SU428412A1 - DETERMINED-PROBABILITY DIGITAL INTEGRATOR - Google Patents

Description

I

The invention relates to the field of computer technology and can be used to create a deterministic-probabilistic digital integrator (UUS) of their machines and homogeneous digital integrating structures, for digital modeling of various objects, as well as in automatic control and management systems.

Deterministic digital integrators are known that perform the Stieltjes integration, which are built on the basis of the simplest formula of rectangles using one-bit pr, irashenii. Their disadvantage is low accuracy and low speed.

Deterministic digital integrators are known that perform Stieltjes integration, which are constructed on the basis of more accurate numerical integration formulas and multi-bit increments. In particular, integrators are built with (using the trapezoid, quadratic and cubic parabolic formulas. Such integrators are distinguished by high accuracy and speed. However, their structure is complex and requires high equipment costs.

Probably integrators are known that perform the Stieltjes integration, which are based on the Monte Carlo statistical method. For high accuracy

Such works of the integrator require the implementation of a very large number of statistical tests, which dramatically reduces their performance.

The aim of the invention is to create a deterministic-probabilistic digital integrator that performs the Stieltjes integration and provides high accuracy of integration, as well as a significant response time at 1o at a low equipment cost.

In addition to the known unit for forming the integrand function and the unit for calculating the integral increment directors, the integrator described here includes a unit for forming the probabilistic flux of the increments of the integral residual, a unit for generating the corrective probability increment of the increments and block | K for the formation of the probabilistic increment of the second integral difference.

The proposed deterministic-probabilistic digital integrator performing

the Stieltjes integration operation differs from the known ones in that the information in it is presented and processed not in a deterministic and probabilistic, but in a mixed deterministic intimated-probabilistic form.

The main parts of the integrand, non-temporal integration and the integral are represented in this integrator in digital form and are calculated by a deterministic method based on the simplest form of ordinary carbonic lines, and the corrective small parts of these Velikchdans, which serve to refine the calculations, are presented in the form of probable streams of single increments and calculated by the statistical Monte Carlo method.

The calculation of the main determining part of the integral is performed very simply with the help of the formula of rectangles based on multi-bit increments with high accuracy. The small refinement part of the integral is calculated by a statistical method with a relatively low accuracy using a small number of statistical tests. However, as the theoretical analysis shows, the overall resulting accuracy of the integral calculation by this method turns out to be very high and comparable to the accuracy that is obtained when calculating the integral based on the formula of quadratic parabolas.

The principle of calculating in the integrator the main part of the integral in deterministic form using the simplest formulas of the rectangles, and the specifying part by the probabilistic method, makes it possible to obtain a simple integrator structure that requires little equipment, and at the same time ensures high accuracy of integration, and T9; kzhe significant performance with a relatively small number of statistical tests.

The proposed deterministic-probabilistic integrator calculates the increment of the Stieltjes integral

Ug (Xi + i - UR (x) dyi, (x), x /

where UR (x) is the integrand function, Yf / (x) is variable integration,

X - (Independent variable-a, z (x) - integral.

Each of the variables / in the proposed integrator is represented as the sum of the quantized private residue Ov:

z (x), r ((l :), Y, (x) UR (X) + O, - 1Ur (x), Un (x) Ua (x + Ol.; (l :).

The quantized parts z (x), ur (x), and yQ (X) are used in deterministic digital form (and the residuals are represented by probability streams of single crystals).

(x (/).,. (O., (x) - (i),

Oh, {g / (x)} - (i, so that

O. ,,,) “Ag iizi (/), y /

t

O, v ,, () Ar / p2 (rp /), t

Qxi {yq (x, n) Aud 2 (l}

where rizi (/) is the probabilistic flow of increments of the remainder of the integral, LRI /) is the probabilistic flow of increments of the remainder of the integrand

functions,

4f / i (/) is the probabilistic flow of increments of the remainder of the variable of integration. Taking into account the representation of each variable in the form of the main quantized value and the remainder, represented by the variable stream of increments, in the construction of the algorithm of the deterministic - probabilistic I1 integrator performing the Stieltjes integration operation, a number of expressions are put.

