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Interval mathematics for practical calculations

In scientific researches, the engineering and mass manufacture frequently should carry spend measurements of any sizes (length, weight, force of a current etc.). At recurrence of measurements of the same object which is carried out with the help of the same measuring device with identical carefulness because of influence of the various factors, identical data never turn out. The casual vibrations of separate parts of the device, physiological changes of sense organs of the executor, various not taken into account changes in environment (temperature, optical, electrical and magnetic properties etc.) concern to number of such factors. Though the result of each separate measurement at presence of casual dispersion cannot beforehand be predicted, it corresponds «to a normal curve of distribution» (figure).

Figure Normal curve of distribution

From figure it is visible, that basic weight of received results will be grouped about some central or average meaning a, which the «true size» measurely of object answers unknown. The deviations in this or that of the party will occur by that less often, than more absolute size of such deviations, and are characterized by size  - average quadratic deviation. On a site from  up to  There is on the average share equal 0,6287 (of 68,27 %) of all weight of made repeated measurements. In borders  is placed on the average 0,9545 (95,45%) all measurements, and on a site  - already 0,9973 (99,73%), so for «three sigma» limits leaves only 0,0027 (0,27%) all number of measurements, i.e. their insignificant share [1].

«Classical» interval arithmetics assumes, that all meanings of an interval are equiprobable [2,3]. Therefore all results received with its help, cover every possible meanings and are «supersufficient».

In the given work new interval arithmetics which is taking into account non-uniformity of distribution of meanings inside an interval is offered.

Let's enter formal concept of an interval a in the following kind:

                                                                        (1)

Where  – middle of an interval (or mathematical expectation),  – width of an interval (or äèñïåðñèÿ). Let's designate set of all such intervals as Iâåð(R).

Let a, b, c - intervals from Iâåð(R). Let's enter the following interval arithmetic operations (in the assumption, that the intervals are the independent normally distributed sizes):

1. Addition of two intervals a, bÎ Iâåð(R): a, bÎ Iâåð(R),

                                                                                     (2)

2. Subtraction of two intervals a, bÎ Iâåð(R):   ñ= a - b,

                                                                            (3)

3. Multiplication of two intervals a, bÎ Iâåð(R):   ñ= a * b,

                                                                        (4)

4. Return interval àÎ Iâåð(R):

                                                                                                     (5)

5. Division of two intervals a, bÎ Iâåð(R):

                                                                               (6)

Let  - independently distributed (independently allocated) interval characterized by mathematical expectation (by middle of an interval)  and äèñïåðñèåé (in width of an interval) .

Function  let is given interval meaning, which argument is in turn interval. Meaning of this function will be an interval, which we shall designate , determined on the formula:

, .                                                                                 (7)

Let  – independently distributed intervals characterized by mathematical expectation (by middle of an interval)  and äèñïåðñèåé (in width of an interval) . Let's designate .

Function  let is given interval meaning, which arguments are in turn intervals. Meaning of this function will be an interval, which we shall designate , determined on the formula:

,.            (8)

Let's compare entered interval arithmetics with «classical» on examples.

1. Operation of addition (subtraction). Intervals  and  let are given.

Then for new interval mathematics we shall receive the following results:

,

,

Similarly for classical interval mathematics:

,

.

Thus, the centres of both intervals coincide, however width of the entered interval (1.582) is less than width of a «classical» interval (2.0).

2. Operation of multiplication. Intervals  and  let are given.

Then for new interval mathematics we shall receive the following results:

Similarly for classical interval mathematics:

,

Thus, width of the entered interval (1.803) is less than width of a «classical» interval (2.50) and the centre of a «classical» interval is displaced on size 0.188.

3. Operation of calculation of a return interval.

In «classical» calculation of a return interval it is supposed, that . In the entered definition of a return interval it is supposed, that , if only .

Example. Intervals  and  let are given.

Then for new interval mathematics we shall receive the following results:

, .

Similarly for classical interval mathematics:

, , does not exist.

Thus, the centre of a «classical» interval  is displaced on size 0.08.

4. Operation division of two intervals.

For example, for intervals  and  we shall receive

, , , .

The centre of the entered interval is equal 1. The centre of a «classical» interval is displaced rather 1, though contains 1.

At  ànd  we shall receive    

, .

Thus, width of the entered interval is equal 1.414 and less than width of a «classical» interval, which is equal 2.666. Besides the centre of a «classical» interval is displaced from 1 on size 0.666.

Example. Intervals  and  let are given.

Then for new interval mathematics we shall receive the following results:

Similarly for classical interval mathematics:

.

Thus, width of the entered interval (0.113) is less than width of a «classical» interval (0.162) and the centre of a «classical» interval is displaced on size 0.030.

Example. Intervals  and  let are given.

Then for new interval mathematics we shall receive the following results: , is similar for classical interval mathematics: , does not exist.

5. Operation of calculation of interval functions

Let  – interval meaning function of interval argument.

Then for  we shall receive: for new interval mathematics

                                                                                (9)

For classical interval mathematics

 where                                                         (10)

As it is visible from the above mentioned formulas for differentially of functions of calculation of meanings of function on new interval mathematics (9) more structurally in view of ending of carried out arithmetic operations. At the same time calculations under the formulas (10) require the decision two optimization of tasks, for the decision generally is necessary for each of which realization of iterative calculations. Thus there are problems of convergence of iterative process and choice of an index point.

References

1.     Ñìèðíîâ Í.Â., Äóíèí-Áàðêîâñêèé È.Â. Êóðñ òåîðèè âåðîÿòíîñòåé è ìàòåìàòè÷åñêîé ñòàòèñòèêè äëÿ òåõíè÷åñêèõ ïðèëîæåíèé. – Ì.: Íàóêà, 1969.– 512 ñ.

2.     Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.

3.     Øîêèí Þ.È. Èíòåðâàëüíûé àíàëèç. – Íîâîñèáèðñê: Íàóêà, 1986. – 224 ñ.