Package com.imsl.math

Class ZerosFunction

java.lang.Object
com.imsl.math.ZerosFunction
All Implemented Interfaces:
Serializable, Cloneable
public class ZerosFunction extends Object implements Serializable, Cloneable
Finds the real zeros of a real, continuous, univariate function, f(x).

ZerosFunction computes n real zeros of a real, continuous, univariate function f. The search for the zeros of the function can be limited to a specified interval, or extended over the entire real line. The algorithm is generally more efficient if an interval is specified. The user supplied function, f(x), must return valid results for all values in the specified interval. If no interval is given, the user-supplied function must return valid results for all real numbers.

The function has two convergence criteria. The first criterion accepts a root, x, if $$\left| f\left(x\right) \right|\le \tau$$ where \(\tau\) = error, see method setError.

The second criterion accepts a root if it is known to be inside of an interval of length at most errorAbsolute, see method setAbsoluteError.

A root is accepted if it satisfies either criteria and is not within minSeparation of another accepted root, see method setMinimumSeparation.

If initial guesses for the roots are given, Müller's method (Müller 1956) is used for each of these guesses. For each guess, the Müller iteration is stopped if the next step would be outside of the bound, if given. The iteration is also stopped if the algorithm cannot make further progress in finding a root.

If no guesses for the zeros were given, or if Müller's method with the guesses did not find the requested number of roots, a meta-algorithm, combining Müller's and Brent's methods, is used. Müller's method is used primarily to find the roots of functions, such as \(f(x) = x^2\), where the function does not cross the y=0 line. Brent's method is used to find other types of roots.

The meta-algorithm successively refines the interval using a one-dimensional Faure low-discrepancy sequence. The Faure sequence may be scaled by setting the bound interval [a,b] using the setBounds method. The Faure sequence will be scaled from (0,1) to (a,b).

If no bound on the function's domain is given, the entire real line must be searched for roots. In this case the Faure sequence is scaled from (0, 1) to \(\left( -\infty, +\infty \right)\) using the mapping $$h(u) = \mbox{xScale} \cdot{\tan{(\pi( u - 1/2))}}$$ where xScale is set by the setXScale method.

At each step of the iteration, the next point in the Faure sequence is added to the list of breakpoints defining the subintervals. Call the points \(x_0=a, x_1=b, x_2, x_3, \ldots\). The new point, \(x_s\) splits an existing subinterval, \([x_p, x_q]\).

The function is evaluated at \(x_s\). If its value is small enough, specifically if $$\left|f\left(x_s\right)\right|\lt \hbox{mullerTolerance}$$ then Müller's method is used with \(x_p\), \(x_q\) and \(x_s\) as starting values. If a root is found, it is added to the list of roots. If more roots are required, the new Faure point is used.

If Müller's method did not find a root using the new point, the function value at the point is compared with the function values at the endpoints of the subinterval it divides. If \(f\left(x_p\right)f\left(x_s\right) \lt 0 \) and no root has previously been found in \([x_p,x_s]\), then Brent's method is used to find a root in this interval. Similarly, if the function changes sign over the interval \([x_s,x_q]\), and a root has not already been found in the subinterval, Brent's method is used.

See Also:

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static interface 
    ZerosFunction.Function
    Public interface for the user supplied function to ZerosFunction.

  • Constructor Summary

    Constructors
    Constructor
    Description
    ZerosFunction()
    Creates an instance of the solver.

  • Method Summary

    Modifier and Type
    Method
    Description
    boolean
    allConverged()
    Returns true if the iterations for all of the roots have converged.
    double[]
    computeZeros(ZerosFunction.Function objectF)
    Returns the zeros of a univariate function.
    int
    getMaxEvaluations()
    Returns the maximum number of function evaluations allowed.
    int
    getNumberOfEvaluations()
    Returns the actual number of function evaluations performed.
    int
    getNumberOfRoots()
    Returns the requested number of roots to be found.
    int
    getNumberOfRootsFound()
    Returns the number of zeros found.
    void
    setAbsoluteError(double errorAbsolute)
    Sets the second convergence criterion.
    void
    setBounds(double lowerBound, double upperBound)
    Sets the closed interval in which to search for the roots.
    void
    setError(double error)
    Sets the first convergence criterion.
    void
    setGuess(double[] guess)
    Sets the initial guess for the zeros.
    void
    setMaxEvaluations(int maxEvaluations)
    Sets the maximum number of function evaluations allowed.
    void
    setMinimumSeparation(double minSeparation)
    Sets the minimum separation between accepted roots.
    void
    setMullerTolerance(double mullerTolerance)
    Sets the tolerance used during refinement to determine if Müllers method is started.
    void
    setNumberOfRoots(int numRoots)
    Sets the number of roots to be found.
    void
    setXScale(double xScale)
    Sets the scaling in the x-coordinate.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Constructor Details

    • ZerosFunction

      public ZerosFunction()
      Creates an instance of the solver.

