Device for determination of number signs in system of remainder classes

FIELD: physics.

SUBSTANCE: device comprises the set of input registers for storage of number composed by the code of symmetric system of remainder classes. Permanent registers are used for storage of interval-position characteristics of constant, i.e., a positive number in symmetric system of remainder classes. Besides, it incorporates the unit for computation of interval-position characteristics and unit to test for accuracy of interval-position characteristics. Also, it includes comparator of interval-position characteristics and two-way binary decoder.

EFFECT: higher response and control over accuracy of sign definition.

3 dwg

 

The invention relates to computing and is designed to perform the operation of determining the sign of the number represented in the system of residual classes.

A device for determining the sign of the number represented in the system of residual classes (A. S. SU # 1552181, BI No. 11, 23.03.1990), which comprises a unit 1 for determining the number of the interval, the group of information inputs of the device 2, first 3 and second 4 of the comparator circuit, the first 5 and second 6 elements OR, the first 7 and second 8 inputs the constants of the device, the first 9 and second 10 output device. The device is based on identifying the affiliation of the interval in which is a number expressed in the system of residual classes (JUICE), to the group of positive or negative intervals on this basis, the JUICE of pithat broken full modular range [0, P-1], where P is the product of all of the bases JUICE. The disadvantage of this device is the biggest challenge and low performance because to determine the sign of the number you want to work with (P/pi)-bit integers.

The closest to the claimed invention is a device for determining the sign of the modular number based on the approximate method (A. S. RU # 2503995, BI # 1, 10.01.2014) containing the input registers for the modules p1p2, ..., pnfor temporary storage of bits �OK parallel adder for summing|Pi/pi|piαiinput bus for supplying the original number, look-up tables for storage of works of constants discharges JUICE|Pi/pi|piαirepresented in binary code. However, this device does not allow to check the correctness of the designated sign in if the number is in close proximity to the point of smashing the entire numeric range JUICE on the area of the positive and negative values.

The technical result of the claimed device for determining the signs of the numbers in the system of residual classes is to improve the performance compared to devices based on the exact methods, and ensuring control of correctness of determination of the sign. Presents the provisions provided through the use of a new interval-positional characteristics of modular arithmetic, which approximates with two sided�n the relative magnitude of the numbers in the modular view.

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

Let the basis set JUICE set pairwise coprime odd modules p1p2, ..., pnandP=Πi=1npi. Then the integer X from the interval [0, P-1] will be presented in the form of independent the least non-negative residue x1, x2, ..., xnandxiXmodpi|X|pi. Positional value of the number X in accordance with the well-known Chinese theorem of residues is determined by the ratio

where B1B2, ..., Bnorthogonal bases of JUICE, each i-th of which the essence is the product of Pi=P/piand|Pi1|pi . Where|Pi1|piis the weight of the orthogonal basis (multiplicative inversion from Pimodulo pi).

The sign of the number in the system of residual classes can be introduced in various ways. The most common way is to use a symmetric JUICE. In this case, if P is an odd number, then the entire numeric range [0, P-1] is split into two equal intervals [0, (P-1)/2] and [(P+1)/2, P-1], and positive numbers are represented in the younger interval and negative at older. Thus, the problem of determining the sign of X, represented in symmetric JUICE is determining its position relative to the split point (P-1)/2. To solve this problem requires the estimation of the positional value of the number X. Since the computation of its absolute value (1) time-consuming because each term has a value of order of the product modules P, and its length can substantially exceed the size of the machine word, the claimed device, also known as the analogue of (A. S. RU # 2503995, BI # 1, 10.01.2014) based on an assessment of relative values.

The relative magnitude of E(X/P) of the modular number X is the ratio of its positional integer Zn�tion to the work of all the modules P, that is,

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, we use a new interval-positional characteristic (TOC)I(X/P)=[X/P_,X/P], which is defined as a segment of directionally with rounded bordersX/P_andX/Psatisfying the conditionX/P_E(X/P)X/P. TOC projects a range of JUICE on pollinterval [0, 1), associating every modulars�th number X with a pair of rounded position numbers - borders which localize its relative value, as shown in Fig.1.

The boundaries of the TOC are represented as binary floating point numbers, and when calculating lower bounds is always rounded up to the bitness of the machine word with a negative ("down"), and when calculating the upper bound is rounded up to the bitness of the machine word in abundance ("up"). This ensures the inclusion I(X/P)∈E(X/P), that is accurate relative value (2) modular number X is localized it to the TOC. The lower bound is calculated by the formula

and the upper limit, by the formula

where xi- the i-th residue number X, | |1- fractional part of the argument, and the arrows correspond to directed rounding up to the bitness of the machine word in the calculation and summation of terms: ↓ - rounding with the disadvantage, ↑ - rounding to excess.

