Evaluation of the Computation Speed

Further important results can be achieved by experimental calculations in solving numerically the differential equation (20) with an explicit use of the Taylor series.

The exact solution y(1) = e¹ can also be approximated by solving the equation (20) with the integration step h = 0.5s ( Tab.8 - two computation steps are necessary in this case) or with the integration step h = 0.1s ( Tab.9 - ten computation steps are necessary).

Only those digits ( in the result of the numerical solution of (20)) are printed which tally the exact value.

Reduced value y(1)	ORD	Absolute error
2.	1	-0.468281828459045310
2.	2	-0.077656828459045312
2.71	3	-0.009514467347934263
2.7182	4	-0.000935637052795313
2.7182	5	-0.000077008038038451
2.718281	6	-0.000005449497788135
2.7182818	7	-0.000000338137442601
2.71828182	8	-0.000000018677261404
2.7182818284	9	-0.000000000929473165
2.71828182845	10	-0.000000000042083559
2.71828182845	11	-0.000000000001747935
2.71828182845904	12	-0.000000000000067057
2.718281828459045	13	-0.000000000000002442
2.718281828459045	14	-0.000000000000000222
2.718281828459045	15	-0.000000000000000222
2.718281828459045	16	-0.000000000000000222

Tab.8: Long real arithmetic

Reduced value y(1)	ORD	Absolute error
2.	1	-0.124539368359045440
2.71	2	-0.004200981850820962
2.718	3	-0.000104565977435356
2.7182	4	-0.000002084323879603
2.718281	5	-0.000000034655339265
2.71828182	6	-0.000000000494185248
2.71828182845	7	-0.000000000006168954
2.71828182845	8	-0.000000000000068612
2.71828182845904	9	-0.000000000000000555
2.718281828459045	10	0.000000000000000222
2.718281828459045	11	0.000000000000000222
2.718281828459045	12	0.000000000000000222

Tab.9: Long real arithmetic

The results in Tab.8 and Tab.9 are again characterized by the saturated absolute errors ESAT ( the computation is terminated when three consecutive absolute error values have not changed). However, the saturated absolute error ESAT is reached using a lower ORD (as compared with Tab.5). This is due to the fact that with the same type of arithmetic higher order Taylor series terms will have no effect when the integration step h is shortened. Up from a certain method order a shorter integration step h causes an underflow so that the adding of further Taylor series terms does not change the result.

Similarly, the results for the integration step h= 0.01s are in Tab.10 and the results for the integration step h= 0.001s are shown in Tab.11.

Reduced value y(1)	ORD	Absolute error
2.7	1	-0.013467999037519162
2.7182	2	-0.000044965899087201
2.718281	3	-0.000000112359411108
2.718281828	4	-0.000000000224644081
2.71828182845	5	-0.000000000000374145
2.71828182845904	6	-0.000000000000001332
2.71828182845904	7	-0.000000000000001332
2.71828182845904	8	-0.000000000000001332

Tab.10: Long real arithmetic

Reduced value y(1)	ORD	Absolute error
2.71	1	-0.001357896223154187
2.718281	2	-0.000000452707280774
2.718281828	3	-0.000000000113170251
2.7182818284590	4	-0.000000000000022982
2.7182818284590	5	-0.000000000000022982
2.7182818284590	6	-0.000000000000022982

Tab.11: Long real arithmetic

It follows from Tab.5, Tab.8 to Tab.11 that the saturated absolute error ESAT is reached with different numbers of Taylor series terms.

A different question is, of course, with what speed can the result be obtained if a shortened integration step h is used (with the corresponding saturated absolute error ESAT).

This speed evidently depends on the technical construction and on the number of operations required. For evaluation of the computation speed the equation (22) was rewritten into the form

$y_{n+1}= y_n + DY1_n + DY2_n + ..... + DYP_n,\hfill (24)$
where
DY1_n= h*y_n
$DY2_n=\frac{ h}{2}*DY1_n$
.
.
$DYP_n=\frac{ h}{P}*DY(P-1)_n.$

The relation between the number of operations ( required for reaching the saturated absolute error ESAT) and the integration step h is given in Tab.12.

h	ORD	NRCS	Addition	Multipl.	DIV
1	18	1	18	18	17
0.5	14	2	28	28	26
0.1	10	10	100	100	90
0.01	6	100	600	600	500
0.001	4	1000	4000	4000	3000

Tab.12

In the column "DIV" ve have the number of division operations, in the column "Multipl." we have the number of product operations and in the column "Addition" the number of addition operations necessary for computing the results according to Tab.5, Tab.8, Tab.9, Tab.10 and Tab.11. In the column "ORD" in Tab.12 there is always the value of the method order with which the absolute error (according to Tab.5, Tab.8, Tab.9, Tab.10, Tab.11) reached its saturated value ESAT.

In the column "NRCS" , to complete the picture, the number of computation steps is shown after which, given the integration step h, the point t_n = 1s has been reached (i. e. the point at which the exact solution is y_n = e¹).

Regardless of the practical construction of addition, multiplication and division it is clear from Tab.12 that the number of operations ( required for reaching the highest accuracy - with the corresponding integration step h) increases as the integration step h shortens.

Shortening the integration step h does not mean, however, only an increase in the number of operations. It is also characterized by an existence of the accumulated error - the error from one step is carried to the following steps.

This fact can be clearly demonstrated by studying the absolute errors in Tab.5, Tab.8 to Tab.11. It is obvious that the optimal integration step h producing the least saturated absolute error ESAT of the computation exists.

It is certainly very advantageous to do the computation with the optimal integration step h if we want to reach a high accuracy. Using the same arithmetic, the accuracy reached with the integration step h = 1s is by two orders better than with the integration step h = 0.001s.

Futhermore, a very important conclusion can be drawn from Tab.12 - the number of operations (addition, division, and multiplication) with the integration step h = 1s is less than the corresponding number of operations with the integration step h = 0.001s.

Briefly, it means that the computation is done most precisely and at the same time most quickly with the optimal integration step h (and the corresponding optimal method order). A computation done with other than optimal integration step h is always slower and less accurate.