TW548533B - Method for fast calculating the square root of integers - Google Patents

Abstract

The present invention provides a method for fast calculating the square root of integers, which is to add 0.5 to a series of integers. After conducting squaring, the series formed by the integers after squaring has the characteristic of two-stage equal-distance series. With this characteristic, in an electronic product, there is established an index table formed by a series of integer progression of the squared numbers within the maximum range for the required square root in the program for conducting squaring operation. Thus, when the electronic product is conducting the squaring operation on the inputted squared number, it can look up the index table to ensure the trying starting number and the series increments, and use the exhausted cyclic operation method to calculate the integer most close to the square root through a small amount of adding operation.

Description

548533 A7 B7 Printed by the Consumer Cooperatives of the Intellectual Property Bureau of the Ministry of Economic Affairs 5. Description of the invention (1) Background of the invention: In recent years, due to the rapid development of the high-tech digital electronics industry, various small-sized and easy-to-move handheld electronic products such as : Electronic dictionaries and Personal Digital Assistants (PDAs) ... emerged at the historic moment. Among them, personal digital assistants have the most market potential in many technological electronic products today. The design and development of these electronic products, Not only is it striding towards the trend of thin, light and short markets, its functions and uses are planned to better meet the needs of ordinary consumers, and gradually combine with general electronic consumer products to become a multi-purpose electronic communication product, such as Combined with general mobile phones, it becomes a personal digital assistant with mobile phone functions. If it is difficult to combine data, Liangzhu becomes a personal digital assistant with Internet messaging functions. The concept of such things is endless. Lift. At present, because the processing speed of the central processing unit of these electronic devices is limited, and their information storage space is also limited by the lack of memory capacity, they cannot provide consumers with faster and richer information. On electronic devices, the existing hardware equipment is used to compress a large amount of data stored and received more effectively to increase its storage space and make up for the lack of its memory capacity. When using such data, it can quickly Decompression to effectively improve its operating efficiency has become an important issue for the design and manufacturing of electronic products. At this stage, the processing speed of the central processing unit of the PDA product is relatively slow. The square calculation is implemented by software. If only integer-level data is needed in the product, then the floating-point function library is used for calculation. Requirements and consumes resources of the host. For example, in a palm-type electronic product generally equipped with an electronic map, the road name is marked by first finding the length of the road, and then uniformly distributing the name of the road on the line segments of the road. According to the traditional method, the length of the road is calculated using the square root function of the floating-point function library. However, when there are many roads, many accurate square root operations are required, and in the end, only the square root closest to the square root can be obtained. Integer, and test results show that its slower speed > does not achieve the effect of satisfying users. (Please read the notes on the back first

--Line · This paper size applies to China National Standard (CNS) A4 (210 X 297 mm) 548533 A7 B7 Printed by the Consumer Cooperatives of the Intellectual Property Bureau of the Ministry of Economic Affairs 5. Description of the invention (2) Outline of the invention: Many disadvantages of traditional PDA products occur when calculating the square root operation. The present invention is based on a sequence of 歹 [] integers plus 0.5 respectively, and then squaring, and the resulting sequence of squared integers 値 has the characteristics of a two-level equal difference sequence, and uses this characteristic to In an electronic product, an index comparison table formed by the square number of a sequence of integer sequences is established based on the maximum range of the square root that it actually needs in a program that can be squared and calculated on one of the programs installed on it. When the product performs square root operation on one of the input square numbers, it can look up the comparison table to determine its trial starting number and sequence increment, and use the exhaustive loop algorithm to calculate the closest value through a small number of addition operations. An integer whose square root. The main purpose of the present invention is to find a simple and fast method, so that an electronic product can quickly obtain an integer closest to its square root during the square root operation process, so as to prevent electronic products from performing unnecessary calculations, resulting in There is no need to waste resources, and the speed of operation is increased. Illustration: Figure 1 shows the overall process of the present invention; Figure 2 shows the process of making each data in the index comparison table; Figure 3 does not show the process of determining the initial number of trials; Figure 4 shows a schematic diagram of the process of finding the square root of an integer using the exhaustive loop algorithm. Detailed explanation: The main theoretical basis of the present invention is that a sequence of integers in a sequence is added by 0.5, and then squared. The resulting sequence of integers 平方 of the squared number has the characteristics of a two-level equal difference series. The present invention uses this characteristic to calculate a maximum number 値 which is not greater than a certain square number, and uses the calculated number 値 as an expected calculation result. For example: a sequence of [(32 + 0 · 5) 2] = 1〇56, [(33 + 0 · 5) 2Μ… 22, ... etc. -3- This paper standard applies to China National Standard (CNS) A4 specification (210 X (297 public love) (Please read the precautions on the back before this page)

