Computer Science and Information Technology: Numerical Precision - Research Paper Example

? Numerical Precision Numerical precision. Floating-point arithmetic is reflected as an obscure issue by alot of persons because it is universal in a computer scheme. Every language almost has a floating-point data type, meaning computers that consist from PCs to supercomputers of floating-point accelerators, compilers would oftenly be needed to compile floating-point procedures and then almost every operating system should answer to floating-point exclusions like excess. This paper discusses the aspects of floating-point computer scientist should know to avoid some petty mistakes than can rather cost him a lot. Goldberg (1991) states that discrete mathematics is what distinguishes old computers from scientific computing. He adds that digital computers does not signify exactly all real numbers and therefore new things would be faced when designing computer algorithms for exact numbers. In order to be concerned with the right resulting solution, Goldberg (1991) asserts that we must consider the memory and the running time of the computer. This issue is further worsened because a lot of computer algorithms add approximations to house discrete computer. Goldberg (1991) adds that Java uses a small unit of IEEE 754 binary floating point standard to signify floating point numbers and explain the results of arithmetic operations. He says that a float is signified by 32 bits and that each mixture of possible bits signify an actual number, meaning that 232 possible real figures can be signified even though there are a lot of markedly actual numbers. IEEE standards do use the inner picture same as scientific code but in binary rather than base 10. This shelters a range from +/-1.40129846432481707e-45 to around +/-3.40282346638528860e+3 and with about 6 or 7 important decimal digits, plus or minus infinity as well as not a number. The number contains ‘s’ denoting plus or minus, 8 bits for advocate ’e’ and then 23 bits for a mantissa ’M.’ Goldberg (1991) adds that the decimal number is represented by the formula given as, (-1)s * m * 2(e-127) , Where; Sign bit ‘s’ (bit 31), Exponent field ‘e’ (bits 30 - 23) and, Mantissa ‘m’ (22 - 0). Floating-point arithmetic is the famous method of representing real figures in the contemporary computers. Faking an immeasurable, real figures with the machine figures is not a job that is forthright, negotiations that are a bit ingenious must be found amid correctness, swiftness, the ease of using it, lively range as well as application and memory. I therefore argue that floating-point arithmetic along well chosen precision, radix or any other limit is a very good negotiation for a lot of numerical implementations. Good memory performance of CPUs is mostly around the locality of the orientation and this is the same with the GPUs though with several significant alterations. The figure below shows a comparison of memory performance of the GPU and CPU. Buck (2005) argues that it is the role of cache that gives the difference between the performance in memory of a GPU and that of a CPU. He adds that the cache in the GPU hurries filtering of surface and therefore they need to be so big as the size of the sieve kernel for the surface sampler hat is a little it tiny and being seeded computation. On he other hand, Buck (2005) vows that the GPU cache formats are enhanced for two sizes and are not wanted. This unlike to the Pentium 4 caches that works at a great clock rate with megabytes of statistics. Moreover, he Pentium 4 cache has the ability to read as well as written memory operations unlike GPU cache for read-only surface statistics. Binary Code Decimal is an example of binary indoctrination of decimal figures where every decimal factor is denoted by a secure number of bits, it is always four to eight. The main asset of Binary Coded Decimal is a more perfect picture and rounding of decimal units and make it informal for human-readable picture. It with an advantage that one decimal numeral can be signified by the use of a four bit design. One may decide to uncompressed binary code decimal where every number is prearranged into a single byte with four bits signifying the number while the rest have no meaning e.g. 91 as 9=0000 1001 and 1=0000 0001. On the other hand, you can as well use packed binary coded decimal where numbers are prearranged into a solitary byte wit one number in the least important bite while the others in the most important bite e.g. 91 as 9=1001 and 1=0001. However, as BCD was not used mostly like in the past, dial-fixed point as well as floating-point formats are of much importance in profitable, monetary as well as manufacturing computing where much errors cannot be endured. The solitary precision floating point is kept in 32-bits and it amass the level of IEEE-754. Mantissa and exponent are the two major mechanisms of a floating point number. Here, mantissa keeps the original figures while the exponent keeps the power to which the mantissa should be raised. In floating-point numbers, mathematicians use it because it contains a sign, an exponent as well as a significant where significant has a precision figure. The exponent is said to be a number while significant may be a fixed-point number consisting integer a portion. In term memory, the BCD storage is not memory well-organized. According to Carpenter, Lindquist, Matthews and Silver (1983), i would recommend one to use the floating-point format or processing because it operates by programming a mantissa along with a signed base-2 advocates into integers as they are put into bits. Another reason why i will advice ne to use floating-point format is because the speed is high due to the application hardware and the exactness with scope scale meaning if the size is small the better the exactness as this will make the rounding error remains constant (Carpenter, Lindquist, Matthews and Silver 1983). Again, floats are very good for numbers put from an incessant field and due to that, they can be used in the image as well as acoustic processing. Floating point illustration is the same as sanctification notation though all things are approved to base two somewhat than ten as the most common one used is ANSI/IEEE standard 754-1985. The idea that floating-point numbers can never be signified by all real numbers and the floating-point operations can never be exactly signify correct mathematical operation can result to shocking circumstances. This is in connection exactness with which computers signify figures. Floating point numbers can introduce rounding errors in the banking industry, which at times leads to the financial benefit for the bank but not for the customer. This is because whenever you do arithmetic operation,you bring in an error. If these errors happen unsystematic, we will always get increasing errors and if we are not very careful in designing the numerical algorithms, the errors can spread a lot of losses. In banks, these errors occur simply because the calculations of dollars and cents here deals with base 10 arithmetic. To avoid all the losses to the customers, it is advisable to use the Java’s Big Decimal library because it denotes exact decimal numbers that protects the sales levy subjects of making use of binary floating point. Another reason for using it is that is is able to keep figures with a random amount of exactness and therefore makes a computer operator to control the grade through which the error could affect the computation. In summary, the research paper shows that a lot of errors can be met if a programmer is not very careful with the way he handles his programming. He must make sure that the correct figures are used to reduce errors during ablation. The scientist should also look at the best method of coding by comparing all of the methods in terms speed, accuracy and performance to see which one can work out better. Besides, he must analyze them for precision for proper programing for less error. References Buck, I. (2005). Taking the plunge into GPU computing. GPU Gems, 2, 509-519. Carpenter, T. P., Lindquist, M. M., Matthews, W., & Silver, E. A. (1983). Results of the third NAEP mathematics assessment: Secondary school. The Mathematics Teacher, 76(9), 652-659. Goldberg, D. (1991). What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys (CSUR), 23(1), 5-48. . Read More