A class of implicit advanced step-point methods with a parallel feature for the solution of stiff initial value problems

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ELSEVIER Computers and Mathematics with Applications 40 (2004) 1199-1224 www.elsevier.com/locate/camwa

A Class of I m p l i c i t A d v a n c e d S t e p - P o i n t M e t h o d s w i t h a Parallel F e a t u r e for t h e S o l u t i o n of Stiff Initial V a l u e P r o b l e m s G. PSIHOYIOS Department of Mathematics and Technology Angila Polytechnic University, East Road, Cambridge CB1 1PT, U.K. g. psihoyios©nt lwor id. c o m

A b s t r a c t - - In this paper, a class of implicit advanced step-point methods that possesses a parallel feature is presented. Their accuracy and stability characteristics are examined in some detail and an experimental nonparallel code has been developed in order to give us a fair indication of their capabilities. Our aim is not to discuss a "parallel implementation" but, to demonstrate the worthiness of the new methods. The experimental code is compared with the powerful MEBDF code on the stiff DETEST problem set and the statistics displayed are obtained from the DETEST evaluation package. Some initial conclusions concerning the efficiency of the new methods are drawn from the analysis of the numerical results. @ 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - stiffness, Initial value problems, Advanced step-point methods stability, PIAS methods, Parallel methods, MEBDF. 1. I N T R O D U C T I O N Parallel methods for the efficient solution of initial value problems were suggested many years ago, e.g., [1,2]. The growing interest and technological developments in parallel computers in recent years is the single most important factor that has rekindled the considerable attention that, these methods enjoy. For example more recently, an interesting survey in 1993 on parallel methods for IVPs by Burrage [3] unified earlier suggestions and provided some new ideas for future developments. Since then, there have been further developments on the subject, but given that we are not discussing a "parallel implemention', any such references are not particularly relevant to this paper. Below, we will give a brief account of the modified extended backward differentiation formulae (MEBDF) since we refer to them frequently in the next sections. M E B D F form an important class of methods that belongs to the wider family of Implicit advanced step-point methods [4] (but, also [5]). M E B D F were proposed by Cash [6] in 1983. In 1988, Considine [7] further refined their implementation and in 1992 an M E B D F code for stiff IVPs was presented by Cash and Considine [8]. The M E B D F approach is as follows. STEP 1. Use a standard BDF to compute the first predictor ~n+k, assuming that approximate solutions yn+j have been computed at xn+j, for 0 < j < k - 1 k-1

~,~+k + ~ ~jy~+j = h~kf (zn+~, fJ,~+k),

(1)

j=0 0898-1221/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2005.01.014

Typeset by .AAdS-TEX

1200

Cl. PsIHoYIOS

where dj and bk are the known BDF coefficients. STEP 2. Use a standard BDF to compute the second predictor tdn+k+l k-2

9n+k+l + ak-19~+k + ~

gjYr~+j+l = ht)kf (x~+k+l, O~+k+l) .

(2)

j=O

STEP 3. Assuming that, the Newton iteration converges (see [8] for details), evaluate fn+k = f (X~+k, 9~+k) ,

f~+k+l = f (X~+k+l, 9~+k+1) •

(3)

STEP 4. Compute a corrected solution of order (k + 1) at x~+k using k-1

Yn+k + Z ajy~+j = h [bk+lf~+k+l q- ~)kfn+k q- (bk -- [)k)fn+k] , j=O

(4)

where, (for reasons of computational efficiency) Dk = bk - bk and as, bk+l, bk are given in [9]. The purpose of this paper is to present in some detail a powerful new class of methods for the solution of stiff IVPs and to gain a fair understanding of their capabilities through an experimental code that has been specially developed for these methods. These new methods possess a parallel feature, which we fully document, but it is beyond the scope of this paper to discuss a "parallel implementation". The paper is constructed as follows. In Section 2, we present the general approach, we explain the paralM feature of the methods and we show how the coefficients of the second predictor have been obtained. In Section 3, we discuss in some detail the accuracy and stability of the methods. In Section 4, we briefly explain the key features of the nonparallel evaluation code. In Section 5, we make an important point about the numerical comparisons between our experimental code and the MEBDF code. In Section 6, we briefly discuss the stiff DETEST package and we present a full account and comments of the numerical results obtained from the stiff DETEST problem set. Finally, in Section 7 we summarize our conclusions. In the Appendix, we give the coefficients of the second predictor in detail.

2. T H E

GENERAL

APPROACH

The new methods follow a basic approach, which we may briefly name as "two steps forward (first and second predictor) and one step back (corrector)". The "two steps forward" feature of advanced step-point methods is a "key point" since it may grant them the possibility of being used efficiently on a parallel computer, if appropriate modifications are made. The idea is straightforward. Since, we are using more than one predictor, it is possible that advanced step-point methods in general can be computed in parallel if the formula for the second predictor does not contain the first predictor (as for example the MEBDF do). The proposed algorithm has the following form. (a) Use a standard BDF to compute the first predictor Y~+k of order k. (b) Use an implicit multistep formula to independently compute the second predictor yn+k+l of order k. (Now, clearly Parts (a) and (b) could be computed in parallel if we so wished.) (c) Compute a corrected solution of order k + 1 at x~+k using the MEBDF corrector. An appropriate name for the new methods seems to be "parallel implicit advanced step-point" methods, or H A S methods for short. The crucial choice for the entire PIAS scheme is the formula for the second predictor. From now on, we could call the two predictors: "first parallel predictor" and "second parallel predictor".

Implicit Advanced Step-Point Methods The PIAS

General

1201

Form

Before arriving at a final choice for the second (parallel) predictor, several alternative formulae were investigated. Thus, after a thorough investigation a class of parallel implicit advanced step-point methods was developed that has the following general form. STEP 1. Use a standard BDF to compute the first predictor 9~+k, assuming that approximate solutions y~+j have been computed at xn+j, for 0 _< j _< k - 1 k-1 (]n-[-k "JY E

ajyn+j =

(6)

hbkf (x~+k, 9~+k) ,

j=o where ~j and/~k are the known BDF coefficients. STEP 2. Use an implicit multistep formula to compute the second predictor Y~+k+l

k-1 Yn+k+, -- EajYn-t-j : h [bk+lf (Xn-l-k-t-l,~]n-Fk+l) -t-bk-,fn-t-k-1 ] , j=O

(7)

where 5j, bk+l are coefficients that will be given in Section 3 and bk-1 is the free coefficient in terms of which the other coefficients are expressed. STEP 3. Assuming that, the Newtonian iteration converges, evaluate

k-1 ]~+k+l

~1

-

hbk+l

~+k+l - ~

j=o

] ~jy,~+j - hbk-lfn+k-1

(8) f~+k= ~

~n+k + E a S Y ~ - j j=O

"

STEP 4. Compute a corrected solution of order (k + 1) at x~+k using k-,

Yn+k + E ajy~+j = h [bk+l]~+k+l ÷ bk]~+k + (bk-/~k)fn+k] , j=O

(9)

where/~k = bk - bk and aj, bk+l, bk are given in [9]. 2.1. T h e Coefficients of t h e S e c o n d P a r a l l e l P r e d i c t o r Let us give an example, for k = 2, as to how we obtain the coefficients of (7). Making the usual localizing assumption the right-hand side of (7) (k = 2, in this case), if expanded in Taylor series, becomes

~

h2 h3 ) y~ + hy~l + -~y~I I + -d-y,~i l l + 5og~

+h [b3 (y" + 3hy~'# + 9h2y~# 2 )+

51 (ytn -~-hy~

h2 H, + O(h4), +TY~)]

(io)

and if we rearrange (10) we get

+

~al + 3/~3 -k bl

Yn + -~al + -~ 3 -~- -~bl h3y Ill (xn) + 0 (h4).

(11)

1202

O. PsIHOYiOS

Expanding y(Xn+3) using Taylor series and equating Taylor series coefficients of the like powers with (11) we have

4 4~

a o : - g + g ~,

I = 51 +50,

9

3 = 51 + b3 + / h , 9 1

= ~

::~

(ZI= 5 6 b3 5

~,

+ 3~,~ +

4g

(12)

g 1, 1~ ~bl,

and the LTE of the second parallel predictor is y(Xn+S)--Yn+3 =

--

6a1+~

(Xn)+O(h4).

3+~bl

If we substitute (12) into the above equation the LTE of the second parallel predictor for k = 2 becomes

LTE ~_ y(Xn-}-3)--Yn+3 -~- (--~ + 8D1) h3yttt(xn) +0 (h4).

(13)

As we see from the above analysis, there is one free coefficient, bl, and all the other coefficients are expressed in terms of bl. The coefficients of (7) are given in the Appendix.

3. A C C U R A C Y

A N D S T A B I L I T Y OF P I A S M E T H O D S

The evaluation of the LTE is a rather tedious task and therefore it will be shown how it is obtained for a particular order, and then conclusions will be drawn for the remaining orders. Let us consider the case for k = 2. FIRST PARALLEL PREDICTOR.

4

2h~ ,

1

(order 2),

and y (z~+2) - G+2

2h3ytH(Xn)+O(h4).

. . . .

9

SECOND PARALLEL PREDICTOR. =

-

Y~+I+

+-~

1

y,~+h

Yn+3 + blYn+l ~ ,

-

(order 2),

and from (13),

CORRECTOR.