· -Line · 548533 A7 B7 V. Description of the invention (n) In the sequence formed by the integer 平方 of the square number, M represents the largest integer not greater than X, so greater than 1056 and less than 1122: the square number should be squared Get the nearest integer 33. According to, a sequence of integers is divided into 0.5, and after squaring, the integer sequence of the squares can be calculated by the following formula: = [(^ + 〇 · 5) 2] = η2 + η, (1- 1) At this time, in the integer sequence of the square number, the tolerance between two adjacent square numbers in the sequence 値 is: R} = 丨-& = 2/7 + 2 · (1-2) When an input square number α is used to calculate the minimum integer / 1 corresponding to a square root, the minimum integer " must meet the following calculation conditions ... η2 + η > a 〇 (1-3) In the invention, in order to determine the unknown number / 1 in (1_3), an exhaustive loop algorithm can be used to perform a tentative operation to obtain the result. The exhaustive loop algorithm first finds a trial starting number. % · ， Take a square ifc greater than 1056 and less than 1122 as an example. If the number 値 is larger than the maximum square number set in the comparison table, the number of squares 値 can be reduced by 100 times first. And take its integer to get the virtual square number of 10 or 11, and then from the comparison table, according to the virtual square number, the largest one in the comparison table is squared Start searching until you find the first serial number 3 that is less than the virtual square number. Because the square number has been reduced by 100 times previously, at this time, you must expand the obtained serial number by 10 times to get its starting number. Starting square number +1) = 302 +30 = 930, and by setting an increment 2 * (仏 +1) = 2X30 + 2 = 62 »as its tolerance in an integer sequence between adjacent square numbers 平方'Using the exhaustive loop algorithm to calculate by force. In this way, only three loop operations are required to easily obtain the calculation result, and find the integer closest to its square root π = 33. The present invention is to add the integers according to the aforementioned sequence. After 0,5, and then squaring, the sequence formed by the integer 値 of the square number has the characteristics of two-level equal difference sequence. In an electronic product, the square root operation can be performed on one of the electronic products installed on it. In the program, the maximum range of the square root of its actual needs is used to establish a 4- This paper size is applicable to China National Standard Complete (CNS) A4 specification (210 X 297 mm) (Please read the precautions on the back before

-1 line. Printed by the Consumer Cooperative of the Intellectual Property Bureau of the Ministry of Economic Affairs 548533 A7 B7 Printed by the Consumer Cooperative of the Intellectual Property Bureau of the Ministry of Economic Affairs. When the electronic product performs square root operation on one of the input square numbers, it can find the starting number of trials and the increment of the sequence by looking up the lookup table to use the exhaustive loop algorithm. Please refer to Figure 1, After a small amount of addition, the square root closest to the integer is calculated. Therefore, in the present invention, when performing integer square rooting, its processing steps mainly include three major parts. The first part is to establish the cable bow comparison table, the second part is to determine the starting number of the trial, and the third part is based on This starting number uses the exhaustive loop algorithm to calculate the nearest square root. In order to make the processing procedure adopted and the effect achieved by the method of the present invention more specific and clear, here is a preferred embodiment, which is described in detail with the accompanying drawings as follows: An electronic product equipped with an electronic map If you want to label road names evenly on one of the roads displayed on the electronic map, you must first calculate the length of the road. Traditionally, the two points A (X!,%) And B (X2., The formula for calculating the distance between 2) is: V (X1 — χ 2) 2 + (less 1-less 2) 2 This formula is calculated by using the squared function of the floating-point function library in this electronic product. Find out the length of the path, but if it has a large number of digits, it needs to perform multiple precise square root operations and then take the integer closest to its square root, so its operation speed is slower. The present invention is to improve the disadvantages that occur in the traditional calculations described above. It is mainly based on the maximum range of the square root actually required when the electronic map is used on the electronic product (that is, the maximum distance displayed on the electronic map). The production process shown in Figure 2 is to create a cable bow of the square number of an integer sequence. 丨 The comparison procedure is as follows: (1) First, read the maximum range of the square root that actually needs 値 SQRTTABLEN. If the input range of the square root is When 値 is large, the number 将该 can be reduced by 10 times; (2) Initialize the number of groups of data stored in the lookup table i = Η4, i = 0; (3) Calculate the lookup table SqrtTab [i] one by one. Data to be stored SqrtTab [i] = i * i, and flatten it (please read the precautions on the back before this page)

