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

Class NewtonCotes

java.lang.Object

com.numericalmethod.suanshu.analysis.integration.univariate.riemann.NewtonCotes

All Implemented Interfaces:

Integrator, IterativeIntegrator, java.io.Serializable

Direct Known Subclasses:

Midpoint, Trapezoidal

public class NewtonCotes
extends java.lang.Object
implements IterativeIntegrator

The Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulae for numerical integration (also called quadrature) based on evaluating the integrand at equally-spaced points. A number of standard numerical quadrature methods are special cases of the more general Newton–Cotes formula, e.g., the Trapezoidal rule (rate = 2), the Midpoint method (rate = 2 and specifying OPEN formula).

This implementation is based on the Euler-Maclaurin formula.

See Also:

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

, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	NewtonCotes.Type There are two types of the Newton-Cotes method: OPEN and CLOSED.

Constructor Summary

Constructors

Constructor and Description
NewtonCotes(int rate, NewtonCotes.Type type, double precision, int maxIterations) Construct an instance of the Newton–Cotes quadrature.

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, \[ \int_a^b\! f(x)\, dx \]
double	next(int iter, UnivariateRealFunction f, double a, double b, double sum0) 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

NewtonCotes

public NewtonCotes(int rate,
                   NewtonCotes.Type type,
                   double precision,
                   int maxIterations)

Construct an instance of the Newton–Cotes quadrature.

Parameters:

rate - the rate of further sub-dividing an integral interval. For example, when rate = 2, we divide [x_i, x_i+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, \[ \int_a^b\! f(x)\, dx \]

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 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 current iteration, counting 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

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