28

5

[-4,

20_,

2 , 1

y~+2 = ~ y ~ + l - ~-~y~ + h "~-~)n+a + ~Y~+2 + ~Yn+2 ,

(order 3),

(14)

and

~+3 ~ f (x,~+3,y(x,~+a))+ (6 - 8 bl) h3~yy'" (xn) , 2h30f

(15) it,

Implicit Advanced Step-Point Methods

1203

If we make the usual localizing assumption and expand the right-hand side of (14), we get

1)

--23 Yn + hytn + h2y~ + h3y~' +

20 (

,,

..-2

h4Y(4) _ '~Y'~5

,,,

(16)

4_h3~,(4)+ 2h3~f ,,'~

Rearranging (16) we get

y,~+2hy~ + 2h2 "yn+ -~h43 "'yn+ 25h% ( 4 ) + 4~6

+-3-~1J h4

-

Y~'+O(hS)"

(17)

Expanding y(Xn+2) using Taylor series and equating the coefficients of the like powers with (17) we finally get

LT E = ~-~sh 4y (4) (x n ) + ( ~--~5

+ o

•

(18)

If we do a similar analysis for the other orders we will obtain analogous results. Thus, we may conclude, considering (18), that the LTE of the scheme has the form LTE = Ahk+2y (k+2) (x~) + Bhk+2 ~-~y (k+l) (x,~) + O (hk+3),

(19)

where .~ is a constant that depends on the known M E B D F coefficients a n d / ~ is also a constant that depends on the M E B D F coefficients and on the coefficients of (7). The PIAS algorithm given in Section 2 displays good stability characteristics, if we appropriately choose the free coefficient, bk-1 from (7). Thus, we choose bk-1 so as to achieve the best possible absolute stability. In Table 1, we give the A(a)-stability of PIAS methods compared to the M E B D F where the coefficients are chosen so that bk = bk - bkStability analysis of these methods shows that low order PIAS methods are A-stable and not L-stable as is the case with MEBDF. T a b l e 1. A ( a ) - s t a b i l i t y o f P I A S m e t h o d s . K

1

2

3

4

5

6

7

Order

2

3

4

5

6

7

8

PIAS

90 °

90 °

90 °

90 °

80 °

62 °

47 °

90 °

90 °

90 °

88.4 °

82.5 °

74.5 °

62 °

MEBDF

bk = bk - bk 4. A S S E S S I N G

THE

POTENTIAL

OF

THE

PIAS

METHODS

In order, to find out whether the "parallel feature" and the good stability characteristics of the new methods are reflected in practice, we developed an experimental code purely for the purpose of getting a glimpse of the potential of the PIAS scheme. This experimental code is based on the M E B D F code on which appropriate modifications were performed. The M E B D F code has been presented in great detail in [7,8] and the reader is referred to these publications for further

1204

G. PSIHOYIOS T a b l e 2. Second p a r a l l e l p r e d i c t o r coefficients u s i n g b a c k w a r d differences. ~6

W5

W4

W3

W2

Wl

W0

1

4(1-~k_1)

1

9--4bk_1 13

2(7--6bk-1) 13

1

288--42%k_1 462

324--259bk_ 1 308

188--175bk_1 154

1

600--107bk_ I 1044

264--61bk_ 1 261

222--77bk_ I 174

I14--85bk_ 1 87

1

6(100--13bk_i) 1115

7800--1237bk_ I 8028

2568--539bk_ 1 2007

1914--619bk_ I 1338

918--659bk_ I 669

1

4500--529bk_ 1 4810

3650--529bk_ 1 2886

4(41--8bk_1)

743--228bk_ 1 481

2(341--238bk_1) 481

1

5

2940--293bk_ 1 5772

IIi

details. It is outside the scope of the present paper to discuss code implementation, especially since our experimental code has been created mainly for evaluation purposes. The only feature of the M E B D F code that we need to mention is that it treats the two BDF Steps 1 and 2 as being entirely separate from each other and holds estimates for the convergence rates for the Newton-Raphson iteration for the first step and second step. The experimental code was developed in order to aid us in estimating the parallel feature of the PIAS methods and it is not a code written for parallel processors. The experimental PIAS code maintains all the main features of the M E B D F code but has also certain distinct characteristics that are summarized below (for the sake of illustrating that, the comparisons that follow are meaningful). • The feature of the M E B D F code that treats the two predicted steps as entirely separate from each other is particularly useful, since such a feature is necessary for the experimental PIAS code. On a parallel machine, the two predictors would be computed separately and simultaneously. In this case, the two predictors are computed serially but maintain their independence from each other. In addition, the M E B D F code has been reprogrammed to accommodate two separate iteration matrices, one for the first parallel predictor and another one for the second parallel predictor. This leads, of course, to a substantially larger number of aacobian evaluations as we will see in Section 6. The corrector uses the same iteration matrix as the first predictor. • The M E B D F code had also to be reprogrammed in order to accommodate the new coemcients for the predictors and the corrector, which needed to be given in backward difference form. Thus, the coefficients of (7), Table 1, had to be rewritten in terms of backward differences according to the formula k-1

Yn+k+l - E ffJiViY'~+k-1 = h[bk+lf(x,~+k+l, Y~+k+l) + bk-lf~+k-1].

(20)

i=O

The M E B D F code uses backward differences and so does the experimental PIAS code. Hence, the coefficients used in the experimental code are taken from (20). The coefficients on the left-hand side of (20) are given in Table 2. It should be stressed that the experimental PIAS code used here is only for the purpose of assessing the capabilities of these new methods and does not show their full potential. It is within our plans though to write a parallel code for the PIAS methods in the future. 5.

A

NOTE

ABOUT

THE

NUMERICAL

COMPARISONS

Since the experimental PIAS code is not a parallel code, it does not take advantage of the methods' property for parallel computation of the second predictor. In practice, this means that,

Implicit Advanced Step-Point Methods

1205

the amount of time displayed, for the solution of any particular problem, is more than it would be if parallel processors were used. It would be thus unfair to present any "nonparallel" numerical results as if they obtained from a genuinely parallel computation. Therefore, in order to take into consideration this property of the PIAS methods we have to reduce the "Time" and "Ovhd." (overhead) columns in the numerical results by an appropriate ratio as we explain below. It seems reasonable to expect that, if the experimental PIAS code was using the parallel property of the methods, the amount of time would be reduced by 1/3, if we accept that the code spends approximately a third of its time for computing each of the three stages, the first predictor, the second and the corrector. Although this may seem like an arbitrary decision, the truth is that we are probably being unfair to the PIAS algorithm since there is some experimental evidence which suggests that the code spends a lot more than 2/3 of its time on the first two stages. In any case and in the interest of fairness, we decided to multiply the time required for the integration, that the D E T E S T package computes in each problem, by 2/3,

• The columns "Time" and "Ovhd." (overhead) in the experimental PIAS code numerical results have been multiplied by 2/3 in order to approximate the parallel property of the methods. 6.

STIFF

DETEST

AND

NUMERICAL

RESULTS

Different codes are based on different discretization methods and thus they commit different errors. Codes for IVPs control an estimate of the local error and not in general an estimate of the global error, the magnitude of which we do not know. Generally, it is hoped that, the global error will be in the range of the error tolerance specified by the user. T h a t is if we keep local truncation error suitably banded then, the global error will also be banded. In practice, though it is normally the case that, the global error is larger than, the specified local error tolerance. When two different codes are applied to the same problem with the same local tolerance they may, at the same time, have a quite different global error. This difference in the global error makes the comparison of different codes a rather difficult issue and that is why it is not sufficient to make a comparison by counting only the number of function evaluations, the number of steps taken to complete the integration and the time required for the integration. The D E T E S T program [10,11] provides us with a reliable estimate of the maximum global error over the whole range of integration. This facility allows DETEST to compute maximum global error = max Ily(x,O - Yn II,

n = 1, 2 , . . . ,

where Yn is the numerical solution generated by the user's code and y(xn) is the true solution at the point xn. D E T E S T is able to compute an accurate approximation to both the maximum global error and the global error at the endpoint by implementing a very accurate IVP integration method that runs at a stricter tolerance than, the one prescribed by the user. D E T E S T has also the ability to display information about the amount of work needed to achieve a specified accuracy, i.e., to keep the maximum global error less or equal to a given bound. Consequently, the comparison between different codes is much fairer because the numerical results correspond to solutions with the same maximum global error. D E T E S T refers to such results as normalized efficiency results. Below, we obtain numerical results for the experimental PIAS and the M E B D F codes on the 30 stiff D E T E S T problems [10,11]. The M E B D F code was chosen for the comparisons not only because it has been used as a basis for the experimental PIAS code but also because it is one of the most powerful stiff IVP solvers available. The stiff D E T E S T program includes 30 stiff problems which belong to 6 different categories. A detailed description of the problems in Groups (A) to (E) can be found in [10] and Group (F) is described in [11]. (A) This is a set of four linear systems (A1)-(A4) with constant coefficients and the eigenvalues of the Jacobian matrix are real.

