## org.apache.commons.math.analysis.integration
Class LegendreGaussIntegrator

    java.lang.Object
      org.apache.commons.math.ConvergingAlgorithmImpl
          org.apache.commons.math.analysis.integration.UnivariateRealIntegratorImpl
              org.apache.commons.math.analysis.integration.LegendreGaussIntegrator

All Implemented Interfaces:
  UnivariateRealIntegrator, ConvergingAlgorithm

    public class LegendreGaussIntegrator

extends UnivariateRealIntegratorImpl

Implements the Legendre-Gauss quadrature formula.

Legendre-Gauss integrators are efficient integrators that can accurately integrate functions with few functions evaluations. A Legendre-Gauss integrator using an n-points quadrature formula can integrate exactly 2n-1 degree polynomialss.

These integrators evaluate the function on n carefully chosen abscissas in each step interval (mapped to the canonical [-1 1] interval). The evaluation abscissas are not evenly spaced and none of them are at the interval endpoints. This implies the function integrated can be undefined at integration interval endpoints.

The evaluation abscissas xi are the roots of the degree n Legendre polynomial. The weights ai of the quadrature formula integrals from -1 to +1 ∫ Li2 where Li (x) = ∏ (x-xk)/(xi-xk) for k != i.

Since:
  1.2
Version:
  $Revision: 811685 $ $Date: 2009-09-05 13:36:48 -0400 (Sat, 05 Sep 2009) $

Field Summary
`private double[]` `abscissas` Abscissas for the current method.
`private static double[]` `ABSCISSAS_2` Abscissas for the 2 points method.
`private static double[]` `ABSCISSAS_3` Abscissas for the 3 points method.
`private static double[]` `ABSCISSAS_4` Abscissas for the 4 points method.
`private static double[]` `ABSCISSAS_5` Abscissas for the 5 points method.
`private double[]` `weights` Weights for the current method.
`private static double[]` `WEIGHTS_2` Weights for the 2 points method.
`private static double[]` `WEIGHTS_3` Weights for the 3 points method.
`private static double[]` `WEIGHTS_4` Weights for the 4 points method.
`private static double[]` `WEIGHTS_5` Weights for the 5 points method.

Fields inherited from class org.apache.commons.math.analysis.integration.UnivariateRealIntegratorImpl
`defaultMinimalIterationCount, f, minimalIterationCount, result, resultComputed`

Fields inherited from class org.apache.commons.math.ConvergingAlgorithmImpl
`absoluteAccuracy, defaultAbsoluteAccuracy, defaultMaximalIterationCount, defaultRelativeAccuracy, iterationCount, maximalIterationCount, relativeAccuracy`

Constructor Summary
`LegendreGaussIntegrator(int n, int defaultMaximalIterationCount)` Build a Legendre-Gauss integrator.

Method Summary
`double` `integrate(double min, double max)` Deprecated.
`double` `integrate(UnivariateRealFunction f, double min, double max)` Integrate the function in the given interval.
`private double` `stage(UnivariateRealFunction f, double min, double max, int n)` Compute the n-th stage integral.

Methods inherited from class org.apache.commons.math.analysis.integration.UnivariateRealIntegratorImpl
`clearResult, getMinimalIterationCount, getResult, resetMinimalIterationCount, setMinimalIterationCount, setResult, verifyInterval, verifyIterationCount`

Methods inherited from class org.apache.commons.math.ConvergingAlgorithmImpl
`getAbsoluteAccuracy, getIterationCount, getMaximalIterationCount, getRelativeAccuracy, resetAbsoluteAccuracy, resetMaximalIterationCount, resetRelativeAccuracy, setAbsoluteAccuracy, setMaximalIterationCount, setRelativeAccuracy`

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Methods inherited from interface org.apache.commons.math.ConvergingAlgorithm
`getAbsoluteAccuracy, getIterationCount, getMaximalIterationCount, getRelativeAccuracy, resetAbsoluteAccuracy, resetMaximalIterationCount, resetRelativeAccuracy, setAbsoluteAccuracy, setMaximalIterationCount, setRelativeAccuracy`

Field Detail

### ABSCISSAS_2

    private static final double[] ABSCISSAS_2

  Abscissas for the 2 points method.

### WEIGHTS_2

    private static final double[] WEIGHTS_2

  Weights for the 2 points method.

### ABSCISSAS_3

    private static final double[] ABSCISSAS_3

  Abscissas for the 3 points method.

### WEIGHTS_3

    private static final double[] WEIGHTS_3

  Weights for the 3 points method.

### ABSCISSAS_4

    private static final double[] ABSCISSAS_4

  Abscissas for the 4 points method.

### WEIGHTS_4

    private static final double[] WEIGHTS_4

  Weights for the 4 points method.

### ABSCISSAS_5

    private static final double[] ABSCISSAS_5

  Abscissas for the 5 points method.

### WEIGHTS_5

    private static final double[] WEIGHTS_5

  Weights for the 5 points method.

### abscissas

    private final double[] abscissas

  Abscissas for the current method.

### weights

    private final double[] weights

  Weights for the current method.

Constructor Detail

### LegendreGaussIntegrator

    public LegendreGaussIntegrator(int n,
                                   int defaultMaximalIterationCount)
                            throws java.lang.IllegalArgumentException

  Build a Legendre-Gauss integrator.

  Parameters:
    `n` - number of points desired (must be between 2 and 5 inclusive)
    `defaultMaximalIterationCount` - maximum number of iterations
  Throws:
    `java.lang.IllegalArgumentException` - if the number of points is not in the supported range

Method Detail

### integrate

    @Deprecated
    public double integrate(double min,
                                       double max)
                     throws ConvergenceException,
                            FunctionEvaluationException,
                            java.lang.IllegalArgumentException

  Deprecated.

  Integrate the function in the given interval.

  Parameters:
    `min` - the lower bound for the interval
    `max` - the upper bound for the interval
  Returns:
    the value of integral
  Throws:
    `ConvergenceException` - if the maximum iteration count is exceeded or the integrator detects convergence problems otherwise
    `FunctionEvaluationException` - if an error occurs evaluating the function
    `java.lang.IllegalArgumentException` - if min > max or the endpoints do not satisfy the requirements specified by the integrator

### integrate

    public double integrate(UnivariateRealFunction f,
                            double min,
                            double max)
                     throws ConvergenceException,
                            FunctionEvaluationException,
                            java.lang.IllegalArgumentException

  Integrate the function in the given interval.

  Parameters:
    `f` - the integrand function
    `min` - the lower bound for the interval
    `max` - the upper bound for the interval
  Returns:
    the value of integral
  Throws:
    `ConvergenceException` - if the maximum iteration count is exceeded or the integrator detects convergence problems otherwise
    `FunctionEvaluationException` - if an error occurs evaluating the function
    `java.lang.IllegalArgumentException` - if min > max or the endpoints do not satisfy the requirements specified by the integrator

### stage

    private double stage(UnivariateRealFunction f,
                         double min,
                         double max,
                         int n)
                  throws FunctionEvaluationException

  Compute the n-th stage integral.

  Parameters:
    `f` - the integrand function
    `min` - the lower bound for the interval
    `max` - the upper bound for the interval
    `n` - number of steps
  Returns:
    the value of n-th stage integral
  Throws:
    `FunctionEvaluationException` - if an error occurs evaluating the function

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