Modification of Power Series Expansion

Modification of Power Series Expansion The available methods to compute the logarithm of a number using digital circuits can be divided in two main groups. On the one hand, we have the look-up table based algorithms and, on the other, iterative methods. The first approach is faster and straightforward, but only useful for low precision. For implementing it, requires large amount of memory for increasing the accuracy. This is due to the size of the look-up table. We only evaluated iterative algorithms that need small look-up tables. The second group is slower, but suitable for high precision. Taylors series expansion is among the most popular methods to manually compute logarithms, but it has a slow convergence and requires slow operations like the division. Hence, they are slow when no embedded multipliers are available. Many studies explore hybrid implementations that take advantages from both groups. Our project required an algorithm that could be implemented on FPGAs from any vendor. It should be platform independent. Our algorithm requires less memory and no multiplier at all to implement exponential and logarithm function. To the best of our knowledge there are only two previous works focused on the exponential  function [8] [9], and only one for the logarithm function [10] (from the same authors of [9]). The first one [8], employs an algorithm that does not exploit the FPGA characteristics, and  consequently presents poor performance. The other two implementations [9, 10] are part of  a common work and are designed suiting with FPGA flexibility (using internal tailored fixed  arithmetic and exploiting the parallelism features of the FPGA) achieving much better results. They are parameterizable implementations that, additionally to single f.p. format, also allow  smaller exponent and mantissa bit-widths and are both based on input range reduction and  table-driven methods to calculate the function in the reduced range. Our ex and ln x units,  based on these units, include the following innovative features: Single precision f.p. arithmetic extension. [9, 10] were designed considering only normalized numbers, not denormalized. Additional logic has been introduced to handle  denormalized numbers at the output of ex and the input of ln x.  ² Redesign of units to deal only with single precision. The feature of bit-width config-urability of the base designs has been removed. Thus, the resources needed have been  reduced because specific units, just for single precision, have been developed.  ² Simplification of constant multiplications. As suggested in [9], conventional multipliers  have been removed where the multiplications involved constant coe ±cients, improving  performance and reducing size.  ² Unsigned arithmetic. In [9, 10] internal fixed arithmetic with sign is used. However,  some operations (like the ones involving range reduction and calculation of the exponent  for the result in ex) are consecutive and related, and the sign of the result can be inferred  from the input sign. For such operations signed arithmetic has been replaced by unsigned  arithmetic with the corresponding logic reduction.  ² Improved pipelining. The speed is enhanced by systematically introducing pipeline  stages to the datapath of the exponential and logarithm units and their subunits   The paper [11] explains about the implementation of power and log function based on a simple modification of power series expansion of Taylor series. In power function implementation, the paper aims at reducing the exponent number to a smaller value. It requires a large amount of block ram and hardware multipliers as well. It becomes platform dependent and the clock frequency may vary from vendor to vendor. The degradation in throughput rate is due to the use of 18 X 18 embedded multipliers in it. The powering unit also requires more number of stages which may be reduced further. In the proposed method, we are going to reduce delay and improve the throughput rate by avoiding the embedded multipliers and block RAMs. In this paper, we are not completely avoid look up tables, but any value of logarithm or exponential can be calculated, by adjusting the look up table values to the desired number [8] C. C. Doss and R. L. Riley, FPGA-Based implementation of a robust IEEE-754 ex-ponential unit, in IEEE Field-Programmable Custom Computing Machines, 2004, pp.  229{238. [9] J. Detrey and F. de Dinechin, A parameterized  °oating-point exponential function for  FPGAs, in IEEE International Conference Field-Programmable Technology, 2005, pp.27{34. [10] ||, A parameterized  °oating-point logarithm operator for FPGAs, in Signals, Systems and Computers, 2005. Conference Record of the Thirty-Ninth Asilomar Conference,  2005, pp. 1186{1190. [11] Pedro Echeverra, Marisa Lopez-Vallejo,†An FPGA Implementation of the Powering Function with Single Precision Floating-Point Arithmetic† â€Å"An FPGA Implementation of the Powering Function with Single Precision Floating-Point Arithmetic† PROPOSED METHOD The proposed method avoids multiplication and division operations, and is thus suitable for implementation in software on processors that lack such instructions (or where the instructions are slow) or in hardware on a programmable logic device or dedicated chip. This method is suitable when shifters are available in abundant. It is an extension to the implementation of sine and cosine explained in CORDIC. The proposed algorithm evaluates the power functions for both positive and negative values. There are some constants by which it is easy to multiply. For example, multiplying by 2n, where n is a positive or a negative integer, can be achieved by simply shifting a number by n places. The shift will be to the left (division) if n is positive, to the right (multiplication) if n is negative. It is nearly as easy to multiply by numbers of the form  ±2n ±1. These simply involve an add (or) subtract a shift. Implementation of EXP:     Ã‚   For implementing y = exp(x). The algorithm is going to generate a sequence of values for x and y, and we are going to make sure that for each pair K Exp(k) 5.5452 256 2.7726 16 1.3863 4 0.6931 2 0.4055 3/2 0.2231 5/4 0.1178 9/8 0.0606 17/16 0.0308 33/32 0.0155 65/64 0.0078 129/128 y=exp(4) · y†²=exp(4) ·exp(-(xk)) =exp(4) ·exp(-x) ·exp(k) =y ·exp(k). In other words, if we subtractkfromx, we have to multiplyyby exp(k). All we have to do now is make sure that exp(k) is a nice number, so we can multiply by it easily, and the rest is straightforward. Note thatkitself does not have to be nice, as we are only subtracting that, not multiplying by it. Here are some nice values of exp(k) and the corresponding (not necessarily nice) values ofk. The flow of algorithm is as follows for positive powers of x: Here in each iteration, we subtract the input from the nearest value of exp(k) as listed in the table. If the difference is negative, we multiply the output by the corresponding exp(k). The process continues withmore entities in our table of k, finally we get the result. In the same way the flow chart is mentioned for nagative powers of x. K Exp(k) 5.5452 256 2.7726 16 1.3863 4 0.6931 2 0.2877 3/4 0.1335 7/8 0.0645 15/16 0.0317 31/32 0.0157 63/64 0.0078 127/128 0.0039 255/256 The flow of algorithm is as follows for negative powers of x: Here in each iteration, we subtract the input from the nearest value of exp(k) as listed in the table. If the difference is positive, we divide the output by the corresponding exp(k). The process continues withmore entities in our table of k, finally we get the result. Implementation of LOG : For implementing Y=log (x), the procedure is similar to the implementation of exponential function

