNING: 'final' parameters are not available in .NET:
//ORIGINAL LINE: protected void setup(final int maxEval, final org.apache.commons.math3.analysis.UnivariateFunction f, final double lower, final double upper) throws org.apache.commons.math3.exception.NullArgumentException, org.apache.commons.math3.exception.MathIllegalArgumentException
protected internal virtual void Setup(int maxEval, UnivariateFunction f, double lower, double upper)
{
// Checks.
MathUtils.CheckNotNull(f);
UnivariateSolverUtils.VerifyInterval(lower, upper);
// Reset.
min = lower;
max = upper;
function = f;
evaluations = evaluations.WithMaximalCount(maxEval).withStart(0);
count = count.WithStart(0);
}
/// <summary>
/// {@inheritDoc} </summary>
/// <exception cref="MathException"> If the Commons method could not evaluate the function;
/// if the Commons method could not converge. </exception>
public override double?getRoot(System.Func <double, double> function, double?xLow, double?xHigh)
{
checkInputs(function, xLow, xHigh);
UnivariateFunction wrapped = CommonsMathWrapper.wrapUnivariate(function);
try
{
return(_ridder.solve(MAX_ITER, wrapped, xLow, xHigh));
}
catch (Exception e) when(e is TooManyEvaluationsException || e is NoBracketingException)
{
throw new MathException(e);
}
}
/// <summary>
/// Prepare for computation.
/// Subclasses must call this method if they override any of the
/// {@code solve} methods.
/// </summary>
/// <param name="maxEval"> Maximum number of evaluations. </param>
/// <param name="f"> the integrand function </param>
/// <param name="lower"> the min bound for the interval </param>
/// <param name="upper"> the upper bound for the interval </param>
/// <exception cref="NullArgumentException"> if {@code f} is {@code null}. </exception>
/// <exception cref="MathIllegalArgumentException"> if {@code min >= max}. </exception>
protected internal virtual void Setup(int maxEval, UnivariateFunction f, double lower, double upper)
{
// Checks.
MyUtils.CheckNotNull(f);
UnivariateSolverUtils.VerifyInterval(lower, upper);
// Reset.
this.min = lower;
this.max = upper;
this.function = f;
this.evaluations.MaximalCount = maxEval;
this.evaluations.ResetCount();
this.iterations.ResetCount();
}
/// <summary>
/// Check whether the interval bounds bracket a root. That is, if the
/// values at the endpoints are not equal to zero, then the function takes
/// opposite signs at the endpoints.
/// </summary>
/// <param name="function"> Function. </param>
/// <param name="lower"> Lower endpoint. </param>
/// <param name="upper"> Upper endpoint. </param>
/// <returns> {@code true} if the function values have opposite signs at the
/// given points. </returns>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
//JAVA TO C# CONVERTER WARNING: 'final' parameters are not available in .NET:
//ORIGINAL LINE: public static boolean isBracketing(org.apache.commons.math3.analysis.UnivariateFunction function, final double lower, final double upper) throws org.apache.commons.math3.exception.NullArgumentException
public static bool IsBracketing(UnivariateFunction function, double lower, double upper)
{
if (function == null)
{
throw new NullArgumentException(LocalizedFormats.FUNCTION);
}
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final double fLo = function.value(lower);
double fLo = function.Value(lower);
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final double fHi = function.value(upper);
double fHi = function.Value(upper);
return((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0));
}
/// <summary>
/// Returns an estimate of the integral of {@code f(x) * w(x)},
/// where {@code w} is a weight function that depends on the actual
/// flavor of the Gauss integration scheme.
/// The algorithm uses the points and associated weights, as passed
/// to the <seealso cref="#GaussIntegrator(double[],double[]) constructor"/>.
/// </summary>
/// <param name="f"> Function to integrate. </param>
/// <returns> the integral of the weighted function. </returns>
public virtual double Integrate(UnivariateFunction f)
{
double s = 0;
double c = 0;
for (int i = 0; i < this.points.Length; i++)
{
double x = this.points[i];
double w = this.weights[i];
double y = w * f.Value(x) - c;
double t = s + y;
c = (t - s) - y;
s = t;
}
return(s);
}
/// <summary>
/// Samples the specified univariate real function on the specified interval.