'丨 Line-This paper size is in accordance with China National Standard (CNS) A4 (210 X 297 mm) 548533 A7 B7 V. Description of the invention) Square root as the serial number; (4) If i < SQRTTABLEN, repeat step (3) ; Otherwise, the end. According to the foregoing steps, if the distance between two points in different positions on the electronic map of this embodiment is required to be the largest 値, which is actually not greater than 30, in order to make one of the index comparison tables smaller, 0 may be taken. The integer between 31 and 31 is used as the square root to establish the index lookup table. If the square root of the integer that is actually needed is relatively large, the lookup table will become larger. At this time, it is only necessary to reduce the maximum square root of the integer by 10 times. In this embodiment, the maximum range of the square root of the input can be defined first: SQRTTABLEN is as follows: #defme SQRTTABLEN 31 According to the foregoing steps, the index comparison table is created as follows: {〇, 1, 4, 9, 16, 25, 36, 49 , 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961}; If the maximum range of the input is too large, if its square number is greater than (31xl0) 2 = 96100, it is only necessary to display the maximum range of the input square root, which is an error. In fact, these numbers are in the actual operation of this embodiment. Does not appear. Secondly, the present invention needs to determine the actual starting number of trials when using the electronic map, as shown in the flow chart in Figure 3. When the electronic product performs a square root operation on one of the input square numbers, if it is judged It is greater than the largest number in the established lookup table. It can be divided by 100 to reduce the square number by 100 times' and follow the steps below to label it with an nBitCount: nBitCount = 0; if squared number> 961 then squared number = squared number / 1〇〇; nBitCount = 1; This paper size is applicable to China National Standard (CNS) A4 (210 X 297 mm) (Please read the back first (Notes again ^ write this page)

-1 Line · Printed by the Consumer Cooperatives of the Intellectual Property Bureau of the Ministry of Economic Affairs 548533 A7 B7 Printed by the Consumer Cooperatives of the Intellectual Property Bureau of the Ministry of Economic Affairs 5. Description of the invention (β) end if;値, start looking for the first number not greater than the square number 値, and follow the steps below to find its serial number in the lookup table, nlndex = SQRTTABLEN; while nlndex > = 0 and square number > = SqrtTab [nIndex] do nlndex = nlndex -1; end while; 嗣 Follow the steps below and multiply the found number by 10 according to the nBitCoimt mark: if nBitCount > 0 then nlndex = nlndex * 10; end if; so In the electronic map of this embodiment, if the horizontal coordinate difference between the two ends of a road is 31 and the vertical coordinate difference is 12, when the square number of the distance between the two endpoints is 312 +122 = 1105, square the square. During the calculation, if it is judged that it is greater than the largest number in the established comparison table, the squared number may be divided by 100 first, and an integer may be taken to obtain a virtual squared number of 11. Mark a flag with nBitCount = 1; From the lookup table SqrtTab, find its serial number nlndex in the lookup table, and get nlndex = 3; 再, and then according to the nBitCount mark, get the initial number of trials nlndex = 30. Finally, the present invention calculates according to the calculated The trial starting number is calculated using the cyclic exhaustive algorithm, as shown in Figure 4, and calculated according to the following steps: nStartData = nlndex * (nlndex + 1); nlncrData = 2 * (nlndex + 1); while nStartData < Square number do nStartData = nStartData + nlncrData; nlncrData = nlncrData + 2; ____- 7- This paper size applies to China National Standard (CNS) A4 (210 x 297 mm) (Please read the precautions on the back before this page) )