1206

G. PSlHOYIOS

(B) This group includes five linear systems (B1)-(B5) with constant coefficients and nonreal Jacobian eigenvalues. These problems according to Enright [10] have been designed to cause a substantial number of stepsize changes. (C) This is a set of five nonlinear triangular systems (C1)-(C5) with real Jacobian eigenvalues. (D) This is a set of six nonlinear systems (D1)-(D6) with real Jacobian eigenvalues. These problems are mainly drawn from chemical kinetics. (E) This set includes five nonlinear systems (E1)-(E5) with nonreal Jacobian eigenvalues. These problems are mainly drawn from physics, control theory, and chemistry. (F) This is a set of five nonlinear systems (F1)-(F5) from chemical kinetics. The DETEST package gives two types of result tables: the 'regular' nonnormalized results and the "normalized efficiency results". As explained above, the second type of results is extremely useful for fairly comparing the performance of different codes. Thus, in the interest of space, only the "normalized efficiency results" will be presented (with the exception of problem (F5)) together with the summary results for each group, which are nonnormalized. The tables below include information of the time which is required for the solver to achieve a maximum global error less than or equal to the "Expected Accuracy", the respective overhead (Ovhd.) in seconds (this is equal to the time taken for solving a problem minus the time taken for the function evaluations), the number of function evaluations (Fcn. Calls), the number of Jacobian evaluations (Jac. Calls), the number of matrix factorisations (Mat. Fact), and the number of integration steps taken (No. of Steps). The logarithm of the estimate of the local tolerance that, the solver should be supplied with in order to keep the maximum global error less than or equal to the "Expected Accuracy", is given by the column "Equiv. Logl0 Tol.". If the DETEST's intrinsic solver fails to complete successfully the integration for a given tolerance then, the amount of output displayed for the normalized efficiency tables may vary. 6.1. R e s u l t s for t h e Stiff D E T E S T G r o u p A P r o b l e m s

The nonnormalized summary results from Table 7 show that, the experimental PIAS code is faster (this is also true for each individual problem). If we look at the normalized efficiency Table 3. Stiff D E T E S T normalized efficiency results for problem (A1).

PIAS

MEBDF

Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

I0"* --3

--2.12

0.020

0.017

100

2

15

No. of Steps 44

10"* --4

--3.18

0.032

0.029

141

1

24

69

10"* --5

--4.23

0,051

0.046

194

5

33

101

10"* --6

--5.28

0.071

0.065

233

4

39

136

10"* --7

--6.33

0.086

0.080

274

3

45

164

10"* --8

--7.38

0,104

0.097

324

3

51

200

10"* --9

--8.44

0,132

0.123

395

3

60

251

10"* --10

--9.49

0.172

0.161

502

2

70

325

10"* --4

--2.75

0.036

0.033

92

1

13

59

10"* --5

--3.71

0,052

0.048

128

2

15

82

10"* --6

--4.67

0.069

0.065

160

1

16

105

10"* --7

--5.64

0.092

0.086

196

2

18

132

10"* --8

--6.60

0.113

0.107

236

2

22

162

10"* --9

--7.56

0.138

0.130

284

2

26

199

10"* --10

--8.52

0.167

0.158

341

2

30

245

10"* --11

--9.49

0.207

0.197

415

2

33

302

1207

Implicit Advanced Step-Point Methods Table 4. Stiff DETEST normalized efficiency results for problem (A2).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 4

-2.99

0.085

0.061

153

1

28

80

10"* - 5

-4.20

0.141

0.106

227

2

40

125

10"* - 6

-5.41

0,214

0.169

302

2

52

181

10"* - 7

-6.63

0.290

0.235

380

3

62

238

10"* - 8

-7.84

0.392

0.319

507

6

84

319

10"* - 9

-9.05

0.487

0.404

589

2

94

398

10"* - 4

-2.48

0.084

0.062

99

1

16

64

10"* - 5

-3.53

0.136

0.109

143

1

19

98

10"* - 6

-4.58

0.203

0.169

200

1

23

137

10"* - 7

-5.63

0.284

0.242

261

2

27

182

10"* - 8

-6.68

0.375

0.327

321

1

31

229

10"* - 9

-7.72

0.466

0.406

405

1

37

287

10"* - 1 0

-8.77

0.566

0.497

487

2

43

353

Table 5. Stiff DETEST normalized efficiency results for problem (A3).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.19

0.027

0.020

120

1

22

10"* - 4

-3.23

0.044

0.034

176

1

34

95

10"* - 5

-4.27

0.069

0.055

246

3

44

139

10"* - 6

-5.30

0,103

0.085

327

5

55

197

10"* - 7

-6.34

0.120

0.100

371

4

60

229

10"* - 8

-7.38

0.149

0.125

454

3

70

281

10"* - 4

-2.64

0.044

0.035

118

1

16

76

10"* - 5

-3.64

0.066

0,053

163

1

20

107

10"* - 6

-4.63

0.092

0.076

209

1

22

141

10"* - 7

-5.62

0.121

0.102

262

1

26

180

10"* - 8

-6.62

0.158

0.134

322

1

30

225

10"* - 9

-7.61

0.199

0.170

391

1

35

276

61

Table 6. Stiff DETEST normalized efficiency results for problem (A4).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.12

0.077

0.054

136

1

22

71

10"* - 4

-3.13

0.132

0.089

199

2

43

111

10"* --5

-4.13

0.202

0.154

264

1

47

159

10"* --6

-5.14

0.279

0.220

338

4

60

211

10"* - 7

-6.15

0.363

0.295

410

3

65

263

10"* - 8

-7.15

0.444

0.365

489

3

75

324

10"* - 4

-2.71

0.142

0.110

137

1

19

93

10"* - 5

-3.62

0.210

0.171

190

1

23

130

10"* - 6

-4.53

0.292

0.247

248

1

26

171

10"* - 7

--5.44

0.384

0.331

307

2

29

215

10"* --8

-6.35

0.482

0.423

368

1

33

262

1208

G. PSIHOYIOS Table 7. Summary results for Group A problems (nonnormalized).

PIAS

MEBDF

Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

-2.00

0.166

0.122

441

6

72

No. of Steps 216

End Pnt. Glb. Err.

Maximum Glb. Err.

0.02

0.03

-3.00

0.280

0.202

641

6

127

340

0.02

0.14

-4.00

0.428

0.331

880

12

158

483

0.17

0.28

-5.00

0.617

0.492

1133

16

199

681

0.12

0.52

-6.00

0.806

0.661

1365

1

226

839

0.00

0.58

-7.00

0.967

0.806

1591

12

248

1009

0.01

1.07

-8.00

1.240

1.027

1985

20

320

1270

0.01

1.28

-9.00

1.520

1.275

2337

8

370

1570

0.01

0.30

-I0.00

1.932

1.637

2981

16

430

2001

0.05

0.06

-2.00

0.223

0.167

340

7

58

218

0.13

0.13

-3.00

0,354

0.282

506

4

71

334

0.02

0.04

-4.00

0.537

0.447

699

7

84

471

0.02

0.03

-5.00

0.745

0.638

901

6

93

614

0.03

0.04

-6.00

0.991

0.863

1122

I0

ii0

783

0.01

0.14

-7.00

1.254

1.106

1353

5

125

958

0.01

0.04

-8.00

1.532

1.351

1675

7

150

1192

0.Ol

0.23

-9.00

1.858

1.642

2021

7

174

1469

0.00

0.04

-i0.00

2.391

2.130

2574

7

201

1869

0.05

0.03

results, i.e., Tables 1-6 we see that MEBDF is more accurate and is also faster only in problem (A2). The experimental PIAS code maintains a good accuracy (meaning that, the "Equiv. Logl0 Tol." is less than, the "Expected Accuracy") and is clearly faster on problem (A4), slightly faster on problem (A1) and as far as problem (A3) is concerned, both codes are approximately equally fast. Table 8. Stiff DETEST normalized efficiency results for problem (B1).

PIAS

MEBDF

Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

i0"* --2

--1.98

0.060

i0"* --3

-2.89

0.084

0.050

273

1

27

0.072

317

1

36

I0"* --4

-3.80

0.130

181

0.114

404

1

47

260

10"* --5

-4.71

0.178

0.159

10"* --6

--5.62

0.237

0.213

516

1

54

350

662

2

62

10"* --7

-6.53

0.312

0.282

465

850

2

70

10"* --8

--7.44

0.410

612

0.373

1089

1

80

800

10"* --9

--8.34

0.528

0.482

1389

1

90

1037

i0"* -3

--2.40

0.091

0.080

226

1

17

150

i0"* --4

--3.33

0.139

0.122

314

1

23

217

10"* --5

-4.26

0.202

0.181

417

1

27

301

10"* --6

-5.19

0.282

0.255

547

1

31

408

10"* - 7

--6.12

0.360

0.327

694

1

33

524 678

128

i0"* --8

--7.05

0.467

0.425

892

i

35

I0"* --9

--7.97

0.626

0.572

1153

1

48

879

i0"* --I0

--8.90

0.791

0.726

1464

1

39

1135

i0"* --ii

--9.83

1.033

0.948

1896

2

50

1471

Implicit Advanced Step-Point Methods 6.2. R e s u l t s

for the Stiff DETEST

Group

1209

B Problems

In t h e n o n n o r m a l i z e d s u m m a r y results from Tab l e 13 t h e e x p e r i m e n t a l H A S

code is faster

(this is also t r u e for each i n d i v i d u a l p r o b l e m ) . T h e n o r m a l i z e d efficiency results, i.e., Tables 8-12,

Table 9. Stiff DETEST normalized efficiency results for problem (B2).

HAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

10"* - 3

-2.14

0.026

0.021

89

10"* - 4

-3.19

0.043

0.034

10"* - 5

-4.24

0.064

0.054

10"* - 6

-5.29

0.088

0.077

10"* - 7

-6.34

0.112

0.098

10"* - 8

-7.40

0.138

10"* - 9 10"* - 1 0

-8.45 -9.50

10"* - 4

No. of Steps

2

13

39

126

1

24

62

173

4

28

87

200

3

33

115

240

4

38

143

0.122

284

3

43

174

0.174 0.222

0.155 0.199

352 436

3 3

52 61

221 283

-2.62

0.044

0.037

74

2

11

49

10"* - 5

-3.60

0.066

0.058

107

2

13

69

10'* - 6

-4.59

0.090

0.081

138

1

15

90

10'* - 7

-5.58

0.118

0.107

167

2

16

113

10"* - 8

-6.56

0.148

0.135

201

2

19

139

10"* - 9

-7.55

0.180

0.165

244

2

23

172

10"* - 1 0

-8.53

0.222

0.204

302

2

27

215

10"* - 1 1

-9.52

0.284

0.263

377

2

31

271

Table 10. Stiff DETEST normalized efficiency results for problem (B3).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.43

0.034

0.027

102

1

18

51

10"* - 4

-3.42

0.053

0.043

140

2

25

73

10"* - 5

-4.41

0.074

0.062

174

2

29

97

10"* --6

-5.39

0.094

0.082

208

2

34

123

10"* - 7

-6.38

0.120

0.105

255

2

41

155

10"* - 8

-7.37

0.149

0.131

307

1

48

194

10"* - 9

-8.36

0.185

0.165

370

1

55

241

i0"* - i 0

-9.35

0.233

0.210

455

I

61

302

I0"* -4

--2.60

0.042

0.036

79

I

Ii

52

i0"* -5

--3.58

0.066

0.057

112

i

14

74

i0"* --6

--4.56

0.091

0.082

146

1

16

98

i0"* --7

--5.54

0.123

0.112

177

1

17

122

I0"* --8

-6.51

0.157

0.144

214

2

20

150

10"* - 9

-7.49

0.193

0.177

269

2

24

188

i0"* --i0

--8.47

0.239

0.220

329

2

28

234

i0"* --ii

--9.45

0.301

0.278

401

2

32

294

1210

C. PSIHOYIOS

show that, the MEBDF code is more accurate and that, the experimental PIAS code is faster maintaining its good accuracy. In problem (B5) both codes are approximately equally fast at tolerances from 10 -6 and above. We would actually expect the MEBDF code for be faster on (B5) since this particular problem has eigenvalues very near the imaginary axis and MEBDF has better stability for k > 5. Table 11. Stiff D E T E S T normalized efficiency results for problem (B4).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

10'* - 3 10'* - 4

-2.77 -3.69

0.054 0.092

0.045 0.080

10"* 10"* 10'* 10'*

152 225

1 2

25 34

78 124

-4.61 -5.53 -6.46 -7.38

0.114 0.138 0.176 0.225

0.100 0.121 0.156 0.203

259 297 362 446

1 2 2 1

40 46 53 61

150 181 230 294

10'* - 9 10"* - 1 0

-8.30 -9.22

0.286 0.360

0.259 0.332

554 683

3 3

70 76

375 478

10"* - 4 10"* - 5

-2.77 -3.73

0.066 0.101

0.058 0.089

115 160

1 1

14 18

75 107

10"* - 6

-4.69

0.142

0.128

210

1

21

143

10"* - 7 10'* - 8 10"* - 9

-5.65 -6.61 -7.56

0.182 0.229 0.301

0.166 0.211 0.280

261 319 396

1 1 1

23 27 31

183 229 287

10"* - 1 0 10"* - 1 1

-8.52 -9.48

0.385 0.495

0.359 0.464

502 641

2 1

36 39

369 479

-5 -6 -7 -8

No. of Steps

Table 12. Stiff D E T E S T normalized efficiency results for problem (B5). Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 2

-2.14

0,094

0.082

263

1

30

134

10"* - 3

-3,09

0.130

0.113

334

4

42

194

10"* - 4

-4,05

0.221

0.200

526

3

49

312

10"* - 5

-5.00

0.281

0.258

580

3

59

376

10"* - 6

-5.95

0.368

0.340

702

3

65

494

t0"* - 7

-6.91

0.496

0.461

954

4

78

667

10"* - 8

-7.86

0.669

0.624

1285

3

95

914

10"* - 9

-8.82

0.857

0.807

1587

3

98

1157

10"* --3

--2.30

0.118

0.107

196

1

17

136

10"* --4

--3.24

0.195

0.179

293

1

24

208

10"* --5

--4.18

0.268

0.249

387

1

26

281

10"* --6

--5.12

0.371

0.347

504

1

30

376

10"* --7

--6.06

0.496

0.468

651

1

33

492

10"* --8

--7.01

0.668

0.633

854

1

39

644

10"* --9

--7.95

0.869

0.825

1122

1

45

855

I0"* --10

--8.89

1.127

1.075

1450

I

46

1114

PIAS

MEBDF

Implicit Advanced Step-Point Methods

1211

Table 13. S u m m a r y results for Group B problems (nonnormalized).

PIAS

Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

End Pnt. Glb. Err.

Maximum Glb. Err.

--2.00

0.237

0.201

818

8

i00

392

0,00

1.55

--3.00

0.350

0.295

1041

9

--4.00

0.588

0.517

1535

16

156

577

0.01

2.24

193

898

0.02

0.70

--5.00

0.759

0.676

1776

10

--6.00

1.005

0.904

2235

17

223

1133

0.01

0.76

258

1499

0.01

0.82

--7.00

1.327

1.206

2885

12

298

1979

0.01

1.08

--8.00

1.767

1.615

3781

12

358

2654

0.12

1.24

--9.00

2.257

2.080

4728

11

382

3406

0.19

0.76

--10.00

3.014

2.794

6176

14

451

4545

0.13

0.64

-2.00

0.274

0.236

566

7

63

368

0.01

0.57

-3.00

0.470

0.416

850

6

88

581

0.01

0.24

-4.00

0.692

0.624

1175

9

101

816

0.01

0.15

-5.00

0.981

0.898

1543

6

116

1112

0.00

0.09

-6.00

1.298

1.200

1961

8

126

1445

0.01

0.09

MEBDF

-7.00

1.711

1.588

2531

8

146

1879

0.05

0.09

-8.00

2.259

2.103

3306

9

181

2470

0.05

0.11

-9.00

2.905

2.718

4239

10

184

3221

0.05

0.10

-10.00

3.822

3.587

5545

10

220

4240

0.06

0.10

6.3. R e s u l t s for t h e S t i f f D E T E S T Group C P r o b l e m s The nonnormalized summary results from Table 19 show that, the experimental H A S code is faster on all five problems (this is also true for each individual problem). The normalized Table 14. Stiff D E T E S T normalized efficiency results for problem (C1).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

10"* - 3

-2.02

0.022

0.018

10"* - 4

-3.05

0.038

10"* - 5

-4.07

0.059

No. of Steps

101

17

17

0.030

147

28

28

79

0.047

210

38

38

113

51

10"* - 6

-5.10

0.080

0.064

256

42

42

150

10"* - 7

-6.13

0.103

0.090

302

50

50

192

10"* - 8

-7.15

0.129

0.112

364

56

56

235

10"* - 9

-8.18

0.164

0.141

454

74

74

300

10"* - 1 0

-9.21

0.205

0.178

560

86

86

376

10"* - 4

--2.75

0.044

0.036

103

16

16

70

10"* --5

--3.73

0.064

0.054

148

19

19

100

10"* --6

--4.71

0.090

0.077

192

20

20

132

10"* --7

--5.68

0.118

0.103

241

23

23

167

10"* --8

--6.66

0.148

0.130

288

26

26

204

10"* --9

--7.64

0.182

0.161

354

31

31

250

10"* --10

--8.61

0.225

0.200

434

38

38

312

i0"* - 1 1

-9.59

0.281

0.249

544

45

45

393

1212

G. PSIHOYIOS Table 15. Stiff D E T E S T normalized efficiency results for problem (C2). Expected Accuracy

PIAS

MEBDF

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-1.97

0.020

0.014

91

14

14

45

10"* - 4

-3.09

0.035

0.027

144

26

26

75

10"* --5

-4.22

0.057

0.046

210

34

34

112

10"* - 6

-5.35

0.080

0.067

253

45

45

152

10"* - 7

-6.48

0.106

0.090

308

49

49

194

10"* - 8

-7.60

0.130

0.115

381

61

61

246

10"* - 9

-8.73

0.170

0.146

468

72

72

310

I0"* -4

-2.54

0.033

0.026

88

13

13

58

I0"* -5

-3.60

0.053

0.044

131

16

16

87

10"* - 6

--4.65

0.079

0.067

177

18

18

119

10"* - 7

-5.70

0.116

0.101

227

21

21

156

10"* - 8

-6.75

0.144

0.126

275

24

24

192

10"* - 9

-7.80

0.176

0.154

339

29

29

239

10"* - 1 0

-8.85

0.224

0.197

422

36

36

303

10"* - 1 1

-9.90

0.288

0.253

548

46

46

395

Table 16. Stiff D E T E S T normalized efficiency results for problem (C3).

