# Device for comparing numbers in system of residual classes based on interval-positional characteristics

FIELD: information technology.

SUBSTANCE: presented positions are provided by using a novel interval-positional characteristic of modular arithmetic, which approximates the relative value of a number in a modular presentation from two sides. The device comprises groups of input registers for storing modular numbers to be compared, units for calculating interval-positional characteristics, a unit for bitwise comparison of modular numbers, units for verifying interval-positional characteristics, a unit for comparing interval-positional characteristics and a two-input binary decoder.

EFFECT: faster operation and enabling verification of a comparison result.

4 dwg

The invention relates to computing and is designed to perform the operation of comparing two numbers represented in the system of residual classes.

A device for comparing numbers expressed in the system of residual classes (A. S. SU # 608155, BI No. 19, 19.01.1976) that contains conversion units 1, 2, each of which consists of register 3, the adder 4, the node of division 5, OR 6 elements, groups of elements And 7 And 8, 9, item OR 10, the pulse distributor 11, the register 12 of the storage modules, the selection unit 13 modules, the comparison block 14, block equality comparisons 15, switches 16, 17, element OR 18. The device is based on the exact method of converting a modular numbers the number system with mixed bases. The disadvantage of this device is the biggest challenge and poor performance.

The closest to the claimed invention is a device for comparison of numbers represented in the system of residual classes, based on the approximate method (A. S. RU # 2503992, BI # 1, 10.01.2014) containing the input registers 1, 9 to store numbers, diagrams determine the signs of the numbers 2 and 8, schematic of the shift polarity 3, 7, look-up tables 5, 6 to store the product of constants and discharges the JUICE, the adder 10, the logical element "EXCLUSIVE OR" 4, the schematic analysis of the sign 11. However, this device does not enable�t to check the correctness of the result of comparison of two nearby modular numbers without taking account of rounding errors generated during the calculation of the approximate relative values of the modular number.

The technical result of the claimed device for comparison of numbers in the system of residual classes is to improve the performance compared to devices based on the transformation of the compared numbers in a positional numeral system with mixed bases, and ensuring control of the correctness of the result of the comparison operation. Presents the provisions provided through the use of a new interval-positional characteristics of modular arithmetic, which approximates two sides relative magnitude of the numbers in the modular view.

Device description: the basis for operation of the claimed device for comparing the numbers in the residue number system is a new method of interval estimation of the relative value of modular numbers. Will consider it.

Let the basis of the system of residual classes (JUICE) is pairwise coprime modules p_{1}p_{2}, ..., p_{n}and P is the product of all modules. Then the integer X from the interval [0, R-1] will be presented in the form of independent the least non-negative residues (deductions) 〈x_{1}, x_{2}, ..., x_{n}〉, and x_{i}≡X mod p_{i}↔|X|p_{i}. Positional value of numbers in accordance with the Chinese theorem of residues is determined by the ratio

where_{1}In_{2}, ..., B_{n}orthogonal bases of JUICE, each i-th of which the essence is the product of P_{i}=P/p_{i}and. Hereis the weight of the orthogonal basis (multiplicative inversion from P_{i}modulo p_{i}). The relative magnitude of E(X/R) of the modular number X is the ratio of its positional value to the product of R modules, i.e.

Since the exact rational value of E(X/P), varying in the interval [0, 1), in General not representable in a computer with limited bit net, there is a task of its approximation. To solve this problem is interval-positional characteristic (TOC), which is defined as cut (a closed real interval) directionally with rounded bordersandsatisfying the condition. Thus, the TOC displays the range of the JUICE on pollinterval [0, 1), associating all the modular number X with a pair of position numbers rounded - borders which localize its relative value, as shown in Fig. 1.