Balance and Tragedy in Shakespeares Othello Essay -- Feminism Literar

Mohandas K. Gandhi once proclaimed, â€Å"There is no occasion for women to consider themselves subordinate or inferior to men† (Gandhi n.p.). Women all throughout the world have been forced to endure innumerable hardships and struggles. Merely accepting women as a rightful component of society and a necessary aspect of culture has taken countless numbers of years. And to this day, unfortunately, gender equality has yet to become a reality for many. Certain judgments and stereotypes have been placed onto women from the very beginning of time. The belief that the female gender should only be seen in society as homemakers is something that is widely accepted by people in a multitude of countries and places. Despite the setbacks, various women have felt the need to fight for their rights and prove that they are an extremely crucial part of all societies. In the nineteenth century, the Cult of Domesticity, also known as the Cult of True Womanhood, was founded. It created s pecific rules that women in the United States and Great Britain were expected to follow. How well one obeyed the rules of the Cult of Domesticity dictated her reputation among fellow citizens (â€Å"From Domestic†¦Ã¢â‚¬  n.p.). The strict guidelines often had negative effects on individuals and prevented them from acting in an honest manner. In Othello, Shakespeare created the character of Emilia to perfectly represent the struggles women faced with fitting into their roles in society. The few female characters in the play were all shown in very different lights in order to demonstrate the varying types of women that could be seen in a normal society during that time. In William Shakespeare’s work Othello, Emilia’s imbalance of logic and emotion, a result of the pressur... ...ose to a balance, the chain reaction of events that led to the ultimate tragedy could not have been halted. If Emilia had come to her final realization earlier, could the tragedy have been stopped? Possibly, but it is impossible to know for sure. Even though it occurred, Emilia did, however, in the final moments of her life, speak with purpose and honesty. Her courage to stand up for her beliefs represents one of the most important themes in Othello and has made a truly lasting impact. Works Cited â€Å"From Domestic Goddesses to Suffragists: The Story of Women Told on Bookbindings, 1820-1920.† Publishers’ Bindings Online. U of Alabama, n.d. Web. 2 Nov. 2010. . Gandhi, Mohandas K. â€Å"Woman’s Status and Role in Society.† Mahatma Gandhi. N.p., n.d. Web. 2 Nov. 2010. .