  • Method Details

    • setBounds

      public void setBounds(double lowerBound, double upperBound)
      Sets the closed interval in which to search for the roots. The function must be defined for all values in this interval.
      Parameters:
      lowerBound - a double containing the lower interval bound. lowerBound cannot be greater than or equal to upperBound.
      upperBound - a double containing the upper interval bound.

      By default the search for the roots is not bounded.

    • setXScale

      public void setXScale(double xScale)
      Sets the scaling in the x-coordinate.

      If no bound on the function's domain is given, the entire real line must be searched for roots. In this case the Faure sequence is scaled from (0, 1) to \(\left( -\infty, \infty \right)\) using the mapping $$h(u) = \mbox{xScale} \cdot{\tan{(\pi( u - 1/2))}}.$$

      Parameters:
      xScale - a double containing the scaling in the x-coordinate. The absolute value of the roots divided by xScale should be about one. xScale must be greater than 0.0. By default, xScale=1.0.

    • setNumberOfRoots

      public void setNumberOfRoots(int numRoots)
      Sets the number of roots to be found.
      Parameters:
      numRoots - an int containing the number of roots to be found. numRoots must be greater than or equal to zero. By default, numRoots=1.

    • getNumberOfRoots

      public int getNumberOfRoots()
      Returns the requested number of roots to be found.
      Returns:
      an int containing the requested number of roots to be found

    • getNumberOfRootsFound

      public int getNumberOfRootsFound()
      Returns the number of zeros found.
      Returns:
      an int containing the number of roots found

    • setMaxEvaluations

      public void setMaxEvaluations(int maxEvaluations)
      Sets the maximum number of function evaluations allowed.

      Methods allConverged and getNumberOfRootsFound can be used to confirm whether or not the number of roots requested were found within the maximum evaluations specified. maxEvaluations must be greater than or equal to 0.0.

      Parameters:
      maxEvaluations - an int containing the maximum number of function evaluations allowed. Once this limit is reached, the roots found are returned.

    • getMaxEvaluations

      public int getMaxEvaluations()
      Returns the maximum number of function evaluations allowed.
      Returns:
      an int containing the maximum number of function evaluations allowed

    • setAbsoluteError

      public void setAbsoluteError(double errorAbsolute)
      Sets the second convergence criterion.

      The second criterion accepts a root if the root is known to be inside an interval of length at most errorAbsolute.

      Parameters:
      errorAbsolute - a double value specifying the second convergence criterion. A root is accepted if the absolute value of the function at the point is less than or equal to errorAbsolute. errorAbsolute must be greater than or equal to 0.0. Default: errorAbsolute = 2.22e-14

    • setGuess

      public void setGuess(double[] guess)
      Sets the initial guess for the zeros.
      Parameters:
      guess - a double array containing the initial guesses for the number of zeros to be found. If a bound on the zeros is also given, the guesses must satisfy the bound condition.

    • setMinimumSeparation

      public void setMinimumSeparation(double minSeparation)
      Sets the minimum separation between accepted roots.
      Parameters:
      minSeparation - a double containing the minimum separation between accepted roots. If two points satisfy the convergence criteria, but are within minSeparation of each other, only one of the roots is accepted. minSeparation must be greater than or equal to 0.0.

      By default, minSeparation = 1.0e-8/xScale.

    • setError

      public void setError(double error)
      Sets the first convergence criterion.

      A root is accepted if it is bracketed within an interval of length error. $$\left| f\left(x\right) \right|\le \tau$$ where \(\tau\) = error.

      Parameters:
      error - a double containing the first convergence criterion. error must be greater than or equal to 0.0. By default, error = 2.0e-8/xScale.

    • setMullerTolerance

      public void setMullerTolerance(double mullerTolerance)
      Sets the tolerance used during refinement to determine if Müllers method is started.

      Müller's method is started if, during refinement, a point is found for which the absolute value of the function is less than mullerTolerance and the point is not near an already discovered root. If mullerTolerance is less than or equal to zero, Müller's method is never used.

      Parameters:
      mullerTolerance - a double containing the tolerance used during refinement to determine when the Müller's method is used. By default, mullerTolerance = 1.0e-8/errorAbsolute

    • getNumberOfEvaluations

      public int getNumberOfEvaluations()
      Returns the actual number of function evaluations performed.
      Returns:
      an int containing the actual number of function evaluations performed

    • computeZeros

      public double[] computeZeros(ZerosFunction.Function objectF)
      Returns the zeros of a univariate function.
      Parameters:
      objectF - contains the function for which the zeros will be found
      Returns:
      a double array containing the zerso of the univariate function.

    • allConverged

      public boolean allConverged()
      Returns true if the iterations for all of the roots have converged.