JPS62182841A - Square root arithmetic system - Google Patents

Abstract

PURPOSE:To perform the square root arithmetic at high speed by obtaining the temporary square root of data from a data table and then obtaining a real square root through the convergent calculation using a recurrence formula. CONSTITUTION:The n-bit data X<2> is supplied and the square root of upper 16 bits of the data X<2> is retrieved out of a data table. Then the lower (n-16)bits of the data X<2> are all set at 0. Thus the square root X1 of the data X<2>1 is obtained. Here the retrieved square root is shifted by (n-16)/2 bits and (n-15)/2 bits when the remaining (n-16) bits are even or odd respectively. Thus a square root X1 is obtained. In the same way, the square root X2 of the data X<2>2 is obtained when lower (n-32) bits of the data (X<2>-X<2>1) are all set at 0. Then the initial value Xe(0) of the repetitive arithmetic is set equal to (X1+X2) with a desired square root set at Xe(n). An equation I is calculated and the convergence of an equation 2 is decided. if the convergence of the equation 2 is confirmed, the Xe(n) of that time point is defined as the square root X of the data X<2>.

Description

[Detailed description of the invention] [Industrial application field] The present invention relates to a square root calculation method.

[Conventional technology]

Conventionally, when calculating a qz square root on a general digital computer that does not have special hardware or arithmetic processor for (L, JJ root calculation), the data is added while shifting the data by 2 bits, and then the data to be the answer is calculated. (however,
During the calculation, the intermediate result) is subtracted and the answer data is corrected depending on the presence or absence of pollo.

FIG. 3 is a flowchart showing this calculation method. Temporary register TReg, root register register R
Clear Reg and set input data (maximum 64 bits) in source register SReg (step 21).
Source register S is added to the contents of temporary register D Reg.
Add the upper 2 bits of Reg and add the root register RR.
Subtract the contents of eg (step 22), check whether there is a pollo (step 23), add the upper two bits of the source register SReg to the temporary register TReg if there is a pollo (step 24), root register R
Shift Reg by 1 bit to the left (step 25), and if there is no pollo, add the upper 2 bits of source register SRag to temporary register TReg, and subtract the contents of root register RReg (step 2B) i, shift root register RReg to the left Add 1 to the shifted one bit (step 27), source register SRegt
- Shift 1 bit to the left further by 1 bit (step 28), from step 22 to step 2B
Check whether the process up to this point has been repeated 32 times (Step 29). After repeating the process 32 times, the root register R
Results are obtained in Reg.

The following table shows source register SReg = 101101110
It shows the progress of the calculation when it is set to 0.

[Problem that the invention seeks to solve]

Although the conventional two-way method has the advantage of consisting only of basic operations such as addition, subtraction, shift, pollo detection, etc., it requires a large number of calculation loops (for example, for 64-bit data, The disadvantage is that the processing time is longer (for 64-bit data, it takes about 1.5m5ec with Intel1's processor 8088.5Hz), and if you use Intel1's processor 8087, it will take longer. , 40-50 p-sec
Although it is faster (at 5 Hz), it has the disadvantage of using 8087, resulting in a significant increase in cost. Another square root calculation method is the Japanese Patent Publication No. 59-2055. R.O.
There is a method using M (Japanese Unexamined Patent Publication No. 58-121449), but this is not suitable for a request where the data length is long (e.g. 84 bits) and the permissible accuracy is I LSB. The drawback is that the ROM increases proportionally.

An object of the present invention is to provide a square root calculation method that can perform square root calculations at high speed without adding special hardware.

[L stage for solving problems]

The 1+i-force root calculation method of the present invention has a data table of qi square roots for data of a predetermined bit length, and a data table of the original data whose bit length is equal to or larger than the predetermined bit length and for which the 41 square root is to be calculated. The square root of the upper predetermined bit length data is indexed from the data table according to the even/odd number of the remaining bit length after removing the predetermined bit length from the upper order of the original data, and the indexed square root is then indexed as the remaining bit length. means for obtaining a temporary square root of the original data by shifting the bit length upward by a bit length that is half the bit length of , and means for calculating the square root of the original data by convergence calculation using the temporary square root as an initial value.