A Day to Remember – I Can Still Remember It Like It Was Yesterday

I can still remember it like it was yesterday. It all started at about 2 p. m. It was the time I went back from school on foot. I was so tried and all I wanted was to just relax and eat. When I arrived at the front gate, I was shocked. The gate and the door were opened. It was because at that time there was nobody in my house as all my family members were not at home I was frightened and did not know what I should do. Should I go inside or call the police? I was hesitant because my family went out for work and they would normally tell me if they took afternoon off. However, I decided to see what was happening. I picked up a hard stick so that if something happened, I could protect myself. Then, I walked slowly to my house. I took a deep breath yet, the feeling of dread weighted heavily on me. I clenched the wooden stick on my hands, mind alert and be ready. I walked slowly into my house with my wooden stick in hands. Suddenly, I had nagging feeling that I was being followed. My senses were very alert and I could hear footsteps behind me but when I turned to glance behind I could see no one. I did not take too much notice on it actually. I thought that it might be my illusion because I was too nervous in such an unexpected situation. I grasped my wooden stick again, be prepared. I pushed the slightly-opened door, looked around if there were something going wrong. My heart started to beat quickly. Darkness always terrified me as the curtains blocked out the sun. I looked for the switch and turned on the light. I glanced around again to do a second check. There was nothing wrong! Everything was arranged properly and it just seemed like nothing happened. â€Å"What's happening and why is the door open but there seems to be no one inside? † i said to myself and all this while my head was filled with images of unrealistic thing from novels and movies. I crept around on tiptoes in order to find out what on earth was going on. As I was getting nearer to the staircase, I heard someone talking vociferously. A wave of horror swept over me. Since everyone should not be at home, who was emitting the sound? Once again, I had vivid recollection of Count Dracula, Frankenstein's monster, mummies, phantoms and zombies but I compelled myself to kicked kick these horrifying images out my mind. When I eventually convinced myself to regain composure, I walked gently quietly up the stairs and snatched a quick peek peep into the dim master room through the crevice of the wall. I could merely see two vague outlines of human figure who stood as still as stone in the room, silhouetted against the light. To my astonishment, they were Mum and Dad. They should not be here right now. I knew them very well. Workaholics like them would only want to come home after they had settled all their jobs. I was absolutely perplexed by their peculiar behaviour that day. As I was dominated by the profound inquisitiveness, I reached the door knob and intended to open the heavy wooden door. Nevertheless, the door banged opened before I succeeded in quenching my thirst for the content of their earlier conversation. I was standing right in front of them but they seemed to be ignorant to of my existence. I wondered if the scenario in sci-fi movie befell me and my parents would not recognise who I was. I had been proven wrong when I heard their quarrel them quarreling. They were just too engrossed in their quarrel and did not notice me. Then, I hid myself at the corner of the stairs and listened to them. â€Å"Can't you find a job? Do you think I can keep on supporting all this home with this small meagre income? † Mum shouted. â€Å"What do you expect me to do since when I don't even received receive any reply after sending numerous copies of job application letters? Dad said desperately. From the quarrel, I found that Dad was dismissed last week due to economic downturn and he pretended to go to work every day just to ensure that we were not worried about him. Although Mum earned a lucrative salary every month, this amount of money would be insufficient because of various types of bills and fees which became the main cause of their quarrel. Brainstorming, rushing in my mind was the thought of whether I should give up the tuitions and my favourite piano lesson to cut down the family expenditures? I knew I had the responsibilities to do it. I came out and approached them slowly. â€Å"Dad†¦Ã¢â‚¬ ¦ mum†¦Ã¢â‚¬ ¦ â€Å"I said that with the tone of depressed with a cautious. â€Å"Oh, why were you standing over here there? What is going on, Jason? † Dad asked. â€Å"I†¦ I'm sorry I have overhead your argument, I have decided to give up all the tuition. I don't want to attend the piano lesson anymore. I am unwilling I could not bear to see all these and it is also my responsibility to help to reduce the family burden. â€Å"I said. â€Å"Why do you think like that? It is just a small matter. We have abilities to let you go to for tuition. Don't worry about it, Jason. † Dad said smilingly. I knew that my dad was only consoleding me, as he did not want to make me unhappy. I just kept quiet and did not say anything. â€Å"Go to bathe now† Dad said again. I followed his order. After I had taken the bath, I went back to my room immediately and sat down on the chair. At the same time, I thought that how could I do. â€Å"Should I give up the tuition or go out to find part-time job? † I was very confused and bothered on to make an appropriate decision but I had to do so. Well, I decided to give up the tuition because I was trusted my own ability that I could still cope with my study if I put more effort and concentration in my study. I did not want my parents to quarrel on this small issue as I still had a chance to prevent it if I could make a concession. It was the first time I saw my parent wrangled in front of me. I was shocked and still not able to believe that incident had happened as my parents were very kind and respect each other. Unbelievable about the incident happen yesterday, as my parent is very kind and harmonious. It was certainly a day to remember

