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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 1. Computer Arithmetic The purpose of computing is insight, not numbers. — R.W. Hamming, [ 23 ] The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathematical quantities, including functions, integrals, solu- tions of algebraic equations, solutions of differential equations (both ordinary and partial), solutions of minimization problems, and so on. The objects of interest typically (but not exclusively) arise in applications, which seek not only their qualitative properties, but also quantitative numerical data. The goal of this course of lectures is to introduce some of the most important and basic numerical algorithms that are used in practical computations. Beyond merely learning the basic techniques, it is crucial that an informed practitioner develop a thorough understanding of how the algorithms are constructed, why they work, and what their limitations are. In any applied numerical computation, there are four key sources of error: ( i ) Inexactness of the mathematical model for the underlying physical phenomenon. ( ii ) Errors in measurements of parameters entering the model. ( iii ) Round-off errors in computer arithmetic. ( iv ) Approximations used to solve the full mathematical system. Of these, ( i ) is the domain of mathematical modeling, and will not concern us here. Neither will ( ii ), which is the domain of the experimentalists. ( iii ) arises due to the finite numerical precision imposed by the computer. ( iv ) is the true domain of numerical analysis , and refers to the fact that most systems of equations are too complicated to solve explicitly, or, even in cases when an analytic solution formula is known, directly obtaining the precise numerical values may be difficult. There are two principal ways of quantifying computational errors. Definition 1.1. Let x be a real number and let x ? be an approximation. The absolute error in the approximation x ? ≈ x is defined as | x ?- x | . The relative error is defined as the ratio of the absolute error to the size of x , i.e., | x ?- x | | x | , which assumes x 6 = 0; otherwise relative error is not defined. 3/15/06 1 c ° 2006 Peter J. Olver For example, 1000001 is an approximation to 1000000 with an absolute error of 1 and a relative error of 10- 6 , while 2 is an approximation to 1 with an absolute error of 1 and a relative error of 1. Typically, relative error is more intuitive and the preferred determiner of the size of the error. The present convention is that errors are always ≥ 0, and are = 0 if and only if the approximation is exact. We will say that an approximation x ? has k significant decimal digits if its relative error is < 5 × 10- k- 1 . This means that the first k digits of x ?...
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- Fall '09
- Numerical Analysis, Peter J. Olver