Numerical Analysis IM.R. O’DonohoeReferences:S.D. Conte & C. de Boor,Elementary Numerical Analysis: An Algorithmic Approach, Third edition,1981. McGraw-Hill.L.F. Shampine, R.C. Allen, Jr & S. Pruess,Fundamentals of Numerical Computing, 1997. Wiley.David Goldberg,What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACMComputing Surveys, Vol. 23, No. 1, March 1991§.The approach adopted in this course does not assume a very high level of mathematics; in particular the levelis not as high as that required to understand Conte & de Boor’s book.1. Fundamental concepts1.1 IntroductionThis course is concerned with numerical methods for the solution of mathematical problems on a computer,usually using floating-point arithmetic.Floating-point arithmetic, particularly the ‘IEEE Standard’, iscovered in some detail.The mathematical problems solved by numerical methods include differentiationand integration, solution of equations of all types, finding a minimum value of a function, fitting curves orsurfaces to data, etc. This course will look at a large range of such problems from different viewpoints, butrarely in great depth.‘Numerical analysis’ is a rigorous mathematical discipline in which such problems, and algorithms for theirsolution, are analysed in order to establish theconditionof a problem or thestabilityof an algorithm andto gain insight into the design of better and more widely applicable algorithms. This course contains someelementary numerical analysis, and technical terms likeconditionandstabilityare discussed, although themathematical content is kept to a minimum. The course also covers some aspects of the topic not normallyfound in numerical analysis texts, such as numerical software considerations.In summary, the purposes of this course are:(1) To explain floating-point arithmetic, and to describe current implementations of it.(2) To show that design of a numerical algorithm is not necessarily straightforward, even for some simpleproblems.(3) To illustrate, by examples, the basic principles of good numerical techniques.(4) To discuss numerical software from the points of view of a user and of a software designer.1.2 Floating-point arithmetic1.2.1 OverviewFloating-point is a method for representing real numbers on a computer.Floating-point arithmetic is a very important subject and a rudimentary understanding of it is a pre-requisitefor any modern numerical analysis course.It is also of importance in other areas of computer science:§A PostScript version is available as the file$CLTEACH/mro2/Goldberg.psonPWF Linux.1

almost every programming language has floating-point data types, and these give rise to special floating-pointexceptionssuch asoverflow. So floating-point arithmetic is of interest to compiler writers and designersof operating systems, as well as to any computer user who has need of a numerical algorithm. Sections 1.2.1,1.2.2, 1.7, 1.13 and 3.3.2 are closely based on Goldberg’s paper, which may be consulted for further detailand examples.

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- Fall '13
- BertrandI.Halperin
- Physics, Numerical Analysis, floating-point number, floating-point arithmetic