CRITERION FOR RICHARDSON’S EXTRAPOLATIONS

OF  RISK  TECHNICAL  SYSTEMS

 

Dr.Sc. Nikolay  Iv. Petrov Assoc. Prof.

Trakian University – St. Zagora, Yambol, Bulgaria

nikipetrov@lycos.com

 

Abstract: We propose estimators of a round off error contained in an approximation for Richardson’s  extrapolation scheme under finite digit arithmetic. We also propose a stopping criterion, based on consideration of the round off error, for Richardson’s extrapolation scheme of  risk  technical  systems (automobile and railway transport, aircrafts, marine and river transport, chemical installations, munitions, information society suffering by terrorism). Usually the error of an approximation is evaluated by a truncation error. However, we can accurately estimate the behavior of this error utilizing both truncation and round off errors under finite digit arithmetic. We emphasizes that the stopping criterion proposed is independent of tolerance.

 

Key-Words: Numerical analysis; Error analysis; Extrapolation scheme

 

1. Introduction

Under finite digit  arithmetic, an approximation contains an accumulated round off error and an accumulated truncation error. The accuracy of the approximation depends on these errors. We know Richardson’s extrapolation scheme as a scheme that reduces the accumulated truncation error [3]. When we use this scheme for the actual numerical calculation, it can be done so that this accumulated truncation error may not exist under finite digit arithmetic. This means that the size of the accumulated truncation error becomes smaller than the machine epsilon by Richardson’s  extrapolation scheme. Therefore, the accuracy of the approximation depends on round off error only. We adopt such situation as stopping criterion for  Richardson’s extrapolation scheme.

 

2. Preliminary Consideration

          We consider to the following initial value problem of ordinary differential equation [1, 2]:

 

(1)                                    

where  and  are an interval of definition, the accurate solution, respectively.

          The discrete approximation  depends on the step size  for . This  can be characterized as follows , where  is an associated sequence for . We consider that the following  Richardson’s extrapolation scheme is applied to  equation (1):

 

(2)                                            ,

 

 where  is put with . It is well known that  is given by the following expansion:

 

(3)                                ,

 

          Therefore, the truncation error  included by  becomes:

 

(4)                                   .

 

We emphasizes that the first term of  is eliminated if  and  are substituted for the scheme (2). Next, we consider about the finite digit arithmetic. It decides to be carried out under finite digit arithmetic when the scheme (2) is applied to the numerical calculation of the practice. Then, we can get the value which valid digits length was decided as. Here, the arithmetic of the  floating point  digit is carried out under finite digit arithmetic. We introduce the following symbol to express   under finite digit arithmetic  . Furthermore, the end digit of calculation is expressive of . In general,  include an accumulated round off error in finite digits. It can be expressed as follows when this quantity is shown with the . It is shown by equation . Therefore,  shows the round off error that appeared in finite digits.   in finite digits is shown with . Smaller quantity than  isn’t  included by this value.

3. The Evaluation of round off error

          We consider about an accumulated round off error when the scheme (2) is applied. This round off error  is formed by the calculation of , and   determine  of  by the calculation of the recursive scheme (2). We give an estimation of   of .  becomes the following formula by using the recursive scheme (2). It is shown by equation:

                                            ,

where  shows the local round off error which appear by the calculation of the scheme (2).

 

          If   is smaller than .  can be shown with equation:

(5)                                          .

          Consequently, we can show  with:

 

(6)                                               ,

 

 where  is given to it by the following amount,

         

                                                                                                                                                              .

 

Therefore,   becomes:

         

                                         .

          The estimation of   to it with . In general, if an associated sequence  fulfills Toeplitz’s condition [2], each   becomes as follows  .

          From equation (6), we suggest the following formula to estimate  . It is shown be equation:

 

(7)                                                     .

 

          This suggests as follows. Under finite digit arithmetic, it can’t get the accuracy of  beyond the accuracy of .

 

4. A stopping criterion of the extrapolation scheme

          Under finite digit arithmetic, when we look for approximation with the scheme (2),  which has the really same value appears. We employ this value as an approximation for (1). Eventually, when it becomes equation:

(8)                                         ,

we adopt  as an approximation for the initial value problem (1).

We consider the stopping criterion that the propose. Here, the following  criterion is considered for the approximation. It is shown be equation:

(9)                                                   .

          We suppose that   satisfies the criterion (10). From the scheme (2), we can conduct the following formula:

(10)                            .

          Here, if we suppose the following approximate formula:

(11)                                  .

          We can get the following approximate formula:

(12)                                           .