The boundaries of the TOC are represented as binary floating point numbers

p> whereand- the mantissa, e - order, which is the same for both borders. So the bottom border is always calculated with rounding down, and the upper bound is rounded up. This ensures the inclusion I(X/R)∈E(X/P), that is accurate relative value (1) modular number X is localized it to the TOC.Absolute error TOC characterizes its diameter

An objective measure of the accuracy of the TOC is its relative error

whereanddetermined by the relationship of the absolute error limits (2) to the value (3) when X≠0. In the limiting case, we have

Let n be the number of modules SAP, R is the product of the JUICE modules, k is the number of bits in the mantissas in the representation of boundaries (2), ε is the limit of permissible relative error of the TOC. We introduce the notation

The algorithm for computing the TOC with an error of (4), not exceeding the permissible limit ε, is formulated as follows.

ALGORITHM 1.

Step 0: preliminary by using the extended Euclid's algorithm is calculated and stored weight orthogonal bases

In addition, computes�I and stored in the memory of the natural vector of powers of two

where ν_{j}=log_{2}(1/2^{j}ψ^{j}) for all j=1, 2, ..., g, with g=log_{2}(1/P)/(1+log_{2}ψ)-1.

Also calculated and stored in the memory matrix is shifted weights orthogonal bases, whose rows are associated with the elements of the vector v, and the columns with modules

Step 1. Computes the upper bounds of the TOC (formula follows directly from the Chinese theorem on residues)

where ↑ indicates that the calculation and summation of terms is performed with rounding to the full ("up").

Step 2. Ifthen transition to step 3, otherwise to step 4.

Step 3. Calculates the lower bound of the TOC

where ↓ indicates that the calculation and summation of terms is rounded with a negative ("down"). Continue to go to step 10.

Step 4. Set the start index offset j=1. Next - go to step 5.

Step 5. Calculated offset upper limit of the TOC

where M_{j,i}- i-th element of the j-th row of the matrix M.

Step 6. Ifthen given accuracy is achieved. In this case, go to step 8, otherwise to step 7.

Step 7. The index j increases needinit and you are returned to step 5.

Step 8. In accordance with the found in an iterative block index offset j is selected from j-I row of the matrix M is calculated and offset the lower border

Step 9. At index offset j is selected from the v vector

Step 10. The algorithm terminates, returning the value of the rangeand.

The scheme of algorithm 1 is shown in Fig. 2. The finiteness of the algorithm (no loops) provides an accepted way to specify a vector v, which ensures thatwhere.

The maximum number of iterations is

It is guaranteed that for g iterations the TOC for the smallest non-zero number X=1 will be computed with a relative error that does not exceed ε, that is, diam (I(X/R)/(1/P)≤ε.

Limitations of the presented algorithm: first, it must satisfy the condition ψ<0,5; secondly, the minimum exponent (order) e_{min}in floating-point format that is used d�I represent the boundaries TOC,
should not be greater than the difference e-ν_{j}where e is the order in exponential notation the boundaries of the TOC, and ν_{j}- the value of the bias powers of two, pick up on the steps 5-6-7. Consider the following example.

EXAMPLE 1.

Let modular system is defined by the set of modules{7, 9, 11, 13}. You want to calculate the TOC for numbers X=〈1,6,10,0〉 and Y=〈4,7,3,2〉 with an error not exceeding 1%, in a four-digit decimal arithmetic, that is, rounded to four significant digits (significant figures, the numbers are all the numbers in his recordings, starting with the first non-zero left). For clarity, we will use decimal and not binary system, so the above algorithm 1 will have a corresponding decimal interpretation.

1. Define all the necessary constants for a given system modules:

- a set of weights for orthogonal bases(5): {6, 5, 9, 10};

- the check digit ψ=4·10^{-4}/0,01=0,04;

is the bias vector of degrees (6): v=(10^{1}, 10^{2}, 10^{3});

- matrix is shifted weights orthogonal bases (7)

2. First, we calculate the TOC number X.

2.1. Calculated the upper limit according to the formula (8):

2.2. As 0,3722>0,04, the refinement of the TOC is not required. The calculated lower boundary by the formula (9)

Thus, the relative led�rank number X=〈1,6,10,0〉 is localized to the interval [0,3722, 0,3725]. To check the converted X in a positional system: X=3354, therefore, 0,3722<E(X/P)=3354/9009<0,3725. Relative error of 0.08%.