PIAS

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

10"* - 3

--1.95

0.020

0.014

10"* - 4

--3.11

0.035

10"* --5

-4.27

10"* - 6

No. of Steps

89

13

13

43

0.027

146

27

27

75

0,058

0.047

207

36

36

113

-5.43

0.080

0.066

261

44

44

153

10"* - 7

--6.59

0.106

0.090

313

49

49

195

10"* --8

--7.75

0.140

0.120

390

63

63

254

10"* --9

--8.91

0.180

0.154

493

80

80

327

i0"* -4

-2.59

0.036

0.029

91

13

13

59

i0"* -5

-3.58

0.053

0.044

130

16

16

86

10"* --6

--4.57

0.075

0.063

172

19

19

116

10"* --7

--5.56

0.106

0.091

215

21

21

149

10"* --8

-6.55

0.135

0.118

261

24

24

183

10"* --9

--7.54

0.164

0.143

321

28

28

226

10"* --10

-8.53

0.203

0.178

396

34

34

282

10"* --11

-9.52

0.259

0.228

491

40

40

355

MEBDF

eificiency results, i.e., Tables 14-18, show that, the experimental PIAS code is clearly faster on problem (CI), slightly faster on problem (C2) and approximately equally fast on problem (C3). On the other hand, the MEBDF code is again more accurate and is also faster on problem (C5) and slightly faster on problem

(C4).

1213

Implicit Advanced Step-Point Methods Table 17. Stiff DETEST normalized efficiency results for problem (C4).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

10"* - 3

-2.52

0.020

0.015

10"* - 4

-3.60

0.034

10"* - 5

-4.68

10"* - 6 10'* - 7 10"* - 8

No. of Steps

93

13

13

43

0.026

153

21

21

71

0.053

0.042

212

30

30

107

-5.75

0.072

0.058

280

40

40

140

-6.83

0.101

0.085

340

46

46

193

-7.91

0.131

0.110

406

66

66

249

10"* - 9

-8.99

0.164

0.140

485

69

69

310

10'* - 1 0

-10.06

0.211

0.180

597

89

89

385

10'* - 3

-2.18

0.021

0.016

58

9

9

35

10"* - 4

-3.10

0.033

0.027

84

13

13

56

10"* - 5

-4.01

0.047

0.039

122

13

13

81

10"* - 6

-4.92

0.069

0.058

163

17

17

106

10"* - 7

-5.83

0.090

0.077

200

19

19

131

10"* - 8

-6.75

0.113

0.098

239

22

22

162

10'* - 9

-7.66

0.141

0.123

285

25

25

199

I0'* -i0

-8.57

0.175

0.154

341

29

29

244

I0"* -II

-9.48

0.220

0.194

416

34

34

300

Table 18. Stiff DETEST normalized efficiency results for problem (C5).

PIAS

MEBDF

Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

10"* --3

--2.41

0.019

0.014

10"* --4

--3.50

0.035

10"* --5

--4.58

10"* --6

No. of Steps

99

14

14

43

0.026

164

25

25

71

0.052

0.041

213

35

35

104

--5.67

0.074

0.060

277

44

44

144

10"* --7

--6.75

0.102

0.084

369

55

55

194

10"* --8

--7.83

0.122

0.101

387

61

61

230

10"* --9

--8.92

0.152

0.130

451

65

65

285

10"* --10

--10.00

0.199

0.170

553

82

82

365

i0"* --3

--2.27

0.021

0.017

56

9

9

34

i0"* --4

--3.17

0.033

0.027

81

ii

ii

52

10"* --5

--4.07

0.048

0.040

119

14

14

75

i0"* --6

--4.97

0.066

0.055

153

18

17

i01

i0'* --7

-5.87

0.084

0.072

187

19

19

125

i0"* --8

--6.76

0.108

0.093

226

20

20

154

i0"* --9

-7.66

0.133

0.116

268

22

22

187

i0"* --i0

--8.56

0.170

0.149

324

26

26

230

i0"* --ii

--9.46

0.211

0.186

395

32

32

283

214

G. PSIHOYIOS Table 19. S u m m a r y results for Group C problems (nonnormalized).

PIAS

MEBDF

Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

E n d Pnt. Glb. Err.

Maximum Glb. Err.

-2.00

0.091

0.070

432

66

66

208

0.00

0.26

--3.00

0,154

0.118

675

119

118

333

0.00

0,46

-4.00

0.246

0.195

975

160

160

489

0.00

0.33

--5.00

0,339

0.276

1196

202

202

662

0.01

0.94

-6.00

0.445

0.370

1454

234

234

840

0.01

1.35

-7.00

0,572

0.483

1768

264

264

1063

0.01

1.92

-8.00

0,709

0.601

2054

338

338

1313

0.01

0.47

-9.00

0,877

0,752

2470

376

376

1618

0.00

1.40

-10.00

1,115

0.961

3061

456

456

2048

0.01

0.90

--2.00

0.114

0.088

314

55

55

199

0.00

0.27

-3.00

0,194

0.158

480

71

71

321

0.00

0.15

-4.00

0.285

0,237

697

82

82

462

0.00

0.23

-5.00

0,411

0.350

911

97

96

614

0.00

0.13

--6.00

0.556

0.482

1135

111

111

776

0.02

0.14

-7.00

0.689

0.602

1360

123

123

951

0.00

0.04

-8.00

0.850

0.744

1678

146

146

1181

0.01

0.10

-9.00

1.085

0.956

2062

176

176

1486

0.00

0.04

-10.00

1,373

1.211

2610

214

214

1886

0.00

0.22

Table 20. Stiff D E T E S T normalized efficiency results for problem (D1).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.25

0.012

0.009

86

13

13

27

10"* --4

-3.45

0,020

0.016

135

23

23

50

10"* - 5

-4.65

0.032

0.026

194

32

32

81

10"* - 6

-5.84

0.053

0.044

299

42

42

131

t0"* - 7

-7.04

0.075

0.063

422

49

49

177

10"* - 8

--8.23

0.106

0.090

542

70

70

261

10"* - 9

-9,43

0.149

0.129

698

96

96

366

10"* - 2

-1.93

0.012

0.009

59

10

10

20

10"* --3

-2.93

0.018

0.013

98

10

10

31

10"* - 4

-3.92

0.027

0.022

118

13

13

46

10"* --5

-4.92

0.037

0.031

157

14

14

67

10'* - 6

-5.92

0.047

0.040

174

15

15

86

10"* --7

--6.91

0.065

0.057

214

19

19

113

10"* - 8

-7.91

0.090

0.078

280

21

21

155

10"* - 9

-8.91

0.133

0.116

401

37

37

234

10"* - 1 0

-9.90

0.174

0.155

473

42

41

287

Implicit Advanced Step-Point Methods

1215

Table 21. Stiff DETEST normalized efficiency results for problem (D2).

PIAS

MEBDF

Expected Accuracy

Equiv, Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.76

0.024

0.018

135

24

24

66

10"* - 4

-3.82

0.038

0.028

207

38

38

96

10"* - 5

-4.88

0.055

0.042

290

46

46

138

10"* - 6

-5.94

0.084

0.067

395

59

59

209

10"* - 7

-7.00

0.105

0.085

439

77

77

254

10"* - 8

-8.06

0.129

0.106

532

84

84

311

10"* - 3

-2.27

0.025

0.019

88

15

15

52

10"* - 4

-3.34

0.037

0.029

128

18

18

77

10"* - 5

-4.41

0.052

0.042

173

21

21

107

10"* - 6

-5.48

0.079

0.065

245

24

24

155

10"* - 7

-6.55

0.108

0.091

309

29

29

202

10"* - 8

-7.62

0.139

0.119

376

32

32

253

Table 22. Stiff DETEST normalized efficiency results for problem (D3).

PIAS

MEBDF

Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

20

20

No. of Steps

10"* - 2

-1.93

0.026

0.020

130

10"* - 3

-2.89

0.044

0.034

194

36

36

95

10"* - 4

-3.85

0.061

0.050

244

45

45

130

10"* - 5

-4.82

0.089

0.073

327

59

59

185

10"* - 6

-5.78

0.121

0.102

391

69

69

236

10"* - 7

-6.75

0.157

0.135

482

80

80

300

10"* - 8

-7.71

0.196

0.170

585

92

92

366

10"* - 9

-8.67

0.243

0.212

680

108

108

448

10"* - 4

-2.80

0.052

0.044

128

20

20

87

10"* - 5

-3.82

0.081

0.069

191

24

24

129

10"* - 6

-4.85

0.117

0.102

253

28

28

174

10"* - 7

-5.87

0.162

0.144

342

30

30

232

10"* - 8

-6.89

0.209

0.187

416

38

38

289

10"* - 9

-7.92

0.261

0.233

514

45

45

364

63

Table 23. Stiff DETEST normalized efficiency results for problem (D4).