/// <para/>
/// The interval is divided equally into <c>n</c> sections and sample points
/// are taken from <c>min</c> to <c>max - (max - min) / n</c>; therefore
/// <c>f</c> is not sampled at the upper bound <c>max</c>.
/// </summary>
/// <param name="f">Function to be sampled</param>
/// <param name="min">Lower bound of the interval (included).</param>
/// <param name="max">Upper bound of the interval (excluded).</param>
/// <param name="n">Number of sample points.</param>
/// <returns>the array of samples.</returns>
/// <exception cref="NumberIsTooLargeException"> if the lower bound <c>min</c> is
/// greater than, or equal to the upper bound <c>max</c>.</exception>
/// <exception cref="NotStrictlyPositiveException"> if the number of sample points
/// <c>n</c> is negative.</exception>
public static double[] sample(UnivariateFunction f, double min, double max, int n)
{
if (n <= 0)
{
throw new NotStrictlyPositiveException <Int32>(new LocalizedFormats("NOT_POSITIVE_NUMBER_OF_SAMPLES"), n);
}
if (min >= max)
{
throw new NumberIsTooLargeException <Double, Double>(min, max, false);
}
double[] s = new double[n];
double h = (max - min) / n;
for (int i = 0; i < n; i++)
{
s[i] = f.value(min + i * h);
}
return(s);
}
public static void Test()
{
double min = -4.0;
double max = 0.0;
int maxEval = 1000;
UnivariateFunction f1 = (UnivariateFunction) new DelegateUnivariateFunction((Func <double, double>)(x => (x + 3.0) * (x - 1.0) * (x - 1.0)));
DifferentiableUnivariateFunction f2 = (DifferentiableUnivariateFunction) new DelegateDifferentiableUnivariateFunction((Func <double, double>)(x => (x + 3.0) * (x - 1.0) * (x - 1.0)), (Func <double, double>)(x => 2.0 * (x - 1.0) * (x + 3.0) + (x - 1.0) * (x - 1.0)));
new BisectionSolver().Solve(maxEval, f1, min, max);
new BrentSolver().Solve(maxEval, f1, min, max);
new RegulaFalsiSolver().Solve(maxEval, f1, min, max);
new IllinoisSolver().Solve(maxEval, f1, min, max);
new PegasusSolver().Solve(maxEval, f1, min, max);
new RiddersSolver().Solve(maxEval, f1, min, max);
new SecantSolver().Solve(maxEval, f1, min, max);
new MullerSolver().Solve(maxEval, f1, min, max);
new MullerSolver2().Solve(maxEval, f1, min, max);
new BracketingNthOrderBrentSolver().Solve(maxEval, f1, min, max);
new NewtonSolver().Solve(maxEval, f2, min, max);
}
/// <summary>
/// {@inheritDoc}
/// </summary>
public override double Integrate(UnivariateFunction f)
{
int ruleLength = this.NumberOfPoints;
if (ruleLength == 1)
{
return(this.GetWeight(0) * f.Value(0d));
}
int iMax = ruleLength / 2;
double s = 0;
double c = 0;
for (int i = 0; i < iMax; i++)
{
double p = this.GetPoint(i);
double w = this.GetWeight(i);
double f1 = f.Value(p);
double f2 = f.Value(-p);
double y = w * (f1 + f2) - c;
double t = s + y;
c = (t - s) - y;
s = t;
}
if (ruleLength % 2 != 0)
{
double w = this.GetWeight(iMax);
double y = w * f.Value(0d) - c;
double t = s + y;
s = t;
}
return(s);
}
/// <summary>
/// Returns an estimate of the integral of {@code f(x) * w(x)},
/// where {@code w} is a weight function that depends on the actual
/// flavor of the Gauss integration scheme.
/// The algorithm uses the points and associated weights, as passed
/// to the <seealso cref="#GaussIntegrator(double[],double[]) constructor"/>.