3. Now we calculate the TOC number Y.

3.1. Calculated the upper limit according to the formula (8)

3.2. Because 0,0029<0,04, the required accuracy of the TOC is not provided. So it is necessary to apply a clarifying iteration. We will take j=1 and calculate the offset upper limit according to the formula (10), choosing as the multiplicative inverses of the first row of the matrix M

3.3. The required accuracy is not achieved, as 0,0279<0,04. So we take j=2 and the newly computed offset upper limit according to the formula (10), but using the second row of the matrix M

3.4. As 0,2777>0,04, the accuracy achieved. The calculated lower boundary in accordance with the expression (11)

3.5. By dividing the offset of the boundaries of the second element of the vector V is obtained resulting TOC I(Y/P)=[0,002773, 0,002777] (leading zeros are not stored in registers of the computer, but reflect the negative value of the order). To check the converted Y in a positional system: Y=25, hence, 0,002773<E(X/P)=25/9009<0,2777. Relative error 0,072%. Therefore, algorithm 1 allowed us to obtain accurate information on the size of the numbers in the modular representation without the use of labor�capacious conversion in a positional system.

Thus, when the interval of approximation of the relative values of the modular number is a natural account of rounding errors, do not require consideration of the specifics of the model used computing machine. This allows a simple way to control the correctness of the result of non-modular operations, regardless of the number of modules JUICE and their capacity.

Let the JUICE modules with p_{1}, R_{2}, ..., p_{n}given unsigned numbers X=〈x_{1}, x_{2}, ..., x_{n}〉 and Y=〈y_{1}, y_{2}, ..., y_{n}〉. Algorithm comparison using interval-positional characteristics is formulated as follows.

ALGORITHM 2.

Step 1. Performed a pairwise comparison of residues to exclude the trivial case: if x_{i}=y_{i}for all i=1, 2, ..., n, then X=Y, and the algorithm terminates.

Step 2. According to the above high-precision algorithm 1 calculates the TOC of the operands, I(X/R) and I(Y/P), respectively.

Step 3. Checked the first formal condition for a correct comparison, ifandthen the condition is satisfied. In this case, proceeds to step 4, otherwise to step 6.

Step 4. If, then the algorithm terminates with the result of X>Y. Otherwise, goes to step 5.

Step 5. If , then the algorithm terminates with the result of X<Y. Failure to comply with this inequality in this step speaks about the violation of the second formal requirements for a valid comparison due to insufficient accuracy of the calculation of the TOC. Thus it is necessary to proceed to step 6.

Step 6. If improving the accuracy of calculating the TOC is not feasible within the capacity of the used data formats, it can convert numbers X and Y from the JUICE in a system with mixed bases and compare the numbers politicheskikh codes, starting with the eldest, or to generate and issue a signal to the impossibility to compare numbers because of insufficient accuracy of the calculation of the TOC. The algorithm thus terminates.

The conjunction of the first and second formal conditions is the symptom of (sufficient condition) the correctness of the result of non-modular operations comparison of numbers, invariant as to the number of modules the JUICE, and to their capacity.

EXAMPLE 2.

You want to compare numbers X=〈1,6,0,5〉 and Y=〈2,3,4,7〉 presented in JUICE with modules {1,9,11,13}.

1. Numbers obviously are not equal, therefore, calculated their TOC in accordance with algorithm 1, rounded to four significant decimal digits:

2. The intervals I(X/P)=[0,03662, 0,03665] and I(Y/P)=[0,03827, 0,03831] ensure the fulfillment of the first formally�CSOs conditions correct comparison, as 0,03665>0,03662 and 0,03831>0,03827.

3. The conditionthat fails because 0,03662<0,03831. Compare the opposite border: 0,03665<0,03827 therefore X<y

4. Check the conversion to the decimal system: X=330, Y=345. Thus, using the TOC obtained the correct result of the comparison of two nearby modular integers.

