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

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

All Implemented Interfaces:

Integrator, IterativeIntegrator

public class Simpson
extends java.lang.Object
implements IterativeIntegrator

Simpson's rule is often an accurate integration rule.

Simpson's is expected to improve on the trapezoidal rule for functions which are twice continuously differentiable. However for rougher functions the trapezoidal rule is likely to be more preferable.

This class implements Simpson's rule as a weighted average of two successive steps of the Trapezoidal rule.

See Also:

• Wikipedia: Simpson's rule
• Wikipedia: Rectangle method

Field Summary

int	maxIterations the maximum number of iterations
double	precision the convergence threshold

Constructor Summary

Simpson(double precision, int maxIterations)
Construct an integrator that uses Simpson's rule.

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 iteration, UnivariateRealFunction f, double a, double b, double sum) 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

maxIterations

public final int maxIterations

the maximum number of iterations

precision

public final double precision

the convergence threshold

Constructor Detail

Simpson

public Simpson(double precision,
               int maxIterations)

Construct an integrator that uses Simpson's rule.

Parameters:

precision - the convergence threshold

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 iteration,
                   UnivariateRealFunction f,
                   double a,
                   double b,
                   double sum)

Description copied from interface: IterativeIntegrator

Compute a refined sum for the integral.

Specified by:

next in interface IterativeIntegrator

Parameters:

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

f - the integrand

a - the lower limit

b - the upper limit

sum - 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