/// </summary>
/// <param name="f"> Function to integrate. </param>
/// <returns> the integral of the weighted function. </returns>
public virtual double Integrate(UnivariateFunction f)
{
double s = 0;
double c = 0;
for (int i = 0; i < points.Length; i++)
{
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final double x = points[i];
double x = points[i];
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final double w = weights[i];
double w = weights[i];
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final double y = w * f.value(x) - c;
double y = w * f.Value(x) - c;
//JAVA TO C# CONVERTER WARNING: The original Java variable was marked 'final':
//ORIGINAL LINE: final double t = s + y;
double t = s + y;
c = (t - s) - y;
s = t;
}
return(s);
}
public static UnivariateFunction Compose(this UnivariateFunction function, UnivariateFunction inner)
{
return(x => function(inner(x)));
}
public static UnivariateFunction Wrap(UnivariateFunction function)
{
return(function);
}
public static UnivariateFunction OutCompose(this UnivariateFunction function, UnivariateFunction outer)
{
return(x => outer(function(x)));
}
/// <summary>
/// {@inheritDoc}
/// </summary>
protected internal override void Setup(int maxEval, DifferentiableUnivariateFunction f, double min, double max, double startValue)
{
base.Setup(maxEval, f, min, max, startValue);
this.functionDerivative = f.Derivative();
}
public MultivariateFunctionAnonymous(BivariateFunction combiner, UnivariateFunction f, ref double initialValue)
{
this.combiner = combiner;
this.f = f;
this.initialValue = initialValue;
}
/// <summary>
/// This method simply calls {@link #bracket(UnivariateFunction, double, double, double,
/// double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
/// with {@code q} and {@code r} set to 1.0 and {@code maximumIterations} set to {@code Integer.MAX_VALUE}.
/// <strong>Note: </strong> this method can take
/// <code>Integer.MAX_VALUE</code> iterations to throw a
/// <code>ConvergenceException.</code> Unless you are confident that there
/// is a root between <code>lowerBound</code> and <code>upperBound</code>
/// near <code>initial,</code> it is better to use
/// {@link #bracket(UnivariateFunction, double, double, double, double,
/// double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)},
/// explicitly specifying the maximum number of iterations.</p>
/// </summary>
/// <param name="function"> Function. </param>
/// <param name="initial"> Initial midpoint of interval being expanded to
/// bracket a root. </param>
/// <param name="lowerBound"> Lower bound (a is never lower than this value) </param>
/// <param name="upperBound"> Upper bound (b never is greater than this
/// value). </param>
/// <returns> a two-element array holding a and b. </returns>
/// <exception cref="NoBracketingException"> if a root cannot be bracketted. </exception>
/// <exception cref="NotStrictlyPositiveException"> if {@code maximumIterations <= 0}. </exception>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
public static double[] Bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound)
{
return(Bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, int.MaxValue));
}
public UnivariateFunctionAnonymous4(BivariateFunction combiner, UnivariateFunction f, UnivariateFunction g)
{
this.combiner = combiner;
this.f = f;
this.g = g;
}
/// <summary>
/// Returns a MultivariateFunction h(x[]) defined by <code>
/// h(x[]) = combiner(...combiner(combiner(initialValue,f(x[0])),f(x[1]))...),f(x[x.Length-1]))
/// </code>
/// </summary>
/// <param name="combiner">Combiner function.</param>
/// <param name="f">Function.</param>
/// <param name="initialValue">Initial value.</param>
/// <returns>a collector function.</returns>
public static MultivariateFunction collector(BivariateFunction combiner, UnivariateFunction f, double initialValue)
{
return(new MultivariateFunctionAnonymous(combiner, f, ref initialValue));
}
/// <summary>
/// Returns the univariate function <para/>
/// <c>h(x) = combiner(f(x), g(x))</c>.
/// </summary>
/// <param name="combiner">Combiner function.</param>
/// <param name="f">Function.</param>
/// <param name="g">Function.</param>
/// <returns>the composite function.</returns>
public static UnivariateFunction combine(BivariateFunction combiner, UnivariateFunction f, UnivariateFunction g)
{
return(new UnivariateFunctionAnonymous4(combiner, f, g));
}
/// <summary>
/// This method simply calls {@link #bracket(UnivariateFunction, double, double, double,
/// double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
/// with {@code q} and {@code r} set to 1.0. </summary>
/// <param name="function"> Function. </param>
/// <param name="initial"> Initial midpoint of interval being expanded to
/// bracket a root. </param>
/// <param name="lowerBound"> Lower bound (a is never lower than this value). </param>
/// <param name="upperBound"> Upper bound (b never is greater than this
/// value). </param>
/// <param name="maximumIterations"> Maximum number of iterations to perform </param>
/// <returns> a two element array holding a and b. </returns>
/// <exception cref="NoBracketingException"> if the algorithm fails to find a and b
/// satisfying the desired conditions. </exception>
/// <exception cref="NotStrictlyPositiveException"> if {@code maximumIterations <= 0}. </exception>
/// <exception cref="NullArgumentException"> if {@code function} is {@code null}. </exception>
public static double[] Bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, int maximumIterations)
{
return(Bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, maximumIterations));
}
/// <summary>
/// This method attempts to find two values a and b satisfying <ul>
/// <li> {@code lowerBound <= a < initial < b <= upperBound} </li>
/// <li> {@code f(a) * f(b) <= 0} </li>
/// </ul>
/// If {@code f} is continuous on {@code [a,b]}, this means that {@code a}
/// and {@code b} bracket a root of {@code f}.