Another square root operation method of the present invention has a square root data table for data of a predetermined bit length, and a bit length that is at least twice the predetermined bit length, and has a data table for square roots of data of a predetermined bit length, and has a data table for data having a square root of data having a predetermined bit length. The square root of the data of the predetermined bit length is indexed from the data table according to the even/odd number of the first remaining bit length obtained by removing the predetermined bit length from the second position of the original data, and further from the original data. A second remaining bit length obtained by removing a bit length twice the predetermined bit length from the upper order of the original data, which is the square root of the upper predetermined bit length data of the lower data excluding the upper predetermined bit length data. index from the data table according to the even/odd number of , and add both square roots thus indexed from the data table after shifting them upward by half the bit length of the first and second remaining bit lengths, It has means for calculating a temporary square root of the original data, and means for calculating the square root of the original data by convergence calculation using the temporary square root as an initial value.

[Effect]

In this way, since no special hardware is required other than preparing a memory for the data table, square root calculations can be performed using a general processor in a processing time equivalent to that of an arithmetic processor. Note that even if the data length increases, the number of convergence calculations increases, and the
The number does not change.

〔Example〕

Next, an embodiment of the present invention will be described with reference to the drawings.

FIG. 1 is a flowchart showing an embodiment of the square root calculation method of the present invention, and FIG. 2 is a diagram showing a data structure.

A memory (not shown) stores a table of square root data for 18-bit data (8 bits for each decimal part). The capacity of this data table is 2X2'
7-dotX2 (separate tables depending on whether (n-IEI) is even/odd) = 256KB 0 In this embodiment, this data table is indexed twice for every 16 bits,
A provisional square root is found, and the square root is found by calculation using this provisional 1i square root as an initial value.

Next, the calculation method of the tree embodiment will be explained with reference to FIGS. 1 and 2. First, data of npiI・X2 (Fig. 2(
1))) (Step l) 0 Next, data x2
The IL square root of the upper 16 bits of is indexed from the data table, and the square root Xl of data x12 (Figure 2 (2)) is determined by setting all the lower (n - 16) bits of data x2 to O (step 2). In this case, the number of remaining bits (n
-16) is an even number or the number of tones, the indexed square root is (n-113)/2 bits for an even number and (n-113)/2 bits for the number of tones.
By shifting n-15)/2 bits, the square root
, then in the same way, the data (X2− X+
2) (The square root of the upper 16 bits in Fig. 2 (3) is indexed from the data table, and the lower (n - 32) bits of the data (X2 L 2) are all 0, resulting in data x22 (
The square root x2 of (4) in FIG. 2 is found (step 3).

Set the square root to be sought as Xe(n), and set the initial value of the repeated operation as Xe(0) = XI + X2 (Step 4)
.

(Step 5), this operation requires one multiplication (Xe(n
) 2) Perform subtraction and m (number of bits of Xe(n)) bit shift and addition, and finally.

If so, return to green from step 5, and if it converges, Xe(n) at that time becomes the square root X of data x2.

Generally, when calculating a 32-bit square root from 64-bit data, the convergence ILs is calculated using the data table described above.
When B, it converges in three operations.

Next, the basis of the convergence calculation formula will be explained.

If the error of the square root X5(0) with the original data X2 is ε, then X2= (Xe(0) + ε) 2= Xe(0)
2+ 2Xe(0) * e + 62...-(1)X
e(0) = XI + X2 Since the data structure of the data table shows that at least the upper 8 bits of the temporary square root x 1 are correct, it can be said that the upper 8 bits of Xe(0) are also correct. In other words, (<2''-8-(2) However, since x2 is also added to Xe(0), the error e is generally considered to be smaller. (2
) by approximating equation (1) (ignoring the term ε2), we get So, Xe(1) = X
Although e(0)+ε may be used, it is generally difficult to perform division with a large bit length, so equation (4) is further approximated.

21-I≦Xe(0)<2”...
(5), so here @, <εi<2ε, ...
(7) From equation (6), Xe(1) = Xe(0) + e,'If this is put into general form, the equation in step 5 is obtained.

Next, a description will be given of how small the error becomes after performing the iterative calculation once.