A diagram of the inventive device for comparing the numbers in the residue number system based on interval-positional characteristics, functioning in accordance with the described principles is shown in Fig. 3. The device comprises a group of input registers 1, 2 storage compare modular numbers, blocks calculate interval-positional characteristics 3, 5, block a bitwise comparison of modular numbers 4, blocks validation interval-positional characteristics 6, 8, a unit for comparing interval-positional characteristics of 7, two-input binary decoder 9.

The group of input registers 1 is designed to store a number X is represented as n-tuple (where n is the number of modules the JUICE) and received via the data bus 10, and contains the registers 1.1, 1.2, ..., 1.n whose outputs are connected to information inputs of the blocks 3 and 4. In turn, the group of input registers 2 is for storing the number Y is also represented by n-tuple and�topaudio via the data bus 11, and contains the registers 2.1, 2.2, ..., 2.n whose outputs are connected to information inputs of the blocks 5 and 4. The output of block 4 is connected to control inputs of blocks 3, 5. The outputs of blocks 3, 5 are connected to the inputs of blocks 6, 8, respectively, and also with the information inputs of the block 7. The outputs of blocks 6, 8 are connected to control inputs of the block 7. The outputs of the block 7 is connected to the inputs of the decoder 9. The outputs of the decoder 9 is connected to the output tyres 12, 13, 14, 15.

The work of the proposed device for comparison of numbers in the residue number system based on interval-positional characteristics is as follows. Compare modular numbers X=〈x_{1}, x_{2}, ..., x_{n}〉 And Y=〈y_{1}at_{2}, ..., y_{n}〉 live according to the data buses 10, 11 and recorded in the group of registers 1, 2, respectively. From the group 1 data registers are fed to the inputs of unit 3 and the first n inputs of the block 4. From the group 2 data registers are fed to the inputs of the block 5, and the second n inputs of the block 4. Unit 4 produces a pairwise comparison of residues (x_{i},y_{i}), i=1, 2, ..., n, and generates a signal of logical zero if all balances are pairwise equal, and the signal of logical units otherwise. This signal is fed to control inputs of blocks 3, 5, in which, if the numbers are not equal (the output of block 4 is formed of a logical unit) is Vice�tion TOC I(X/R) and I(Y/P), respectively.
If the output of block 4 is formed a logical zero, then each of the outputs of blocks 3, 5 is also set to zero. Calculated the TOC, each of which is represented as two binary floating point numbers, respectively,and,served on the blocks 6, 8, respectively, and the block 7. Unit 6 compares bordersand: ifthen the corresponding control input of the block 7 signal of the logical unit. Ifthen the corresponding control input of the block 7 signal is logic zero. The same applies to the block 8. In block 7 compares the opposite borders of the TOC and the result is fed to the inputs of the decoder 9: if,,,then on the first and second outputs of the block 7 are formed a signal of logical zero; otherwise, ifthen at the first output unit 7 is formed a signal of logical zero and the second output of the block 7 is formed a signal of logical one; otherwise, ifthen at the first output unit� 7 is formed a signal of logical units,
and on the second output unit 7 is formed a signal of logical zero; otherwise, and if at least one of the control inputs of the block 7 is set to logical zero, both outputs of the block 7 are formed a signal of logical units. Decoder 9 operates as follows: if both its inputs are installed from the logical zero ("00" code), then the signal on bus 12, indicating that X=Y; if the first input is set to logical zero and the second logical unit (code "01"), then the signal on bus 13, indicating that X>Y; and if the first input is set to logical unit and the second logical zero (code "10"), the signal on bus 14, indicating that X<Y if both its inputs are installed from the logical units (code "11"), then the signal on bus 15, indicating that the result of the comparison of the numbers X and Y cannot be established due to insufficient accuracy of the calculation of their interval-positional characteristics.

An example of operation of the inventive device for comparing the numbers in the residue number system based on interval-positional characteristics shown in Fig. 4. Performed comparison of the numbers X=〈6,8,9,4〉 and Y=〈4,6,7,2〉 presented in JUICE with modules {7,9,11,13}. Interval-positional characteristics were calculated in blocks 3 and 4 with OK�plenium to four significant decimal digits.