/// <para>
/// The algorithm checks the sign of \( f(l_k) \) and \( f(u_k) \) for increasing
/// values of k, where \( l_k = max(lower, initial - \delta_k) \),
/// \( u_k = min(upper, initial + \delta_k) \), using recurrence
/// \( \delta_{k+1} = r \delta_k + q, \delta_0 = 0\) and starting search with \( k=1 \).
/// The algorithm stops when one of the following happens: <ul>
/// <li> at least one positive and one negative value have been found -- success!</li>
/// <li> both endpoints have reached their respective limites -- NoBracketingException </li>
/// <li> {@code maximumIterations} iterations elapse -- NoBracketingException </li></ul></para>
/// <para>
/// If different signs are found at first iteration ({@code k=1}), then the returned
/// interval will be \( [a, b] = [l_1, u_1] \). If different signs are found at a later
/// iteration ({code k>1}, then the returned interval will be either
/// \( [a, b] = [l_{k+1}, l_{k}] \) or ( [a, b] = [u_{k}, u_{k+1}] \). A root solver called
/// with these parameters will therefore start with the smallest bracketing interval known
/// at this step.
/// </para>
/// <para>
/// Interval expansion rate is tuned by changing the recurrence parameters {@code r} and
/// {@code q}. When the multiplicative factor {@code r} is set to 1, the sequence is a
/// simple arithmetic sequence with linear increase. When the multiplicative factor {@code r}
/// is larger than 1, the sequence has an asymtotically exponential rate. Note than the
/// additive parameter {@code q} should never be set to zero, otherwise the interval would
/// degenerate to the single initial point for all values of {@code k}.
/// </para>
/// <para>
/// As a rule of thumb, when the location of the root is expected to be approximately known
/// within some error margin, {@code r} should be set to 1 and {@code q} should be set to the
/// order of magnitude of the error margin. When the location of the root is really a wild guess,
/// then {@code r} should be set to a value larger than 1 (typically 2 to double the interval
/// length at each iteration) and {@code q} should be set according to half the initial
/// search interval length.
/// </para>
/// <para>
/// As an example, if we consider the trivial function {@code f(x) = 1 - x} and use
/// {@code initial = 4}, {@code r = 1}, {@code q = 2}, the algorithm will compute
/// {@code f(4-2) = f(2) = -1} and {@code f(4+2) = f(6) = -5} for {@code k = 1}, then
/// {@code f(4-4) = f(0) = +1} and {@code f(4+4) = f(8) = -7} for {@code k = 2}. Then it will
/// return the interval {@code [0, 2]} as the smallest one known to be bracketing the root.
/// As shown by this example, the initial value (here {@code 4}) may lie outside of the returned
/// bracketing interval.
