dev.nm.analysis.integration.univariate.riemann.newtoncotes

Class Simpson

java.lang.Object

dev.nm.analysis.integration.univariate.riemann.newtoncotes.Simpson

All Implemented Interfaces:

Integrator, IterativeIntegrator

public class Simpson
extends Object
implements IterativeIntegrator

Simpson's rule can be thought of as a special case of Romberg's method. It is the weighted average (or extrapolation) of two successive iterations of the Trapezoidal rule. 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.

Integrating a scalar function

Supply a function that can be evaluated at the finite interval's endpoints and interior grid points. precision compares successive integral estimates by relative error; maxIterations limits grid refinement, not the number of function evaluations. Refinement increases evaluation cost quickly. The returned estimate does not include an error certificate; compare refinements for sensitive calculations. The integrator retains iteration state, so use separate instances for concurrent calculations.

Example

 import dev.nm.analysis.function.rn2r1.univariate.AbstractUnivariateRealFunction;
 import dev.nm.analysis.integration.univariate.riemann.newtoncotes.Simpson;

 AbstractUnivariateRealFunction square = new AbstractUnivariateRealFunction() {
     public double evaluate(double x) { return x * x; }
 };
 double integral = new Simpson(1e-10, 10).integrate(square, 0.0, 1.0);
 assert Math.abs(integral - 1.0 / 3.0) < 1e-10;
 

See Also:

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

Constructor Summary

Constructors

Constructor and Description
Simpson(double precision, int maxIterations) Construct an integrator that implements Simpson's rule.

Method Summary

Modifier and Type	Method and Description
int	getMaxIterations() Get the maximum number of iterations for this iterative procedure.
double	getPrecision() Get the convergence threshold.
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_a^b\!
double	next(int iteration, UnivariateRealFunction f, double a, double b, double sum) Compute a refined sum for the integral.

Methods inherited from class java.lang.Object

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

Constructor Detail

Simpson

public Simpson(double precision,
               int maxIterations)

Construct an integrator that implements 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, \[ \int_a^b\! f(x)\, dx \]

Specified by:

integrate in interface Integrator

Parameters:

f - a univariate function

a - the lower limit

b - the upper limit

Returns:

\(\int_a^b\! f(x)\, dx\)

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 current iteration, counting 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

getMaxIterations

public int getMaxIterations()

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:

getMaxIterations in interface IterativeIntegrator

Returns:

the maximum number of iterations

getPrecision

public double getPrecision()

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:

getPrecision in interface Integrator

Returns:

the precision