org.spaceroots.rkcheck

Class CheckOrderConditions

java.lang.Object

org.spaceroots.rkcheck.CheckOrderConditions

public class CheckOrderConditions
extends java.lang.Object

This application checks the order conditions of Runge-Kutta processes.

It reads the description of a Runge-Kutta method from an XML file having a structure similar to this example:

 <?xml version="1.0" encoding="ISO-8859-1" ?>
 <!DOCTYPE Runge-Kutta PUBLIC "-//spaceroots.org//DTD Runge-Kutta V1.2//EN"
                              "http://www.spaceroots.org/Runge-Kutta.dtd" >
 
 <Runge-Kutta name="Higham and Hall" >
 
   <time-steps>
     <zero/>
     <rational> <p>2</p> <q>9</q> </rational>
     <rational> <p>1</p> <q>3</q> </rational>
     <rational> <p>1</p> <q>2</q> </rational>
     <rational> <p>3</p> <q>5</q> </rational>
     <one/>
     <one/>
   </time-steps>
 
   <internal-weights>
     <row>
       <!-- empty first row -->
     </row>
     <row>
       <rational> <p>  2</p> <q>  9</q> </rational>
     </row>
     <row>
       <rational> <p>  1</p> <q> 12</q> </rational>
       <rational> <p>  1</p> <q>  4</q> </rational>
     </row>
     <row>
       <rational> <p>  1</p> <q>  8</q> </rational>
       <zero/>
       <rational> <p>  3</p> <q>  8</q> </rational>
     </row>
     <row>
       <rational> <p> 91</p> <q>500</q> </rational>
       <rational> <p>-27</p> <q>100</q> </rational>
       <rational> <p> 78</p> <q>125</q> </rational>
       <rational> <p>  8</p> <q>125</q> </rational>
     </row>
     <row>
       <rational> <p>-11</p> <q> 20</q> </rational>
       <rational> <p> 27</p> <q> 20</q> </rational>
       <rational> <p> 12</p> <q>  5</q> </rational>
       <rational> <p>-36</p> <q>  5</q> </rational>
       <integer>5</integer>
     </row>
     <row>
       <rational> <p>  1</p> <q> 12</q> </rational>
       <zero/>
       <rational> <p> 27</p> <q> 32</q> </rational>
       <rational> <p> -4</p> <q>  3</q> </rational>
       <rational> <p>125</p> <q> 96</q> </rational>
       <rational> <p>  5</p> <q> 48</q> </rational>
     </row>
   </internal-weights>
 
   <estimation-weights>
     <rational> <p>  1</p> <q> 12</q> </rational>
     <zero/>
     <rational> <p> 27</p> <q> 32</q> </rational>
     <rational> <p> -4</p> <q>  3</q> </rational>
     <rational> <p>125</p> <q> 96</q> </rational>
     <rational> <p>  5</p> <q> 48</q> </rational>
     <zero/>
   </estimation-weights>
 
   <error-weights>
     <rational> <p> -1</p> <q> 20</q> </rational>
     <zero/>
     <rational> <p> 81</p> <q>160</q> </rational>
     <rational> <p> -6</p> <q>  5</q> </rational>
     <rational> <p> 25</p> <q> 32</q> </rational>
     <rational> <p>  1</p> <q> 16</q> </rational>
     <rational> <p> -1</p> <q> 10</q> </rational>
   </error-weights>
 
 </Runge-Kutta>

This file describes the various coefficients tables of the process. In addition to the elements displayed in this example, coefficients can also be entered as real numbers (like <real>-0.5</real> for example) with an arbitrary precision. Real numbers are converted to rational numbers during parsing, using continued fractions. For simple numbers, this is fine and exact arithmetic can still be used afterwards, however, this is not true if too few digits are given. This feature should be used with care and avoided if possible. The (<error-weights>...</error-weights>) table is optional, it is used only for embedded methods with error control.

Given this description, the application checks the homogeneity conditions and the order conditions, increasing the order until it finds contitions that are not met anymore by the coefficients arrays. It then displays the order of the method. If there is an <error-weights>...</error-weights> entry, the order of the error estimation is also displayed.

Checking is done using exact arithmetic by default. However, if a method involves real numbers for which not enough digits are given or which are not really rational numbers, it is likely that all tests will fail and the method will be declared to be of order 0 ! In this case, it is advised to use a tolerance for the final tests. Be aware however that for high orders, the order conditions typically involve small numbers like 1/7983360, so the tolerance should really be small in order to avoid too optimistic results about the order of a given method. For reliable results, it is far better to have an exact representation of the coefficients and to use only exact arithmetic.

Constructor Summary

Constructors

Constructor and Description
CheckOrderConditions()

Method Summary

Modifier and Type	Method and Description
static void	main(java.lang.String[] args) Entry point of the application.

Methods inherited from class java.lang.Object

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

Constructor Detail

CheckOrderConditions

public CheckOrderConditions()

Method Detail

main

public static void main(java.lang.String[] args)

Entry point of the application.

Parameters:

args - application arguments. This application supports the following arguments :

• -help to display the list of supported arguments
• -verbose to display the residuals for conditions not met
• -tolerance to set the tolerance for use when non-exact checking is desired
• the uri of the XML file to check