The drawing shows a block diagram of a deterministic-probabilistic digital integrator performing the Stieltjes integration operation in accordance with the derived algorithm.

The proposed device comprises a block / formation -different function; block 2 calculating the deterministic part

increments of the integral; block 3 for calculating the total increment of the integral; a unit 4 for forming a probability stream of increments of the remainder of the integral; block 5 forming a correction probabilistic flow of increments on the remainder of the integrand; a unit 6 for forming a correction probabilistic flow of increments on the second difference of the integration variable; block 7 the formation of the total corrective

probabilistic incremental flow; an auxiliary difference calculation unit 8; an auxiliary difference increment generation unit 9; a probability flow generation unit 10 of increments of a second integral difference; bus // incrementing variable; bus 12 increments of integrand functions; SCHYNY 13 13 Vertical flux of increments of the second difference of the integration variable;

bus 14 | Vertical flux of increments of the remainder of the variable integration and bus 15 prob T | BUT | CT1 of the flow of the increments of the remainder of the integrand.

The proposed integrator circuit consists

from three groups of functional blocks.

The first group includes the / -3 blocks and performs the operation of calculating the increment of the integral Vz; (Xj + i). In this case, in the / block, the value of the integrand is calculated.

the function UR (xi) at the point Xi in block 2 determines the deterministic part of the increment of the integral (Xg + 1), and in block 3 the total value of the increment of the integral is calculated. The second irpynna includes blocks 4 and 5 and performs the operation of calculating the probabilistic unit increment stream (i), which represents the remainder of the integral O., {Z (x). The operation of calculating the stochastic flux t) yy (/) is performed in a block;. Block 5 is used to compute the correction probabilistic flux of increments t | 1g (/), which is determined and used further in the third group of blocks. The third block npyiona include blocks 6-10 and is used to calculate the probabilistic flux of unit increments (/), representing the second difference of the integral, n the total corrective probabilistic flux of unit increments T1g (/). which is used in the first block group to calculate the probability component of the integral increment. In block 6, a corrective probability flow of increments ii2i (/) is calculated. In block 7, the stream 11i (/), coming from the third group of blocks, and the stream (/) are summed. As a result, the stream i, - (/) is formed. Block 8 is used to calculate the value of V 2z (l: g +); 8, block 9 determines the auxiliary probabilistic flux of unit increments TisiO) —Block 10 serves to sum the fluxes Г1г (/) and TisiC /) and determine the probabilistic flux of increments (i) representing the second difference of the integral. The input of the deterministic-probabilistic digital integrator information is received in five as the values of Y (, (X, -) I (X (), represented in digital form, and the stochastic flows of individual increments of TTGC /), r.id) and (i). The increment of the integrand function Vy (l:, -) goes to the input block /, which is summed with the previous value. the integrand function yj, () stored in the block register, as a result of which the image is the value of the integrand function yp (xi) at the 1st integration point. Forming the value y (x,) is fed to the input of block 2. The second input of block 2 is incremented by the integration variable Vi / gC;,). In block 2, the values) and 7yg (xj) are multiplied, resulting in the deterministic part of the increment of the integral VZa (. The increment 72j (; + i) is sent to one of the INPUTS of block 3. The second input of block 3 is given a probable increment of T1g (/), which is summed in the reversible counter of block 3. The obtained value, then summed in block 5 with the deterministic increment part of the integral V 2 (Xi + i), forms the total increment of the integral Vz (:, - + i) issued on one From three outputs of the integrator. The input of block 4 receives probabilistic flows The values of Tjyif /) of the range 14 and (j} of the area 15 and the value of Ur (X {) from the output of block 1. In addition, a random value is fed to the input of block 4) -i (/). The magnitude of Ur (Xr) is placed in the reversible counter of block 4 and summed with the stream of increments R (R). In each m / -th cycle, the result of the summation is compared with a random number | .i - (/), which is in the additional register of block 4. If the sum forming in the reversible counter of block 4 is more than a random variable ji (/) then the comparison circuit (not shown in the drawing) of block 4 yields at the output a positive or negative unit iB depending on the sign of r, / i (j). In the opposite case, as well as at ijid) O, zero is output. The probabilistic flux