In the consistent case, to calculate formulas (3) and (4) requires O(n) elementary operations floating-point parallel O(log n). For comparison, the known conversion algorithms the code system of residual classes in the system with mixed bases require, respectively, O(n2) and O(n) operations.

Absolute error TOC characterizes its diameter, equal to the difference between boundaries

Let n be the dimension of the basis of the JUICE, and k is the number of bits in the mantissas in the binary representation of the boundaries of the TOC, then in the calculation according to the formulas (3) and (4) diameter (5) does not exceed n2-k. If necessary, a more accurate calculation of the TOC instead of the formulas (3) and (4) can be used the original high-accuracy algorithm (Isupov, K. S., an Algorithm for computing interval-positional characteristics to perform non-modular operations in residue number systems // Bulletin of SUSU. A series of Computer technologies, control and radio electronics". - 2014. - Vol. 14, No. 1. - P. 89-97). This algorithm is based on fast and error-free division of boundaries TOC, presents the normalized binary floating point numbers, the natural powers of two and allows us to calculate the TOC with the relative error, defined for X≠0 the ratio of the diameter (5) to the exact relative value (2), not exceeding a priori specified limit ε, thereby to obtain accurate information on the size of the numbers in the modular representation without the use of multi-bit arithmetic and time-consuming conversion in a positional system.

Due to the direction of the rounding error arising in the calculation of the boundaries of the TOC, lead only to increase in diameter (5), not rendering in General influence on the property of the inclusion E(X/P)∈I(X/P). But because the area W�of acini boundaries is limited to the interval [0, 1), in some cases this property may be violated. This occurs when the number X is very small relative to P, or Vice versa, is in the immediate vicinity of the point P-1. In the first case incorrectly calculated the lower bound of the TOC, and the second upper. In any case, diam I(X/P)<0, i.e. the lower bound greater than the upper. This TOC is called wrong by Kougeru or just wrong. The first formal condition for a correct determination of the sign - correct TOC number X, represented in symmetric JUICE. If this condition is satisfied, then the final conclusion about the correctness of the sign is formulated based on the review of the second formal condition, which consists in the absence of intersection (collision) TOCI(X/P)=[X/P_,X/P]and TOCI(P12P)=[P12P_, P12P], localizing the relative magnitude of the constants (P-1)/2, which is the largest positive number in a symmetric JUICE (median modular range). In terms of interval estimates this intersection is determined by the interval

If the diameter (5) of this interval is less than zero, the TOC does not intersect the standard set-theoretic sense, i.e., do not contain common points. In the degenerate case it may be that X=(P-1)/2. Therefore, the second formal condition for a correct computation of the sign of the number is defined as follows:

Let the symmetric JUICE modules with p1p2, ..., pngiven a number X=〈x1, x2, ..., xn〉. The algorithm for determining the sign sgn(X) of X based on the use of technology interval-positional characteristics is formulated as follows.

ALGORITHM.

Step 0. Pre-calculated and stored in computer memory TOCI(P12P)presented in the form of two binary numbers floating point P12P_andP12P. In addition, pre-calculates the next set of multiplicative inverses - weights orthogonal bases JUICE

Step 1. For the number X is calculatedI(X/P)=[X/P_,X/P]by the formulas (3) and (4), or using high-precision algorithm (Isupov, K. S., an Algorithm for computing interval-positional characteristics to perform non-modular operations in residue number systems // Bulletin of SUSU. A series of Computer technologies, control and radio electronics". - 2014. - Vol. 14, No. 1. - P. 89-97).

Step 2. Checked the first formal condition for a correct determination of the sign: ifX/Pmi> X/P_then the condition is satisfied. In this case, proceeds to step 3, otherwise to step 5.

Step 3. IfX/PP12P_then X is a positive number in the symmetric JUICE. In this case, the algorithm terminates with the result sgn(X)=0. Otherwise it proceeds to step 4.

Step 4. IfX/P_>P12Pthen X is a negative number in the symmetric JUICE. In this case, the algorithm terminates with the result sgn(X)=1. Otherwise, the diameter of interval (6) is non-negative (broken second formal condition for a correct determination of the sign of the number) and you must go to step 5.

Step 5. If improving the accuracy of calculating the TOC in step 1 is not feasible within the capacity of the used data formats, you must convert the number X from the JUICE in the numeral system with mixed bases to opredelit his mark on the basis of comparison of numbers obtained politicheskogo code with the corresponding figures calculated in advance politicheskogo code number (P-1)/2, or to generate and output a signal of the inability to determine the sign of X due to insufficient accuracy of his calculations of the TOC. The algorithm thus terminates.

EXAMPLE.

You want to determine the sign of the modular X=〈6, 8, 10, 1〉, presented in a symmetrical JUICE.