Order: Line · 548533 A8 B8 C8 D8 force, patent application scope nlndex = nlndex + 1; end while; In this electronic map, the trial starting number nlndex = 30, and the starting number of its loop operation 値 nStartData = 30 * (30 + 1) = 930, the loop increment nlncrData = 2 * (30 + 1) = 62, because the starting number 値 nStartData is less than the square root number 1105, so through the process of the foregoing loop operation, when nlndex = 33 The number 値 nStartData = 1120 is greater than the squared number 1105. At this time, the loop operation is ended, and the calculated result nIndeX = 33 is the integer closest to its square root. By using the method of the present invention, when performing the operation of integer squared, it is experimentally verified that the operation efficiency of the squared algorithm of the present invention is about 8 times that of the traditional floating-point library function. Therefore, the method of the present invention can indeed make the In the process of performing the square root operation, the electronic product quickly obtains an integer closest to its square root. This prevents the electronic product from performing unnecessary calculations, causing unnecessary waste of resources, and effectively increasing the speed of the square root operation. The above description is only a preferred embodiment of the method of the present invention, but it is not intended to limit the scope of the method of the present invention. According to those skilled in the art, any equivalent changes or modifications made in accordance with the technical content disclosed in the present invention without departing from the spirit disclosed in the present invention should be included in the scope of the patent application described below. (Please read the precautions on the back first! This page. Printed by the Consumer Cooperatives of the Intellectual Property Bureau of the Ministry of Economic Affairs This paper is sized for China National Standard (CNS) A4 (210 X 297 mm)

Claims (1)

548533 6. Application for Patent Scope 1. The method of fast square root of an integer is mainly used in an electronic product to perform a square root operation program on one of the electronic products installed, according to the maximum range of the square root of its actual needs. Integer " add 0.5 respectively, and after squaring, the following sequence formed by the integer 値 of the squared number {^}: = [(^ + 0.5) 2] = n2 ^ n has a two-level equal difference sequence Characteristics of a series of integers of a series of squares, a cable bow 丨 comparison table, 俾 the electronic product for one of the square number input, square root calculation processing, you can look up the comparison table to determine the beginning of the test The number ~ and the tolerance between two adjacent square numbers in the sequence 序列: {"} U = 2m2, and finally, the exhaustive loop algorithm is used to calculate the integer closest to its square root. 2. The method of fast squared integer as described in item 1 of the scope of patent application, wherein the cable bow comparison table is based on the maximum range of the square root actually required on the electronic product, and one of the comparison tables is calculated according to the following steps. First, read the maximum range of the square root you entered. 'If the range of the square root is larger than the actual need, you can reduce the number to a certain integer multiple; initialize the comparison. The number of groups stored in the table; Calculate the square number of the integer sequence to be stored in the comparison table one by one, and use its square root as the serial number; printed by the employee consumer cooperative of the Intellectual Property Bureau of the Ministry of Economic Affairs until the data stored in the comparison table The number of groups is larger than the maximum range of the square root. 3. The method of fast square root of integer as described in item 1 of the scope of patent application, wherein when the maximum range of the square root entered is too large, which is greater than the actual need, the electronic product only needs to display the maximum range of the square root entered.値 is wrong. 4. The method of fast squared by integer as described in item 2 of the scope of patent application, where the actual requirement is that the paper size applies the Chinese National Standard (CNS) A4 specification (210 X 297 mm) 548533 A8 B8 C8 D8 VI. Application The maximum range of the square root of the patent range may be the maximum distance displayed on the electronic device. (Please read the precautions on the back first, then this page) 5. The method of fast square root of integer as described in item 2 of the scope of patent application, wherein the program that can perform square root calculation can be an electronic map program. 6. The method of fast square root of integer as described in item 5 of the scope of patent application, wherein the maximum range of the square root actually required can be the maximum distance displayed on the electronic map. 7. The method for fast squared integers as described in item 2, 3, 4, 5, or 6 of the scope of patent application, where the certain integer multiple may be 10. 8. The method for fast squared integers as described in item 1, 2, 3, 4, 5, or 6 of the scope of the patent application, wherein the initial number of trials is calculated according to the following steps: First, for the input square number If it is judged that it is greater than the largest number in the established comparison table, it can be divided by a certain integer multiple to reduce its square number by another integer multiple and mark it with a mark; then from the established comparison The last number in the table, looking forward to find the first number not greater than the square number, and find its serial number in the comparison table; 嗣 then multiply the found serial number by the other Integer multiples, that is, the starting number line of the test. Πι 0 Printed by the Consumer Cooperatives of the Intellectual Property Bureau of the Ministry of Economic Affairs. 9. The method of fast squared integers as described in item 8 of the scope of patent application. Based on the trial starting number A, set the starting number of its loop operation 运算, · + 1), and according to its loop increment 2 * (Λ / + 1), use the force mouth method to loop the calculation until the calculated The number 値 is greater than the square of the input At this time, the loop operation starts and ends, and the result calculated at this time is the integer closest to its square root. 10. The method of fast squared by an integer as described in item 9 of the scope of the patent application, wherein the other integer multiple may be 100. -10-This paper size applies to China National Standard (CNS) A4 (210 X 297 mm)