Consider the error after performing the iterative calculation once. The error at this time is due to the approximation error in equation (4) and the error in equation (6).

First, speaking only about equation (10), considering the influence of the approximation from equation (2), considering from equation (7), it is about twice
(The difference becomes larger. In other words, even if both are combined, the size of the error will be approximately 1/12B in one calculation. For example,
Considering that the square root of 32 bits is calculated from the 64-bit data X2 in the example of FIG. 1, the initial value and the error after repeated calculations are ε. 2ψ゛2 (initial value error) ε, < 232-19 (error after one calculation) ε2<2
7'=L (after 2nd calculation, !'1 difference) ε3<2μm2te (error after 3rd calculation) e, 1<20 (error after 4th calculation), as mentioned earlier. , generally the error ε is 232−
Since it is seven times smaller than 9 (the effect of X2), it is thought that it will converge in about three operations.

In this way, when finding a 32-bit square root from 64-bit data, you need to index the data table twice, multiply 32 bits by 32 bits, subtract 84 bits - 64 bits, shift 1m bits, and add 32 bits. A solution with ILSB accuracy can be obtained in about three times. This processing time is I
Approximately 150 pse at nte18086.5MH2
c, which is about 10 times faster than the conventional method.

In addition, the bit length for indexing the data table, the bit length of the data in the data table, and the number of times the data table is indexed should be selected based on the memory capacity of the data table, the length of data to be handled, the accuracy of answers, etc. It is.

Further, even if the square root of the 18-hit data is calculated using only the upper 12 bits (lower 4 bits are regarded as O), the accuracy is almost the same, and the tolerance of the data table becomes 1/10.

〔Effect of the invention〕

As explained above, the present invention provides a data table for data of a predetermined data length, calculates a temporary square root of the data from this, and calculates the square root by convergence calculation using a recurrence formula. , it has the effect of allowing high-speed qz-cut calculations without the need for special hardware.

[Brief explanation of drawings]

FIG. 1 is a flowchart showing an embodiment of the square root calculation method of the present invention, FIG. 2 is a diagram showing the data structure in the embodiment of FIG. 1, and FIG. 3 is a flowchart showing a conventional example of the square root calculation method. be. 1-6: Step

Claims (4)

[Claims] (1) A square root data table for data with a predetermined bit length, and a square root of the data with the predetermined bit length, which has a bit length equal to or greater than the predetermined bit length and is higher than the original data for which the square root is to be calculated. Indexing from the data table according to the even/odd number of the remaining bit length after removing the predetermined bit length from the upper part of the data, and further shifting the indexed square root upward by a bit length that is half of the remaining bit length. A square root calculation method comprising: means for calculating a temporary square root of original data; and means for calculating a square root of original data by convergence calculation using the temporary square root as an initial value. (2) Convergence calculation: Xe(K+1) =Xe(K)+(X^2-Xe(K)^2)/(2^m
) However, X^2: Original data Xe(K): Temporary square root in the K-th convergence calculation Xe(K): Square m of Xe(K): Bit length of Xe(K). Square root calculation method described in Section 1. (3) A square root data table for data with a predetermined bit length, and a square root of data with the predetermined bit length that is higher than the original data for which the square root is to be found, and which has a bit length that is twice or more than the predetermined bit length. is indexed from the data table according to the even/odd number of the first remaining bit length after removing the predetermined bit length from the upper part of the original data, and then the lower order data obtained by removing the upper predetermined bit length data from the original data. The square root of the data with the predetermined bit length in the upper part of the data is obtained by removing the bit length twice the predetermined bit length from the upper part of the original data.
index from the data table according to the even/odd number of remaining bit lengths, and shift both square roots indexed from the data table in this way upward by half the bit length of the first and second remaining bit lengths, respectively. A square root operation method comprising: a means for calculating a temporary square root of the original data by adding the values after the calculation, and a means for calculating the square root of the original data by convergence calculation using the temporary square root as an initial value. (4) Convergence calculation: Xe(K+1) =Xe(K)+(X^2-Xe(K)^2)/(2^m
) However, X^2: Original data Xe(K): Temporary square root in the K-th convergence calculation Xe(K): Square m of Xe(K): Bit length of Xe(K). Square root calculation method described in Section 3.