com.numericalmethod.suanshu.analysis.integration.univariate.riemann
Class EulerMaclaurin

java.lang.Object
  com.numericalmethod.suanshu.analysis.integration.univariate.riemann.EulerMaclaurin

All Implemented Interfaces:

Integrator, IterativeIntegrator

Direct Known Subclasses:

Midpoint, Trapezoidal

public class EulerMaclaurin
extends java.lang.Object
implements IterativeIntegrator

This class implements a specialized group of the Newton-Cotes integration methods using the Euler-Maclaurin formula.

This class forms the basis for a few standard numerical quadrature methods with equally spaced interpolation points. The extraploation method uses Neville's algorithm.

By customizing this class, one can construct various standard numerical quadrature algoritms, such as

• the Trapezoidal rule, by setting rate = 2.
• the Midpoint method, by setting rate = 2 and specifying OPEN formula.

See Also:

• Wikipedia: Newton–Cotes formulas
• Wikipedia: Euler–Maclaurin formula
• Wikipedia: Trapezoidal rule
• Wikipedia: Rectangle method

Nested Class Summary

static class

EulerMaclaurin.NewtonCotesType Newton-Cotes methods can be classified into two categories: OPEN and CLOSED.

Field Summary

int	maxIterations the maximum number of iterations for this iterative procedure For those integrals that do not converge, we need to put a bound on the number of iterations.
double	precision the convergence threshold The iterative procedure converges when the relative difference between two successive sums is less than precision.
int	rate the rate of sub-dividing an interval Starting with the whole interval (b-a), we divide each sub-interval into rate many intervals.
EulerMaclaurin.NewtonCotesType	type indicate whether the two end points are included for computation

Constructor Summary

EulerMaclaurin(int rate, EulerMaclaurin.NewtonCotesType type, double precision, int maxIterations)
Construct an instance of the Euler-Maclaurin formula.

Method Summary

double	h() Get the discretization size for the current iteration.
double	integrate(UnivariateRealFunction f, double a, double b) Integrate function f from a to b.
int	maxIterations() Get the maximum number of iterations for this iterative procedure.
double	next(int iter, UnivariateRealFunction f, double a, double b, double sum0) Compute a refined sum for the integral.
double	precision() Get the convergence threshold.

Methods inherited from class java.lang.Object

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

Field Detail

rate

public final int rate

the rate of sub-dividing an interval

Starting with the whole interval (b-a), we divide each sub-interval into rate many intervals. For example, when rate = 2, we double the number of intervals, hence abscissas, in each iteration.

Note: the number of abscissas grows exponentially.

type

public final EulerMaclaurin.NewtonCotesType type

indicate whether the two end points are included for computation

precision

public final double precision

the convergence threshold

The iterative procedure converges when the relative difference between two successive sums is less than precision.

maxIterations

public final int maxIterations

the maximum number of iterations for this iterative procedure

For those integrals that do not converge, we need to put a bound on the number of iterations.

Note that the number of abscissas grows exponentially with each iteration, i.e., rateiterations. Thus, maxIterations should not be too big. Be careful when choosing maxIterations, as it may severely affect the performance.

Constructor Detail

EulerMaclaurin

public EulerMaclaurin(int rate,
                      EulerMaclaurin.NewtonCotesType type,
                      double precision,
                      int maxIterations)

Construct an instance of the Euler-Maclaurin formula.

Parameters:

rate - the rate of further sub-dividing the integral interval.

For example, when rate = 2, we will divide [xi, xi+1] into two equal length intervals. This is equivalent to the Trapezoidal rule.

type - specifying whether to use CLOSED or OPEN formula

precision - the precision required, e.g., 1e-8

maxIterations - the maximum number of iterations

Method Detail

integrate

public double integrate(UnivariateRealFunction f,
                        double a,
                        double b)

Description copied from interface: Integrator

Integrate function f from a to b.

 / b
 |   f(x) dx
 / a
 

Specified by:

integrate in interface Integrator

Parameters:

f - a univariate function

a - lower limit

b - upper limit

Returns:

Σ(f(x))

next

public double next(int iter,
                   UnivariateRealFunction f,
                   double a,
                   double b,
                   double sum0)

Description copied from interface: IterativeIntegrator

Compute a refined sum for the integral.

Specified by:

next in interface IterativeIntegrator

Parameters:

iter - the index/count for the iterations; it counts from 1

f - the integrand

a - the lower limit

b - the upper limit

sum0 - the last sum

Returns:

a refined sum

h

public double h()

Description copied from interface: IterativeIntegrator

Get the discretization size for the current iteration.

Specified by:

h in interface IterativeIntegrator

Returns:

the discretization size

maxIterations

public int maxIterations()

Description copied from interface: IterativeIntegrator

Get the maximum number of iterations for this iterative procedure. For those integrals that do not converge, we need to put a bound on the number of iterations to avoid infinite looping.

Specified by:

maxIterations in interface IterativeIntegrator

Returns:

the maximum number of iterations

precision

public double precision()

Description copied from interface: Integrator

Get the convergence threshold. The usage depends on the specific integrator.

For example, for an IterativeIntegrator, the integral is considered converged if the relative error of two successive sums is less than the threshold.

Specified by:

precision in interface Integrator

Returns:

the precision