of unit increments (iiziQ) thus formed represents the remainder of the integral z (Xj) and is output at one of the three outputs of the deterministic-probabilistic integrator. At the input of block 5, probabilistic currents 11p / (/) and Tigj (/) are given, as well as random magnitudes, ui (/). The flow l /; / (/) is summed up in a reversible counter (not indicated in the drawing) of block 5, and forming a value is compared in each / -th cycle with a random variable -ii (/). If the sum in module 5 is larger in absolute magnitude than | Л1 (/), and if 1Тд, - (/) is different from zero, then the comparison circuit of block 5 gives the output unit, the sign of which coincides with the sign) In the opposite case output is H | ul. The auxiliary probabilistic flux of increments 1T1g (/) thus formed is directed to the input of block 7. At the input of block 6, the value of Ur (Xr) is given. the probabilistic flux of increments fqiU) and the random value 2 (/). The values of yp (Xi) and C2 (/) are compared in each / -th cycle. If | Ur (.G;)) J.i2 (/), then the result of the comparison is the unit 6, the sign of which coincides with the sign of the symbol (/). Otherwise, the output of the block 6 is zero. The resulting ia output of block 6 is a probabilistic flow of increments of zid fed to the input of block 7, to the second input of which a probable flow of increments of r | and (/) is applied. Both Tiii (/) and ii2, - (/) sums | costes in block 7, resulting in a probability stream of unit increments (i) supplied from the output of block 7 to the inputs of blocks 3 and W. The inputs of block 8 are fed - from the output of block 2, the deterministic component of the increment of the integral is 72 liters (l:, q1) and from the output of block 3 - the total increment of the integral (xi +). In the register of block 8, the increment (xi + i) is delayed by the time of the complete integrator step, and in the adder of block 8, the difference of the values of У2д (i) to z (Xi) is formed. The difference obtained at the output of block 8 is fed to the input of block 9, the second input of which is supplied with a random value | m3 (/). If the specified difference is greater than the module (is (V), then the output of block 9 forms a unit with the sign of the difference, and otherwise case - N) ul. The probabilistic increment tray (/) thus obtained in block 9 is directed to the input of block 10, at the second input of which the probabilistic flux r) g (/) enters. Summing up in block 10, the streams (j) and yrg (/) form a probability tfzid increment stream}, which represents the second difference of the integral and is output at one of the three integrator outputs. The subject matter of the invention is a deterministic-probabilistic digital integrator comprising an integrator function generating unit, the output of which is connected to a deterministic calculating unit. the inputs of which are connected to the probabilistic flux of the increments of the remainder of the variable integration and the probabilistic flux of the increments of the remainder of the integrand f actions, - and the output of the block of formation of the integrand; a unit for forming a probability correction flow of increments on the remainder of the integrand, the inputs of which are connected to the probabilities of the incremental flow of the remainder of the variable integration and the probable flow of the increments of the remainder of the integrand; a unit for forming a probability correction flow of increments of the second difference of the integration variable, the input of which has a probability spread of the second difference of the variable of integration, and the output of the integrator function block; a unit for the formation of a total correction probabilistic flow of increments, the inputs of which are connected to the blocks of forming a corrective probabilistic flow of increments on the remainder of the neutral integral function and on the formation of a corrective probable flow of increments on the second difference of the integration variable; an auxiliary difference calculating unit, to the inputs of which the outputs of the calculating unit of the deterministic part of the integral increment and the unit for calculating the total integral increment are connected; an auxiliary difference increment forming unit of the auxiliary difference, to the input of which an auxiliary difference calculation unit output is connected; the probability unit} of the 1st increment stream of the second integral difference, the inputs of which are connected to the outputs of the formation unit of the total correction probabilistic increment stream and the incremental difference formation unit of the auxiliary difference; the unit for calculating the total increment of the integral, the inputs of which are connected to the unit for calculating the deterministic part of the increment of the integral and the unit for generating the total correction probabilistic flow of increments, and the output is connected to the unit for calculating the auxiliary difference.