The Ptolemaic, Seleucid, and Antigonid kingdoms

Alexander, son of king Philip II and Olympias was born in 356 B. C. E, and at age 19, became king of Macedon. With a great army at his disposal and his brilliant military mind, he started his conquest. From 337-323 B. C. E Alexander conquered Greece, Egypt, Mesopotamia, and a large part of Asia. Alexander ruled from Macedonia to the northern part of India. He defeated the Persians numerous times with his incredible military strategies, even when his armies were outnumbered by almost 200,000 Persian soldiers.The sheer size of his empire was unlike anything ever seen before but when he died in 323 B.  C. E, the three main areas he conquered (Greece/Macedonia, Egypt, and Mesopotamia/ Asia) were split up and became their own kingdoms. The area that was once Greece and Macedonia became the Antigonid Kingdom, the Egyptian are became the Ptolemaic Kingdom and the Asia/ Mesopotamia area became the Seleucid Kingdom. The Ptolemaic Kingdom came about when one of Alexander’s cavalry com panions, Ptolemy, took his loyal troops to the Egyptian area to take over the wealthy land. Once there, Ptolemy diverted Alexander’s body to Egypt, which was originally being sent to Macedonia for burial.Once in Alexandria, Ptolemy took Alexander’s body and built a massive and imposing tomb for his late king. After defending Egypt from some of Alexander’s other generals who wanted Egypt, the Ptolemies ruled Egypt as their god/kings. This only took place after they struck up a bargain with the Egyptian priests and in which the Egyptians could continue living under the same Egyptian god belief system while the Ptolemies ruled as kings. Along with that agreement, the Ptolemies promised to allow the priests to maintain the up keeping of their temples and deceased pharaohs.The Ptolemies departed from the beliefs and rules set forth by Philip and Alexander by allowing the Queens of the rulers more power then they ever had before. The perfect example of this was Cleopa tra VII who even challenged the growing power of Rome. The Ptolemies also didn’t really blend with the Egyptians, which was a major idea Alexander wanted to make happen (the blending of Page 2 cultures). This can be seen when the Ptolemies stayed in Alexandria conducting business and trading with Rome, where the Egyptians maintained their old religion.The Seleicid Kingdom is named after one of Ptolemy’s generals, Seleusid. After the death of Alexander, Seleusid entered Babylon in 311 B. C. E and captured the imperial treasure there. With the vast wealth at his disposal, Seleusid claimed the land as the old heartland of Persia. Unfortunately because the area was so large, Seleusid had to trade back northern India and eventually lost Asia Minor. Seleusid imported Greek and Macedonian bureaucrats and colonists. North of Babylonia a new city was built and it was named Seleucia and would eventually become its capital.Knowing that trade with the eastern countries were vital to their survival, Seleucid’s major cities were built around trade routes that connected the countries. Around 20,000 Macedonians came to Seleusid and they felt as if they were world conquerors and felt they needed to be compensated. The Macedonians were given a significant amount of farmland because when they arrived, Seleusid realized that these people were the backbone of their society because they had no religion already set up in the area. Unlike the past rulers, Philip and Alexander, Seleusid was very worried about expanding, rather he was worried about losing land.This was because when Seleusid came to Babylon to start his new Kingdom, he had only brought a few military forces with him. This was a problem because he controlled so much land that he didn’t have enough military personnel to control the kingdom. If I was in Seleusid’s position I would have asked for everyone in the old countries such as Greece and Macedonia to come over. Doing so would give m e a much larger army and then I would not have to worry about losing land, rather I could expand. The last area of land left was Alexander’s home country, Macedonia, and Greece which he conquered.This land would become Antigonids once Alexander died and would be situated in Greece. The kings of Antigonids were descendants of Antigonus the one eyed, who was a general in Alexander’s army when Alexander died. Unfortunately, Antigonus died without conquering the land, but his descendants eventually claimed Macedonia and Greece as Antigonids. At the time of Alexander’s death, many of Alexanders treasures that he took were taken back to Page 3 Macedonia. It was said that the last shipment taken back to Macedonia before Alexander’s death took 110 warships just to carry all the treasure back.So the country was very wealthy but the wealth hardly even changed society. In fact, hardly anything had changed in Macedonia since Philip II was king. The army in the kingdo m still consisted of only Macedonia nobles who fought for the king and unfortunately in the 280’s B. C. E the Gauls (group from the north) invaded Antigonids and caused them serious damage. Along with the damage from the north, in the south, the Greeks, who had never really accepted the Macedonians, started revolting against the Antigonids.The Antigonids branched away from their past rulers, Philip II and Alexander by never expanding and never blending the neighboring cultures together. The Greeks never accepted the Antigonids’ rule and although total warefare never broke out, it damaged the Kingdom as a whole. Not only did the Kingdom fail to expand, they couldn’t even keep their own citizens in tact. If I were any of the kings in the Antigonids kingdom, I would have tried to resolve the differences with Greece and the Gauls because if they could have become allies, there would have been room for expansion.