/// </para> </summary>
/// <param name="function"> function to check </param>
/// <param name="initial"> Initial midpoint of interval being expanded to
/// bracket a root. </param>
/// <param name="lowerBound"> Lower bound (a is never lower than this value). </param>
/// <param name="upperBound"> Upper bound (b never is greater than this
/// value). </param>
/// <param name="q"> additive offset used to compute bounds sequence (must be strictly positive) </param>
/// <param name="r"> multiplicative factor used to compute bounds sequence </param>
/// <param name="maximumIterations"> Maximum number of iterations to perform </param>
/// <returns> a two element array holding the bracketing values. </returns>
/// <exception cref="NoBracketingException"> if function cannot be bracketed in the search interval </exception>
public static double[] Bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, double q, double r, int maximumIterations)
{
if (function == null)
{
throw new ArgumentNullException("LocalizedFormats.FUNCTION");
}
if (q <= 0)
{
throw new NotStrictlyPositiveException(q);
}
if (maximumIterations <= 0)
{
throw new NotStrictlyPositiveException("LocalizedFormats.INVALID_MAX_ITERATIONS", maximumIterations);
}
VerifySequence(lowerBound, initial, upperBound);
// initialize the recurrence
double a = initial;
double b = initial;
double fa = double.NaN;
double fb = double.NaN;
double delta = 0;
for (int numIterations = 0; (numIterations < maximumIterations) && (a > lowerBound || b > upperBound); ++numIterations)
{
double previousA = a;
double previousFa = fa;
double previousB = b;
double previousFb = fb;
delta = r * delta + q;
a = FastMath.Max(initial - delta, lowerBound);
b = FastMath.Min(initial + delta, upperBound);
fa = function.Value(a);
fb = function.Value(b);
if (numIterations == 0)
{
// at first iteration, we don't have a previous interval
// we simply compare both sides of the initial interval
if (fa * fb <= 0)
{
// the first interval already brackets a root
return(new double[] { a, b });
}
}
else
{
// we have a previous interval with constant sign and expand it,
// we expect sign changes to occur at boundaries
if (fa * previousFa <= 0)
{
// sign change detected at near lower bound
return(new double[] { a, previousA });
}
else if (fb * previousFb <= 0)
{
// sign change detected at near upper bound
return(new double[] { previousB, b });
}
}
}
// no bracketing found
throw new NoBracketingException(a, b, fa, fb);
}
/// <summary>
/// Force a root found by a non-bracketing solver to lie on a specified side,
/// as if the solver was a bracketing one. </summary>
/// <param name="maxEval"> maximal number of new evaluations of the function
/// (evaluations already done for finding the root should have already been subtracted
/// from this number) </param>
/// <param name="f"> function to solve </param>
/// <param name="bracketing"> bracketing solver to use for shifting the root </param>
/// <param name="baseRoot"> original root found by a previous non-bracketing solver </param>
/// <param name="min"> minimal bound of the search interval </param>
/// <param name="max"> maximal bound of the search interval </param>
/// <param name="allowedSolution"> the kind of solutions that the root-finding algorithm may
/// accept as solutions. </param>
/// <returns> a root approximation, on the specified side of the exact root </returns>
/// <exception cref="NoBracketingException"> if the function has the same sign at the
/// endpoints. </exception>
public static double ForceSide(int maxEval, UnivariateFunction f, BracketedUnivariateSolver <UnivariateFunction> bracketing, double baseRoot, double min, double max, AllowedSolution allowedSolution)
{
if (allowedSolution == AllowedSolution.ANY_SIDE)
{
// no further bracketing required
return(baseRoot);
}
// find a very small interval bracketing the root
double step = FastMath.Max(bracketing.AbsoluteAccuracy, FastMath.Abs(baseRoot * bracketing.RelativeAccuracy));
double xLo = FastMath.Max(min, baseRoot - step);
double fLo = f.Value(xLo);
double xHi = FastMath.Min(max, baseRoot + step);
double fHi = f.Value(xHi);
int remainingEval = maxEval - 2;
while (remainingEval > 0)
{
if ((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0))
{
// compute the root on the selected side
return(bracketing.Solve(remainingEval, f, xLo, xHi, baseRoot, allowedSolution));
}
// try increasing the interval
bool changeLo = false;
bool changeHi = false;
if (fLo < fHi)
{
// increasing function
if (fLo >= 0)
{
changeLo = true;
}
else
{
changeHi = true;
}
}
else if (fLo > fHi)
{
// decreasing function
if (fLo <= 0)
{
changeLo = true;
}
else
{
changeHi = true;
}
}
else
{
// unknown variation
changeLo = true;
changeHi = true;
}
// update the lower bound
if (changeLo)
{
xLo = FastMath.Max(min, xLo - step);
fLo = f.Value(xLo);
remainingEval--;
}
// update the higher bound
if (changeHi)
{
xHi = FastMath.Min(max, xHi + step);
fHi = f.Value(xHi);
remainingEval--;
}
}
throw new NoBracketingException("LocalizedFormats.FAILED_BRACKETING", xLo, xHi, fLo, fHi, maxEval - remainingEval, maxEval, baseRoot, min, max);
}