1. Calculate the constants:

AndPXI(P-12P)=[0,49,0,50];

- a set of weights for orthogonal bases(7):{6, 5, 9, 10}.

2. Calculated TOC number X by the formulas (3) and (4) rounded up to two digits

Thus, the obtained TOC I(X/P)=[0,52, 0,56], which is correct, so is the first formal condition for a correct determination of the sign of the number is made.

3. The conditionX/PP12P_not performed (0,56>0,49), proceed to the next step.

4. Compare the opposite border of the TOC: 0,52>0,50 therefore, X is a negative number in the symmetry�hexadecimal JUICE and sgn(X)=1.

5. Checking: P=9009, (P-1)/2=4504, convert to the decimal system gives X=4850. Thus, the number X lies in the second half of the full range, therefore, is negative in the symmetric residue number system.

A diagram of the inventive device for determining the signs of the numbers in the system of residual classes, functioning in accordance with the presented algorithm, shown in Fig.2. The device contains a group of input registers 1 to store the number, the sign of which it is necessary to define non-volatile registers 2, 3 storage respectively lowerP12P_and topP12Pborders interval-positional characteristics ofI(P12P)that localizes the relative magnitude of the largest positive number (P-1)/2 in the symmetric JUICE, the evaluation unit interval-positional features 4, block validation in�ervolino-positional features 5, the comparison unit interval-positional characteristics of the 6, two-input binary decoder 7. The group of input registers 1 is designed to store a number X is represented in two's complement (in the symmetric residue number system) in the form n-tuple (where n is the number of modules the JUICE) and input coming via the data bus 8, and contains the registers 1.1, 1.2, ..., 1.n whose outputs are connected to information inputs of the block 4. The outputs of block 4 are connected with inputs of block 5, as well as with the first two inputs of the block 6. The output of the non-volatile register 2 is connected with the third input unit 6, and the output of the non-volatile register 3 is connected to the fourth input of the unit 6. The output unit 5 is connected to the control input unit 6. The outputs of the block 6 are connected to the inputs of the decoder 7. The outputs of the decoder 7 is connected to the output busbars 9, 10, 11.

The work of the proposed device for determining the signs of the numbers in the system of residual classes is as follows. In advance and once the computed interval-positional characteristicI(P12P)=[P12P_,P 12P]approximating with two sides relative magnitude of the constants (P-1)/2, where P is the product of all modules of SAP. In the non-volatile registers 2 and 3 recorded values of the lower and upper boundaries,P12PandP12Prespectively. Number X=〈x1, x2, ..., xn〉 presented in more modular code that arrives on the input data bus 8 and is recorded in group 1 input registers. From the group 1 data registers are fed to the inputs of block 4, which calculates the TOC I(X/P). Calculated the TOC, which is represented as two binary numbers floating pointX/P_andX/Pserved on blocks 5 and 6. Unit 5 production�t comparison of the boundaries of X/P_andX/P: ifX/PX/P_then the corresponding control input unit 6 a signal of logical units. IfX/P<X/P_then the corresponding control input unit 6 a signal of logical zero. In block 6 by successive comparison of the boundaries of the TOC:X/PwithP12P_andX/P_/maths> withP12Pand the result is fed to the inputs of the decoder 9: ifX/PP12P_then on the first and second outputs of the block 6 are formed a signal of logical zero; ifX/P_>P12Pthen at the first output unit 6 is formed a signal of logical zero and the second output unit 6 is formed a signal of logical units; otherwise, and if the control input unit 6 set logical zero, both outputs of the block 6 are formed a signal of logical units. Decoder 7 operates as follows: if both its inputs are installed from the logical zero ("00" code), then the signal on bus 9, indicating that X is non - negative; if the first input is set to logical zero, and �Thor - logical unit (code "01"), then the signal on the bus 10, indicating that X is a negative number; if both the inputs of the decoder 7 has a strong logical units (code "11"), then the signal on bus 11, indicating that the sign of the number X cannot be determined due to insufficient accuracy of the calculation interval-positional features. Code "10", mounted at the input of the decoder 7 is prohibited and indicates a hardware failure.

An example of operation of the inventive device shown in Fig.3. In this example, was determined by the sign of the number X=〈0, 1, 8, 3〉, presented in a symmetrical JUICE with modules{7, 9, 11, 13}. Interval-positional characteristic was evaluated in unit 4, rounded to two significant decimal digits after the decimal point.