PIAS

MEBDF

Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

16

16

No. of Steps

10"* - 4

-3.05

0.018

0.016

91

10"* - 5

-4.23

0.029

0.026

148

24

24

75

10"* - 6

-5.41

0.048

0.044

210

39

39

117

10"* - 7

-6.59

0.061

0.055

269

45

45

146

10"* - 8

--7.77

0.072

0.066

303

47

47

173

10"* - 4

-2.61

0.022

0.020

68

11

11

45

I0"* - 5

-3.64

0.031

0.029

97

13

13

61

10"* - 6

-4.66

0,043

0.039

127

14

14

82

10"* - 7

-5.68

0,063

0.059

176

19

19

115

10"* - 8

-6.70

0.077

0.072

201

21

21

136

10"* - 9

-7,73

0.089

0.082

238

23

23

162

10"* - 1 0

-8.75

0,107

0.100

280

26

26

194

48

216

G. PSlHOYIOS Table 24. Stiff D E T E S T normalized efficiency results for problem (D5).

PIAS

MEBDF

Expected Accuracy 10'* - 3 10'* - 4 10"* - 5 10"* - 6 10"* - 7 10"* - 8 10"* - 9 10"* - 2 10"* - 3 10"* - 4 10"* - 5 10"* - 6 10"* - 7 10"* - 8 10"* - 9

Equiv. Logl0 Tol. -2.96 -4.08 -5.20 -6.32 -7.44 -8.56 -9.68 -2.09 -3.17 -4.24 -5.32 -6.40 -7.48 -8.55 -9.63

Time

Ovhd.

0.013 0.018 0.028 0.045 0.066 0.102 0.127 0.011 0.017 0.023 0.031 0.041 0.063 0.094 0.118

0.010 0.015 0.024 0.039 0.059 0.092 0.114 0.009 0.014 0.020 0.028 0.037 0.058 0.087 0.109

Fcn. Calls 119 155 227 324 447 662 817 61 91 122 155 202 293 414 488

Jac. Calls 21 26 37 52 73 101 115 14 18 18 19 21 23 33 34

Mat. Fact 21 26 37 52 73 101 115 14 18 18 19 21 23 33 34

No. of Steps 46 61 97 152 214 325 407 23 42 58 79 103 154 230 270

Table 25. Stiff D E T E S T normalized efficiency results for problem (D6). Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

i0"* -4

-3.05

0.024

0.021

132

24

24

68

I0"* -5

-4.29

0.040

0.035

202

35

35

104

i0"* -6

-5.54

0.064

0.057

293

48

48

160

10"* - 4

-2.79

0.033

0.030

15

15

64

10"* - 5

-3.95

0.048

0.044

144

18

18

93

10"* - 6

-5.11

0.071

0.065

219

20

20

134

PIAS

MEBDF

97

No. of Steps

Table 26. S u m m a r y results for Group D problems (nonnormalized).

PIAS

MEBDF

Logl0 Tol.

Time

Ovhd.

Fen. Calls

Jac. Calls

Mat. Fact

No. of Steps

E n d Pnt. Glb. Err.

Maximum Glb. Err.

--2.00

0.090

0.071

541

95

95

244

0.65

1.05

--3.00

0.143

0.115

811

146

146

374

0.34

0.62

-4.00

0.207

0.170

1090

194

194

517

1.63

2.67

-5.00

0.311

0.260

1529

261

261

764

1.57

2.85

-6.00

0.437

0.373

1969

317

317

1048

2.29

3.04

-7.00

0.555

0.478

2423

377

377

1303

3.75

4.69

-8.00

0.692

0.601

2904

446

446

1620

1.25

3.05

-9.00

0.924

0.808

3793

569

569

2196

1.73

3.21

--i0.00

1.196

1.042

4653

755

755

2792

1.61

3.71

--2.00

0.118

0.096

434

78

78

236

0.70

0.70

-3.00

0.185

0.155

618

95

95

354

0.82

0.85

-4.00

0.264

0.226

847

109

109

494

2.60

3.29

--5.00

0.370

0.323

1127

121

121

671

1.60

2.01

--6.00

0.507

0.448

1449

141

141

905

3.77

4.93

--7.00

0.635

0.567

1703

160

160

1079

2.73

4.09

--8.00

0.807

0.724

2155

185

185

1406

5.37

7.29

--9.00

1.066

0.958

2787

244

244

1809

0.48

1.74

--i0.00

1.309

1.180

3334

265

264

2191

0.92

2.43

Implicit Advanced Step-Point Methods 6.4.

R e s u l t s for t h e S t i f f D E T E S T

1217

Group D Problems

The picture changes a little for this group. The nonnormalized summary results from Table 26 show that, the experimental HAS code is faster (this is also true for individual problems, e.g., on problems (D2), (D3), (D4), (D6)). The normalized efficiency results, i.e., Tables 20-25, are also affected since, the superior accuracy of the MEBDF code is not as clear anymore and the MEBDF code is challenged for the first time in problem (D5) where both codes have approximately the same accuracy. As far as, the speed is concerned the two codes are approximately equally fast on problems (D2), (D5), and (D1) (on this last problem MEBDF is faster in high tolerances) and the experimental HAS code is faster on D6, D4 and, also (D3) (high tolerances). 6.5. Results for the Stiff D E T E S T Group E P r o b l e m s

The nonnormalized summary results from Table 32 show that, the experimental PIAS code is faster (this is also true for each individual problem). The normalized efficiency results, i.e., Table 27. Stiff D E T E S T normalized efficiency results for problem (El).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.56

0.028

0.025

120

19

19

57

10"* - 4

-3.54

0,043

0.038

154

27

27

82

10"* - 5

-4.52

0,059

0.053

192

34

34

109

10"* - 6

-5.49

0.074

0.068

231

38

38

135

10"* - 7

-6.47

0,091

0.084

271

42

42

166

10"* - 8

-7.45

0.110

0.102

317

43

43

200

10"* - 9

-8.43

0,134

0.125

380

49

49

247

10"* - 4

-2.57

0.039

0.036

88

11

11

58

10"* - 5

-3.59

0.057

0,053

124

13

13

82

10"* - 6

-4.61

0,076

0.072

162

17

17

107

10"* - 7

-5.63

0.095

0,090

193

19

19

131

10"* - 8

-6.65

0.114

0.109

230

21

21

158

10"* - 9

-7.67

0.136

0.130

275

23

23

196

10"* - 1 0

-8.69

0,181

0.173

345

26

26

251

Table 28, Stiff D E T E S T normalized efficiency results for problem (E2).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

i0"* --i

--2.66

0.027

0.021

233

39

39

84

iO** -2

--3.61

0.035

0.028

273

46

46

113

i0"* --3

--4.57

0.052

0.044

365

59

59

170

i0"* --4

--5.52

0.072

0.062

463

67

67

233

10"* - 5

-6.48

0.092

0,080

541

79

79

352

10'* - 6

-7.43

0,111

0,098

627

89

89

352

10"* --7

-8.39

0.132

0.117

742

91

91

417

10"* - 8

-9.34

0.164

0.147

873

108

108

522

10"* --1

--2.87

0.035

0.029

198

23

23

93

10"* --2

-3.88

0.046

0.040

227

23

23

117

10"* - 3

--4.88

0.065

0.057

270

26

26

154

I0"* -4

--5.89

0.081

0.072

338

28

28

198

i0"* --5

-6.90

0.Iii

0.i00

424

32

32

259

i0"* --6

-7.90

0.133

0.121

495

33

33

311

I0"* --7

-8.91

0.178

0.163

607

42

42

404

10"* - 8

-9.91

0.227

0.208

771

55

55

525

1218

C. PsmoYIos Table 29. Stiff D E T E S T normalized efficiency results for problem (E3).

PIAS

MEBDF

Expected Accuracy

Equiv. Logl0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.41

0.018

0.016

113

18

18

51

10"* - 4

-3.49

0.030

0.027

171

30

30

78 115

10"* - 5

-4.57

0.046

0.042

234

41

41

10"* - 6

-5.66

0.071

0.066

330

52

52

174

10"* - 7

-6.74

0.089

0.083

387

63

63

216

10"* - 8

-7.82

0.117

0.111

486

69

69

286

10"* - 9

-8.91

0.140

0.133

548

82

82

332

10"* - i 0

-9.99

0.222

0.206

891

151

151

529

10"* - 3

-2.69

0.026

0.024

97

13

13

53

10"* - 4

-3.60

0.040

0.037

140

14

14

81

10"* - 5

-4.50

0.054

0.051

179

17

17

108

10"* - 6

-5.41

0.067

0.064

211

20

19

129

10"* - 7

-6.31

0.087

0.083

248

21

21

156

10"* - 8

-7.22

0.114

0.109

294

26

26

196

10"* - 9

-8.12

0.134

0.128

360

30

30

242

10"* - 1 0

-9.02

0.162

0.156

428

37

36

292

10"* - 1 1

-9.93

0.197

0.190

507

38

38

356

Table 30. Stiff D E T E S T normalized efficiency results for problem (E4).