 

          Hence,  of  is shown as follows . We note that the formula (11) is the assumption that the following assumption is equal to . It is denoted that   can’t be adopted as an approximation when the assumption of the formula (11) isn’t  satisfied in the criterion (10). Actually, the problem (1)  that the assumption of the formula (11) isn’t  satisfied in the problem, which has a kind of singularity. We control interval I so that step size can fulfill (11) for such problem [4, 5].

          Next, we consider the round off error for the criterion  (10)  by using the scheme  (2),  and   are shown as follows equation:

(13)                                               .

          We understand the following fact from the recursive formula (5).  It is shown be equation:

(14)                                      .

Therefore, we obtain the following conclusion by using the recursive formula (5). If   is satisfied the criterion (9),  becomes  .

 

5. Our Stopping  criterion

          The stopping  criterion that we propose is the model of the criterion (9). We consider it about this  criterion (9) as a value under finite digit arithmetic.    under finite digit arithmetic can be shown as follows equation:

(15)                                    .

          From the consideration of the preceding paragraph, when  satisfies the criterion (9),   is not present in the calculated digits. Therefore, it becomes equation:

 

(16)                                        .

           From the expansion (4), it becomes . Therefore,  can be shown as the equation:

(17)                                         .

          The above shows that if  and  satisfy the criterion (9), and  aren’t included. And, under finite digit  arithmetic, is not included in finite digits . In the formula (2), when  fulfill the criterion (9), it can indicate  .  Therefore, we can get the following three formula ,  and .          We can get the formula (8) from these things. Because, if an extrapolation scheme is applied,  becomes    and they are not included in finite digits . On the other hand,   becomes .   Next, we consider the criterion (9) for the scheme (2). As for  included in  becomes the formula (11) again. If  is supposed toward the coefficient  in the formula (11) and  is made fully small, it becomes equation:

(18)                                          .

          Therefore, it becomes  if the formula (18) is fulfilled. This can achieve the above assumptions by making interval I small. Backward, how I is made small, estimate can do it when coefficients  of , are very big value in comparison with the assumptions when the formula (18) doesn’t achieve it. In such case, we regard the problem (1) as having numerical singularity.

          As for   included in   becomes the following formula by the recursive formula (5) and the expansion (6):

(19)                                                  .

This denotes that it becomes:

(20)                                               .

          Therefore, it was shown that the scheme  (2) satisfied the criterion  (9). The approximation fulfill the stopping criterion (8) which we proposed, doesn’t encompass a truncation error, and it encompasses only a round off error. The accuracy of this approximation can be estimated by the (7).

 

6. Conclusion

          The conclusions obtained are summarized as follows:

1. The proposed stopping criterion does not depend on the tolerances, hence, the error contained in the approximation is equal to the estimate of the round of the error risk technical systems.

          2. It is possible to find singularity properties of the problem (1), based on the new stopping criterion and controlling interval.

3. As well as the error of approximation is equivalent to accumulated round of error obtained by the new stopping criterion, we confirm a reduction of the error.      

                

REFERENCES

          [1]. Nagasaka H. and M. Murofushi. On the Initial Stepsize an Extrapolations Algorithm for IVP in ODE. Numerical Algorithms, Vol. 3, 1992, p.p. 321-334.

          [2]. Brezinski C. Extrapolation Methods - theory and practice. North – Holland, 1991.

          [3]. Bulirsch R. and J. Stoer. Numerical Treatment of Ordinary Differential Equation by Extrapolation Methods, Numer. Math.,Vol. 8, 1966.         

          [4]. Лазаров, И., М. Андреева, Н. Стоименов  Проблеми при оценяване риска за безопасността при практическото обучение. Научни трудове, том 40, серия 4.2 от Научна конференция “РУ-СУ’03”, Русе, Русенски университет“Ангел Кънчев”, Печатна база при РУ”Ангел Кънчев”-Русе, 31.Х - 01.ХІ.2003, с.205-209.

[5]. Лазаров, И.  Изследване влиянието на антропогенните фактори върху безопасността при практическото обучение. Научни трудове от Научна конференция “РУ-СУ’04”, Русе, Русенски университет“Ангел Кънчев”, Печатна база при РУ”Ангел Кънчев”-Русе, 29.Х - 30.Х.2004.

 

Contacts:

Nikolay Iv.Petrov, Assoc.Prof., Dr.Sc..

Trakian University - Stara Zagora, Yambol, Bulgaria;

 8600 Yambol, Gr.Ignatiev Str. 38;

phone 046/ 66-91-81; e-mail: nikipetrov@lycos.com

 

 

 

 

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