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package org.apache.commons.math3.ode.nonstiff;
import org.apache.commons.math3.util.FastMath;
This class implements the Luther sixth order Runge-Kutta
integrator for Ordinary Differential Equations.
This method is described in H. A. Luther 1968 paper
An explicit Sixth-Order Runge-Kutta Formula.
This method is an explicit Runge-Kutta method, its Butcher-array
is the following one :
0 | 0 0 0 0 0 0
1 | 1 0 0 0 0 0
1/2 | 3/8 1/8 0 0 0 0
2/3 | 8/27 2/27 8/27 0 0 0
(7-q)/14 | ( -21 + 9q)/392 ( -56 + 8q)/392 ( 336 - 48q)/392 ( -63 + 3q)/392 0 0
(7+q)/14 | (-1155 - 255q)/1960 ( -280 - 40q)/1960 ( 0 - 320q)/1960 ( 63 + 363q)/1960 ( 2352 + 392q)/1960 0
1 | ( 330 + 105q)/180 ( 120 + 0q)/180 ( -200 + 280q)/180 ( 126 - 189q)/180 ( -686 - 126q)/180 ( 490 - 70q)/180
|--------------------------------------------------------------------------------------------------------------------------------------------------
| 1/20 0 16/45 0 49/180 49/180 1/20
where q = √21
See Also: Since: 3.3
/**
* This class implements the Luther sixth order Runge-Kutta
* integrator for Ordinary Differential Equations.
* <p>
* This method is described in H. A. Luther 1968 paper <a
* href="http://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99876-1/S0025-5718-68-99876-1.pdf">
* An explicit Sixth-Order Runge-Kutta Formula</a>.
* </p>
* <p>This method is an explicit Runge-Kutta method, its Butcher-array
* is the following one :
* <pre>
* 0 | 0 0 0 0 0 0
* 1 | 1 0 0 0 0 0
* 1/2 | 3/8 1/8 0 0 0 0
* 2/3 | 8/27 2/27 8/27 0 0 0
* (7-q)/14 | ( -21 + 9q)/392 ( -56 + 8q)/392 ( 336 - 48q)/392 ( -63 + 3q)/392 0 0
* (7+q)/14 | (-1155 - 255q)/1960 ( -280 - 40q)/1960 ( 0 - 320q)/1960 ( 63 + 363q)/1960 ( 2352 + 392q)/1960 0
* 1 | ( 330 + 105q)/180 ( 120 + 0q)/180 ( -200 + 280q)/180 ( 126 - 189q)/180 ( -686 - 126q)/180 ( 490 - 70q)/180
* |--------------------------------------------------------------------------------------------------------------------------------------------------
* | 1/20 0 16/45 0 49/180 49/180 1/20
* </pre>
* where q = √21</p>
*
* @see EulerIntegrator
* @see ClassicalRungeKuttaIntegrator
* @see GillIntegrator
* @see MidpointIntegrator
* @see ThreeEighthesIntegrator
* @since 3.3
*/
public class LutherIntegrator extends RungeKuttaIntegrator {
Square root. /** Square root. */
private static final double Q = FastMath.sqrt(21);
Time steps Butcher array. /** Time steps Butcher array. */
private static final double[] STATIC_C = {
1.0, 1.0 / 2.0, 2.0 / 3.0, (7.0 - Q) / 14.0, (7.0 + Q) / 14.0, 1.0
};
Internal weights Butcher array. /** Internal weights Butcher array. */
private static final double[][] STATIC_A = {
{ 1.0 },
{ 3.0 / 8.0, 1.0 / 8.0 },
{ 8.0 / 27.0, 2.0 / 27.0, 8.0 / 27.0 },
{ ( -21.0 + 9.0 * Q) / 392.0, ( -56.0 + 8.0 * Q) / 392.0, ( 336.0 - 48.0 * Q) / 392.0, (-63.0 + 3.0 * Q) / 392.0 },
{ (-1155.0 - 255.0 * Q) / 1960.0, (-280.0 - 40.0 * Q) / 1960.0, ( 0.0 - 320.0 * Q) / 1960.0, ( 63.0 + 363.0 * Q) / 1960.0, (2352.0 + 392.0 * Q) / 1960.0 },
{ ( 330.0 + 105.0 * Q) / 180.0, ( 120.0 + 0.0 * Q) / 180.0, (-200.0 + 280.0 * Q) / 180.0, (126.0 - 189.0 * Q) / 180.0, (-686.0 - 126.0 * Q) / 180.0, (490.0 - 70.0 * Q) / 180.0 }
};
Propagation weights Butcher array. /** Propagation weights Butcher array. */
private static final double[] STATIC_B = {
1.0 / 20.0, 0, 16.0 / 45.0, 0, 49.0 / 180.0, 49.0 / 180.0, 1.0 / 20.0
};
Simple constructor.
Build a fourth-order Luther integrator with the given step.
Params: - step – integration step
/** Simple constructor.
* Build a fourth-order Luther integrator with the given step.
* @param step integration step
*/
public LutherIntegrator(final double step) {
super("Luther", STATIC_C, STATIC_A, STATIC_B, new LutherStepInterpolator(), step);
}
}