PIAS

MEBDF

Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

No. of Steps

10"* - 2

-2.79

0.053

0.040

223

39

39

104

-3.71

0.075

0.059

288

53

53

146

10"* - 4

-4.63

0.107

0.086

371

68

68

204

10"* - 5

-5.56

0.150

0.123

479

88

88

285

10"* - 6

-6.48

0.200

0.166

580

100

100

366

10"* - 7

-7.40

0.240

0.206

673

104

104

437

10"* - 8

-8.32

0.295

0.255

810

120

120

534

10"* - 9

-9.24

0.378

0.330

1002

142

142

679

10"* - 2

-2.40

0.055

0.043

157

22

22

83

10"* - 3

-3.34

0.078

0.064

198

24

24

117

10"* - 4

-4.28

0.116

0.100

266

26

26

172

10"* - 5

-5.22

0.168

0.147

359

32

32

235

10"* - 6

-6.16

0.217

0.191

467

39

39

307

10"* - 7

-7.10

0.284

0.254

568

45

45

382

10"* - 8

-8.04

0.346

0.310

673

53

53

468

10"* - 9

-8.98

0.449

0.404

852

66

66

601

10"* - 1 0

-9.92

0.553

0.499

1044

77

77

741

P I A S c o d e is c l e a r l y f a s t e r o n p r o b l e m s

(E5), and that, the two codes are approximately

(E2), (E4),

equally fast on problems (El) and (E3). As far

as, t h e a c c u r a c y is c o n c e r n e d , w e h a v e t h e f i r s t p r o b l e m , n a m e l y c o d e is, n o t o n l y f a s t e r , b u t a l s o m o r e a c c u r a t e .

are approximately

Mat. Fact

10"* - 3

Tables 27-31, show that, the experimental

HAS

Jac. Calls

(E2), where the experimental

On problems

(E5) and (E3) the codes

equally accurate and only on problems (El) and (E4) does the MEBDF

remain more accurate.

code

1219

Implicit Advanced Step-Point Methods Table 31. Stiff D E T E S T normalized efficiency results for problem (E5). Expected Accuracy

Equiv. Logl0 ToE

Time

Ovhd.

10"* --3

--2.55

0.018

0.014

82

15

15

41

10"* --4

--3.53

0.028

0.022

122

24

24

59

10"* --5

--4.50

0.038

0.030

158

31

31

76

10"* --6

--5.48

0.047

0.039

186

30

30

96

10"* --7

--6.46

0.059

0.050

222

31

31

118

10"* --8

--7.44

0.077

0.066

272

43

43

149

10"* --9

--8.41

0.103

0.088

347

56

56

199

10"* - 3

-2.22

0.020

0.016

63

11

11

35

PIAS

MEBDF

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 4

-3.26

0.031

0.025

84

14

14

51

10"* - 5

-4.29

0.044

0.036

122

16

16

68

10"* - 6

-5.33

0.058

0.049

144

18

18

87

10"* - 7

-6.36

0.072

0.063

167

18

18

108

10"* - 8

-7.40

0.091

0.080

205

21

21

137

Table 32. S u m m a r y results for Group E problems (nonnormalized).

PIAS

MEBDF

Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

End Pnt. Glb. Err.

Maximum Glb. Err.

-2.00

0.112

0.091

652

108

108

269

2.99

64.65

-3.00

0.165

0.135

852

148

148

382

3.63

38.29

-4.00

0.242

0.202

1124

205

205

547

3.69

32.62

--5.00

0.344

0.294

1469

255

255

771

2.98

37.60

-6.00

0.470

0.408

1863

294

294

1046

2.96

69.93

-7.00

0.587

0.518

2146

336

336

1276

0.52

43.03

-8.00

0.720

0.641

2586

368

368

1558

4.38

2.49

-9.00

0.922

0.826

3174

425

425

1974

43.81

16.48

-10.00

1.223

1.103

4067

570

570

2630

438.16

82.11

-2.00

0.147

0.120

557

90

90

267

1.74

24.86

-3.00

0.204

0.175

658

86

86

369

8.89

140.68

-4.00

0.299

0.264

892

95

95

525

8.03

116.15

-5.00

0.421

0.378

1118

116

115

698

7.75

105.28

-6.00

0.535

0.484

1395

125

125

881

7.97

81.44

-7.00

0.701

0.640

1695

145

144

1115

8.42

70.83

-8.00

0.860

0.788

2055

166

166

1388

9.41

104.53

-9.00

1.143

1.051

2605

207

206

1794

43.82

62.35

--10.00

1.424

1.315

3223

240

240

2264

438.18

64.71

6.6. R e s u l t s for t h e Stiff D E T E S T G r o u p F P r o b l e m s

N o t enough successful integrations to form normalized statistics In Group F the experimental PIAS code remains faster in the nonnormalized summary results as seen in Table 38 (and on each of the five individual problems). The normalized efficiency results, i.e., Tables 33-37, show that, the two codes are approximately equally fast on problems (F1), (F2), (F3), and that, the experimental PIAS code is clearly faster and also more accurate on

220

C-. PSIHOYIOS Table 33. Stiff D E T E S T n o r m a l i z e d efficiency r e s u l t s for p r o b l e m (F1).

PIAS

MEBDF

Expected Accuracy

Equiv. L o g l 0 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

No. of Steps

10"* - 3

-2.19

0.045

0.038

10"* - 4

-3.40

0.075

0.066

214

36

36

97

340

53

53

155

10"* - 5

-4.60

0,131

0.116

526

87

87

256

10"* - 6

-5.81

0.192

0.172

715

120

120

370

10"* --7

-7.02

0.251

0.225

889

151

151

476

10"* - 3

--2.09

0.048

0.042

142

23

23

79

10"* --4

-3.17

0.090

0.080

245

34

34

138

10"* - 5

-4.26

0.128

0.117

317

38

38

190

10"* - 6

-5.34

0.179

0.165

431

44

44

260

10"* - 7

-6.42

0.250

0.232

593

52

52

353

10"* --8

-7.50

0.339

0.316

790

62

62

476

Table 34. Stiff D E T E S T n o r m a l i z e d efficiency r e s u l t s for p r o b l e m (F2).

PIAS

MEBDF

Expected

Equiv.

Accuracy

L o g l 0 Tol.

Time

Ovhd.

Fcn,

Jac,

Mat.

No. of

Calls

Calls

Fact

10"* --3

-2.14

0.008

0.006

67

Steps

13

13

32

10"* --4

-3.18

0.016

0.012

106

10"* - 5

--4.23

0.023

0.018

144

19

19

52

25

25

10"* - 6

--5.27

0.033

0.027

192

33

75

33

106

10"* --7

--6.31

0.040

0.033

227

10"* --8

--7.36

0.048

0.040

260

37

37

129

42

42

10"* - 9

-8.40

0.060

0.050

312

154

46

46

188

10"* - 1 0

-9.44

0.075

0.064

10"* - 4

-2.61

0.014

0.011

379

55

55

237

64

10

9

10"* - 5

-3.60

0.022

0.018

40

92

12

12

57

10"* - 6

-4.58

0.030

0.025

121

13

13

76

10"* - 7

-5.56

0.040

0.034

152

14

14

100

10"* - 8

-6.55

0.051

0.044

185

18

18

125

10"* - 9

-7.53

0.062

0.053

220

21

21

152

10"* - 1 0

-8.51

0.077

0.067

258

24

24

184

i0"* -11

-9.49

0.095

0.084

313

26

26

228

T a b l e 35. Stiff D E T E S T n o r m a l i z e d efficiency r e s u l t s for p r o b l e m (F3).

PIAS

MEBDF

Expected

Equiv.

Accuracy

Log10 Tol.

Time

Ovhd.

Fcn.

Jac.

Mat.

No. of

Calls

Calls

Fact

i0"* -3

-2.19

0.038

0.030

Steps

133

24

24

69

i0"* --4

--3.22

0.061

I0"* -5

--4.26

0.094

0.048

194

35

35

104

0.077

262

45

45

10"* --6

--5.29

153

0.139

0.117

349

63

63

218

10"* --7

--6.32

0.185

0.158

446

79

79

285

10"* - 4

--2.76

0.066

0.054

133

19

19

88

10"* --5

-3.72

0.097

0.081

187

23

23

125

10"* --6

--4,67

0.136

0.116

242

25

25

164

10"* --7

--5,62

0.192

0.168

315

30

30

218

10"* --8

--6.58

0.244

0.215

383

35

35

269

10"* - 9

--7,53

0.291

0.257

452

39

39

319

Implicit Advanced Step-Point Methods

1221

Table 36. Stiff D E T E S T normalized efficiency results for problem (F4). Expected Accuracy

Equiv. Log10 Tol.

Time

Ovhd.

Fcn. Calls

10"* 0

-4.03

0.141

0.131

10"* - 1

-5.08

0.188

10"* - 2

-6.14

10"* - 3

-7.20

10"* - 4

PIAS

MEBDF

Jac. Calls

Mat. Fact

No. of Steps

765

103

103

253

0.172

1097

170

170

434

0.269

0.246

1429

243

243

614

0.347

0,322

1844

248

248

800

-8.25

0.491

0.454

2367

391

391

1120

10"* - 1

-5.72

0.283

0.267

941

103

103

451

10"* - 2

-6.62

0.353

0.333

1171

119

119

588

10"* - 3

-7.52

0.451

0.427

1413

145

145

768

10"* - 4

-8.42

0.522

0.497

1562

146

146

901

Table 37. Results for the stiff D E T E S T problem (F5) (nonnormalized).

PIAS

MEBDF

problem

(F4).

Log10 Tol.

Time

Ovhd,

Fcn, Calls

Jac. Calls

Mat. Fact

No. of Steps

End Pnt. Glb. Err.