The complexity of the proposed device is evaluated as follows. To compute lower bounds TOCX/P_by the formula (3) in block 4 must be completed n modular multiplications digits xifor multiplicative inversion|Pi1|pi n of divisions of the received products into modules piwith 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. Therefore, the computation of lower bounds TOC requires 3n+1 arithmetic operations with floating point. If the lower bound has already been computed, to compute upper bounds of theX/Pby the formula (4) in block 4 does not need to repeatedly multiply the multiplicative inverse in residue number xiremains to perform one switch ALU mode of rounding "up", n divisions, n-1 additions and one discarding the decimal part of the sum total of 2n+1 operations. In total, the sequential calculation of TOC in accordance with formulas (3) and (4) must perform a total of 5n+2 arithmetic operations with floating point. One comparison operation position numbers you want to run in unit 5 and a maximum of two comparison operations position numbers you want to execute in the block 6. The delay unit 7 is determined only by the time the binary decoder and its� significant impact on the complexity. Thus, in total, to determine the sign of the number represented in n-modular JUICE requires 5(n+1) arithmetic floating point operations, provided sufficient accuracy of calculation of the TOC. On average, n(n-1) operations on residues required for converting the number of n-modular JUICE in the numeral system with mixed bases in accordance with the algorithm presented in the publication scientists N. M. Yassine and W. R. Moore (Improved mixed-radix Conversion for Residue Number Architectures // Circuits, Devices and Systems, IEEE Proceedings, 1991, Vol.138, Issue 1, P. 120-124). The comparison of digits of the received code on the system with mixed bases with the corresponding figures predefined constants (the largest positive number in a symmetric JUICE) will require in the worst case even n operations. Thus, to determine the sign of the number with the device based on the conversion method of modular representations in the numeral system with mixed bases required to perform, on average, n2arithmetic operations. Consequently, the effect of improved performance from the use of the claimed device can reach n2/5(n+1) times, where n is the number of modules the JUICE.

A device for determining the signs of the numbers in the residue number system that contains the group of n input registers for storing numbers, represented in symmetric system ostalo�tion classes characterized in that it comprises first and second non-volatile registers for storing respectively the lowerP12P_and topP12Pborders interval-positional characteristics ofI(P12P)represented as binary floating point numbers and bring the two sides relative magnitude of the largest positive number (P-1)/2 in the symmetric residue number system, where P is the product of all n modules in the system of residual classes,

the computing unit interval-positional characteristics, which operates on the principle of binary arithmetic and logical unit with a float switch selectable rounding modes and has n information inputs and two outputs,

block validation interval-positional characteristics, which has two data inputs and one output,

the comparison unit interval-positional characteristics, having four data inputs, one control input and two outputs, the binary decoder with two inputs and four outputs,

moreover, the outputs of the group of input registers connected to information inputs of the computing unit interval-positional features,

the first and second outputs of the computing unit interval-positional characteristics are connected respectively with the first and second inputs of the block validation interval-positional characteristics, and also with the first and second information inputs of the block comparison interval-positional characteristics,

the outputs of the first and second non-volatile registers are connected respectively with the third and fourth information inputs of the block comparison interval-positional characteristics

the output of the validation interval-positional characteristics is connected to the control input of the compare unit interval-positional characteristics,

the first and second outputs of the block comparison interval-positional characteristics are connected respectively with the first and second binary inputs decoder,

the binary outputs of the decoder are the outputs of the device to determine the sign of the number in the system of residual classes: "X≥0", "X<0", "Badge is not defined"

A device for determining the signs of the numbers in the residue number system that contains the group of n input registers for storing the number presented in the symmetric residue number system, characterized in that it comprises first and second non-volatile registers for storing respectively the lowerand to topborders interval-positional featuresrepresented as binary floating point numbers and bring the two sides relative magnitude of the largest positive number (P-1)/2 in the symmetric residue number system, where P is the product of all n modules in the system of residual classes,
the computing unit interval-positional characteristics, which operates on the principle of binary arithmetic and logical unit with a float switch selectable rounding modes and has n information inputs and two outputs,
block validation interval-positional characteristics, which has two data inputs and one output,
the comparison unit interval-positional characteristics, having four data inputs, one control input and two outputs, the binary decoder with two inputs and four outputs,
moreover, the outputs of the group of input registers�s are connected to information inputs of the computing unit interval-positional characteristics,
the first and second outputs of the computing unit interval-positional characteristics are connected respectively with the first and second inputs of the block validation interval-positional characteristics, and also with the first and second information inputs of the block comparison interval-positional characteristics,
the outputs of the first and second non-volatile registers are connected respectively with the third and fourth information inputs of the block comparison interval-positional characteristics,
the output of the validation interval-positional characteristics is connected to the control input of the compare unit interval-positional characteristics,
the first and second outputs of the block comparison interval-positional characteristics are connected respectively with the first and second binary inputs decoder,
the binary outputs of the decoder are the outputs of the device to determine the sign of the number in the system of residual classes: "X≥0", "X<0", "Badge is not defined".



 

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