The complexity of the proposed device depends on the values compare modular numbers: the smaller the value, the greater the number of iterations required to calculate the TOC in accordance with algorithm 1 (and the longer work units 3 and 5). Give assess the complexity for the case when the numbers are large enough to perform steps 4-9 of algorithm 1 is not required. In this case, to calculate the upper boundary of the TOCin step 1, you must perform n multiplications modular bits of x_{i}for multiplicative inversionn of divisions of the received products into modules p_{i}with rounding down (switching the rounding mode arithmetic and logical unit (ALU) requires one operation - load preset mask control register), n-1 additions stacked and one operation of receiving a fractional part of the resulting amount. In total, the computation of lower bounds TOC requires 3n+1 arithmetic operations of floating point operations. One operation positional comparison is required to perform step 2 under the condition that the constant ψ calculated in advance. If the lower bound has already been computed, it is possible to compute upper and lower borderin step 3 of algorithm 1 does not need to re-multiply multiplicative� inversion on the remains of a number x_{
i}remains to do any switching in ALU rounding mode "down", n divisions, n-1 additions and one discarding the decimal part of the sum total of 2n+1 operations. In addition, you need to install the ALU in the rounding mode to "nearest". Thus, the sequential calculation of TOC by algorithm 1 for the case when performing a lookup of steps is not required, you must complete a total of 5n+4 floating point operations. Hence the complexity of the proposed device is evaluated as follows. Let the JUICE contains n modules. Then to compare the numbers X and Υ, provided that they are not equal, you need to perform the following operations: n operations positional comparison of pairs of residues must be completed in block 4; 5n+4 operations must be performed for simultaneous calculation of TOC in blocks 3 and 5; one of the operation position of the comparison must be performed simultaneously in blocks 6 and 7; two operations positional comparisons (in the worst case) should be performed in block 7. Thus, in the best case, the comparison of two unequal numbers are represented in n-modular JUICE, will be implemented in n+5n+4+1+2=6n+7 operations. To compare numbers using devices based on the conversion method of modular representations in the numeral system with mixed bases requires about 2n^{2}operations. Therefore, the effect of higher�Oia performance from the use of the claimed device can reach on average of 0.33 n times,
where n is the number of modules the JUICE.

A device for comparison of numbers in the residue number system based on interval-positional characteristics containing the first and second group of input registers, each of which consists of n registers that store compare modular numbers X and Υ, characterized in that it comprises first and second blocks calculation interval-positional characteristics, each of which has n information inputs, one control input and two outputs, the block bitwise comparison of modular numbers having 2n information inputs and one output, the first and second blocks of the validation interval-positional characteristics, each of which has two data inputs and one output, the block of comparison interval-positional characteristics, having four data inputs, two control inputs and two outputs, the binary decoder with two inputs and four outputs, and the outputs of the input registers of the first group are connected with the first n information inputs of the block bitwise comparison of modular numbers and information inputs of the first computing unit interval-positional characteristics, the outputs of the input registers of the second group are connected with the second n information inputs of the block bitwise comparison of modular numbers and information inputs of the second �Loka calculate interval-positional characteristics, the output of the bitwise comparison of modular numbers connected to control inputs of the first and second blocks calculation interval-positional characteristics of the first and second outputs of the first computing unit interval-positional characteristics are connected to the inputs of the first block validation interval-positional characteristics, as well as first and second inputs of the block comparison interval-positional characteristics of the first and second outputs of the second computing unit interval-positional characteristics are connected to the inputs of the second block validation interval-positional characteristics, and also with the third and fourth inputs of the block comparison interval-positional characteristics, the output of the first block validation interval-positional characteristics connected to the first control input of the compare unit interval-positional characteristics, the output of the second block validation interval-positional characteristics connected to the second control input of the compare unit interval-positional characteristics of the first and second outputs of the block comparison interval-positional characteristics connected with the first and second binary inputs of the decoder, the binary outputs of the decoder are the outputs of the device to compare the numbers in SIS�EME residual classes based on interval-positional characteristics: "X=Y" "X>Y", "X<Y", "the result of the comparison is undefined".

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