--2.00

0.024

0.020

113

18

18

58

0.00

0.03

--3.00

0.036

0.030

158

29

29

83

0.00

0.09

--4.00

0.054

0.046

223

41

41

118

0.00

0.13

--5.00

0.081

0.071

311

54

54

171

0.03

0.16

--6.00

0.116

0.103

396

65

65

234

0.91

0.00

--7.00

0.160

0.143

537

88

88

318

0.62

0.00

--8.00

0.258

0.228

981

153

153

521

7.19

0.00

-9.00

0.514

0.445

2332

356

356

1084

15.67

0.00

--10.00

4.944

4.193

27771

3732

3732

11402

254.58

0.00

--2.00

0.032

0.027

93

17

17

58

0.00

0.16

--3.00

0.045

0.039

126

21

21

82

0.00

0.04

--4.00

0.067

0.060

173

25

25

114

0.01

0.03

-5.00

0.097

0.087

236

31

31

155

0.02

0.03

--6.00

0.152

0.140

344

35

35

225 '

0.06

O.00

-7.00

0.174

0.160

402

43

43

270

2.21

0.00

-8.00

0.305

0.278

743

86

86

457

3.32

0.00

-9.00

0.854

0.735

2710

399

399

1189

13.86

0.00

-10.00

5.640

4.668

20663

3351

3351

9589

6.37

0.00

On the first three problems the MEBDF

note that in problem

Maximum Glb. Err.

code remains more accurate.

(Please

( F 5 ) , i.e., T a b l e 37, t h e r e a r e n o t e n o u g h s u c c e s s f u l i n t e g r a t i o n s t o f o r m

normalized statistics.) The

"summary

same pattern, experimental

o f r e s u l t s o v e r all g r o u p s " , as s e e n i n T a b l e 39, t a b l e d i s p l a y s o n c e m o r e t h e

i.e., t h a t , t h e e x p e r i m e n t a l

P I A S c o d e is f a s t e r .

code has more function and Jacobian

W e m a y also o b s e r v e t h a t , t h e

calls than the MEBDF

code.

surprising since we would expect the PIAS code to have some more Jacobian

T h i s is n o t

calls due to the

extra iteration matrix that, it requires and due to this some more function evaluations.

1222

G. PSIHOYIOS Table 38. Summary results for Group F problems (nonnormalized).

PIAS

MEBDF

Logl0 Tol.

Time

Ovhd.

Fcn. Calls

-2.00

0.112

0,094

525

-3.00

0.178

0.149

803

-4.00

0.392

0.344

1761

-5.00

0.573

0.504

-6.00

0,789

0.699

-7.00

0.984

-8.00 -9.00 -10.00

Jac. Calls

Mat. Fact

No. of Steps

End Pnt. Glb. Err.

Maximum Glb. Err.

95

95

258

0.07

0.08

139

139

394

1.64

0.27

275

275

765

0.31

5754.96

2491

407

407

1189

0.56

35578.80

3161

552

552

1609

1.60

6090.21

0.882

3913

581

581

1998

0.70

24477.29

1.323

1,186

5118

839

839

2688

7.19

15987.75

1.935

1.731

7768

1191

1191

3979

15.67

16439.78

6,798

5,872

34365

4810

4810

15038

254.58

17858.96

-2.00

0,146

0.122

432

81

81

247

0.12

0.16

-3.00

0,232

0.199

637

98

97

383

0.42

0.15

-4.00

0.487

0.433

1618

190

190

794

1.02

0.41

-5.00

0.678

0.615

1918

205

205

1068

1.36

39224.95

-6.00

0.935

0.856

2341

241

241

1359

0,52

73249.16

-7.00

1.170

1.075

3003

284

284

1770

2.21

47821.59 33614.29

-8,00

1.585

1.460

3878

380

380

2375

3.34

-9.00

2.314

2.088

6258

677

677

3421

13,86

16786.10

-10.00

7.601

6,485

25353

3766

3766

12534

6.37

29117.39

Table 39. Summary results over all groups (nonnormalized).

PIAS

MEBDF

Log10 Tol.

Time

Ovhd.

Fcn. Calls

Jac. Calls

Mat. Fact

-2.00

0.809

-3.00

1.269

-4.00

2.103

1.757

-5.00

2.942

2.501

-6.00

3,949

3,415

-7,00

4.993

4.372

--8.00

6.450

-9.00

8.434

--10.00 -2.00

No. of Steps

0.646

3409

378

536

1587

1.012

4823

567

834

2400

7365

862

1185

3699

9594

1151

1547

5200

12047

1430

1881

6881

14726

1582

2104

8628

5.671

18428

2023

2669

11103

7.472

24270

2590

3323

14743

15.215

13.356

55303

6621

7472

29054

1.022

0.829

2643

318

425

1535

-3.00

1.639

1.385

3749

360

508

2342

-4.00

2.564

2.232

5928

492

661

3562

-5.00

3.606

3,201

7518

551

746

4777

-6.00

4,822

4.334

9403

636

854

6149

--7.00

6.160

5.579

11645

725

982

7752

-8.00

7,893

7,170

14747

893

1208

10012

-9.00

10.370

9.413

19972

1321

1661

13200

-10.00

17.920

15.910

42639

4502

4905

24984

7. C O N C L U S I O N S H a v i n g s e e n t h e n u m e r i c a l c o m p a r i s o n s i n S e c t i o n 6, w e a r e i n a p o s i t i o n t o d r a w s o m e i n t e r esting conclusions regarding the behaviour and the potential of the HAS

scheme.

• T h e e x p e r i m e n t a l P I A S c o d e is c l e a r l y f a s t e r o n t h e v a s t m a j o r i t y o f t h e t e s t p r o b l e m s . The MEBDF the MEBDF

c o d e is m o r e a c c u r a t e o n m o s t o f t h e t e s t p r o b l e m s . I t is n o t s u r p r i s i n g t h a t , c o d e is m o r e a c c u r a t e t h a n t h e e x p e r i m e n t a l H A S

code on certain problems

Implicit Advanced Step-Point Methods

1223

since MEBDF is L-stable whereas the PIAS methods are only A-stable. It has to be mentioned though that, the accuracy of the experimental H A S code is very satisfactory. • The experimental HAS code displays an increased accuracy on the nonlinear test problems Groups D, E, and F. This seems a particularly promising feature if we consider the prospect of improving its speed. Keeping in mind though that this form of the H A S code is an experimental code, there is little doubt that, we can substantially improve its speed when we will a t t e m p t to create a proper parallel variable order/variable stepsize code in the

near future. We also believe that, we will be able to improve its accuracy behaviour.

APPENDIX Coefficients of the second parallel predictor (7) of the PIAS methods. bk+l

56

55

~.4

2 -- bk--1 6--bk_ 1 5 12--bk_ 1 13 3(40+%k--1 )

154 2(30--bk_1) 87

1800--611bk_I 348

10(42--bk_l) 669 5(56--bk_1) 481

78400--20159bk_ 1 9620

88200--25961bk_ 1 13360

10(--2940+739b~_1) 2007

4(--7840+1583bk_1) 1443

15(3920--551bk_1) 1924

Coefficients of the second (parallel) predictor (7) of the PIAS methods. k

53

52

51

1

50

1

2

9-4bk_1

3

4(--l+bk-1)

5

5

4(9--4bk-1) 13

4(--8+5bk--1) 13

9--4bk--1 13

4

3600-- 26 S l b k _ 1 924

-- 800+661 {)k-- 1 154

3 (300 -- 371 bk - 1 ) 308

-- 286-- 427b k _ 1 462

5

8(--300~-97bk-- 1) 261

3(76--17bk-- 1 )29

6(--36+7bk--1) 67

600--107bk-- 1 1044

6

15(245-43bk-1) 223

4 ( - 1 7 6 4 + 2 6 5 b k - 1) 669

5(5880--809bk-- 1 ) 8028

6(-- 100+13bk--1) 1115

7

8(--4704+565bk-- 1) 1443

5(15680-- 1723bk-- 1) 5772

12(--800+835k-- 1) 2405

2940--293bk- 1 5772

REFERENCES 1. W.L. Miranker and L. Liniger, Parallel methods for the integration of ordinary differential equations, Math. Comp. 21, 303-320, (1967). 2. M.A. Franklin, Parallel solution of ordinary differential equations, IEEE Trans. on Comp. 27, 413-420, (1978). 3. K. Burrage, Parallel methods for initial value problems, Appl. Num. Math. 11, 5-25, (1993). 4. G.-Y. Psihoyios and J.R. Cash, A stability result for general linear methods with characteristic function having real poles only, BIT Num. Math. 38 (3), 612-617, (1998). 5. G. Psihoyios, Advanced step-point methods for the solution of initial value problems, Ph.D. thesis, Imperial College, University of London (1995). 6. J.R. Cash, The integration on stiff IVPs in ODEs using modified extended BDF, Computers Math. Applic. 9, 645-657, (1983). 7. S. Considine, Modified linear multistep methods for the numerical integration of stiff IVPs, Ph.D. thesis, Imperial College (1988). 8. J.R. Cash and S. Considine, An MEBDF code for stiff IVPs, A C M TOMS 18II, 142-155, (1992).

224

G. PSIHOYIOS

9. J.R. Cash, On the integration of stiff systems of ODEs using extended BDF jour Num. Math. 34, 235-246, (1980). 10. W.H. Enright, T.E. Hull and B. Lindberg, Comparing numerical methods for stiff systems of ODEs, B I T 15, 10-48, (1975). 11. W.H. Enright and J.D. Pryce, Two Fortran packages for assessing initial value methods, A C M T O M S 13, 1-27, (1987).