Package com.imsl.math

Class Quadrature

java.lang.Object
com.imsl.math.Quadrature
All Implemented Interfaces:
Serializable, Cloneable
public class Quadrature extends Object implements Serializable, Cloneable
Quadrature is a general-purpose integrator that uses a globally adaptive scheme in order to reduce the absolute error. It subdivides the interval [A, B] and uses a \((2k+1)\)-point Gauss-Kronrod rule to estimate the integral over each subinterval. The error for each subinterval is estimated by comparison with the k-point Gauss quadrature rule. The subinterval with the largest estimated error is then bisected and the same procedure is applied to both halves. The bisection process is continued until either the error criterion is satisfied, roundoff error is detected, the subintervals become too small, or the maximum number of subintervals allowed is reached. The Class Quadrature is based on the subroutine QAG by Piessens et al. (1983).

If the function to be integrated has endpoint singularities then extrapolation should be enabled. As described above, the integral's value is approximated by applying a quadrature rule to a series of subdivisions of the interval. The sequence of approximate values converges to the integral's value. The \(\epsilon\)-algorithm can be used to extrapolate from the initial terms of the sequence to its limit. Without extrapolation, the quadrature approximation sequence converges slowly if the function being integrated has endpoint singularities. The \(\epsilon\)-algorithm accelerates convergence of the sequence in this case. The class EpsilonAlgorithm implements the \(\epsilon\)-algorithm. With extrapolation, this class is similar to the subroutine QAGS by Piessens et al. (1983).

The desired absolute error, \(\epsilon\), can be set using setAbsoluteError. The desired relative error, \(\rho\), can be set using setRelativeError. The method eval computes the approximate integral value \(R \approx \int_a^b f(x) dx\). It also computes an error estimate E, which can be retrieved using getErrorEstimate. These are related by the following equation: $$ \left| \int_a^b f(x) dx -R \right| \le E \le \max\left\{\epsilon,\rho \left| \int_a^b f(x) dx \right| \right\} $$

See Also:

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static interface 
    Quadrature.Function
    Public interface function for the Quadrature class.

  • Constructor Summary

    Constructors
    Constructor
    Description
    Quadrature()
    Constructs a Quadrature object.

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    eval(Quadrature.Function objectF, double a, double b)
    Returns the value of the integral from a to b.
    double
    getErrorEstimate()
    Returns an estimate of the relative error in the computed result.
    int
    getErrorStatus()
    Returns the non-fatal error status.
    void
    setAbsoluteError(double errorAbsolute)
    Sets the absolute error tolerance.
    void
    setExtrapolation(boolean doExtrapolation)
    If true, the epsilon-algorithm for extrapolation is enabled.
    void
    setMaxSubintervals(int maxSubintervals)
    Sets the maximum number of subintervals allowed.
    void
    setRelativeError(double errorRelative)
    Sets the relative error tolerance.
    void
    setRule(int rule)
    Set the Gauss-Kronrod rule.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • Quadrature

      public Quadrature()
      Constructs a Quadrature object.

  • Method Details

    • setAbsoluteError

      public void setAbsoluteError(double errorAbsolute)
      Sets the absolute error tolerance.
      Parameters:
      errorAbsolute - a double scalar value specifying the absolute error tolerance. The default value is 1.0536712127723714E-8.

    • setRelativeError

      public void setRelativeError(double errorRelative)
      Sets the relative error tolerance.
      Parameters:
      errorRelative - a double scalar value specifying the relative error tolerance. The default value is 1.0536712127723714E-8.

    • setExtrapolation

      public void setExtrapolation(boolean doExtrapolation)
      If true, the epsilon-algorithm for extrapolation is enabled. The default is false (extrapolation is not used).
      Parameters:
      doExtrapolation - a boolean, true if the epsilon-algorithm for extrapolation is to be enabled, false otherwise

    • setRule

      public void setRule(int rule)
      Set the Gauss-Kronrod rule.
      Rule Data points used
      1 7 - 15
      2 10 - 21
      3 15 - 31
      4 20 - 41
      5 25 - 51
      6 30 - 61
      The default is rule 3.
      Parameters:
      rule - an int specifying the rule to be used. The default is 3.

    • setMaxSubintervals

      public void setMaxSubintervals(int maxSubintervals)
      Sets the maximum number of subintervals allowed. The default value is 500.
      Parameters:
      maxSubintervals - an int specifying the maximum number of subintervals to be allowed. The default is 500.

    • getErrorEstimate

      public double getErrorEstimate()
      Returns an estimate of the relative error in the computed result.
      Returns:
      a double specifying an estimate of the relative error in the computed result

    • getErrorStatus

      public int getErrorStatus()
      Returns the non-fatal error status.
      Returns:
      an int specifying the non-fatal error status:
      Status Meaning
      0 No error.
      1 Maximum number of subdivisions allowed has been achieved. One can allow more subdivisions by using setMaxSubintervals. If this yields no improvement it is advised to analyze the integrand in order to determine the integration difficulties. If the position of a local difficulty can be determined (e.g. singularity, discontinuity within the interval) one will probably gain from splitting up the interval at this point and calling the integrator on the subranges. If possible, an appropriate special-purpose integrator should be used, which is designed for handling the type of difficulty involved.
      2 The occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved. The error may be under-estimated.
      3 Extremely bad integrand behavior occurs at some points of the integration interval.
      4 The algorithm does not converge. Roundoff error is detected in the extrapolation table. It is presumed that the requested tolerance cannot be achieved, and that the returned result is the best which can be obtained.
      5 The algorithm does not converge. Roundoff error is detected in the extrapolation table. It is presumed that the requested tolerance cannot be achieved, and that the returned result is the best that can be obtained.
      6 The integral is probably divergent, or slowly convergent. It must be noted that divergence can occur with any other status value.

    • eval

      public double eval(Quadrature.Function objectF, double a, double b)
      Returns the value of the integral from a to b.
      Parameters:
      objectF - an implementation of Function containing the function to be integrated
      a - a double specifying the lower limit of integration
      b - a double specifying the upper limit of integration, either or both of a and b can be Double.POSITIVE_INFINITY or Double.NEGATIVE_INFINITY