Notes on Numerical Stability
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چکیده
Correctness in the presence of error (e.g., when floating point computations are performed) takes on a different meaning. For many problems for which computers are used, there is one correct answer and we expect that answer to be computed by our program. The problem is that most real numbers cannot be stored exactly in a computer memory. They are stored as approximations, floating point numbers, instead. Hence storing them and/or computing with them inherently incurs error. The question thus becomes “When is a program correct in the presense of such errors?” Let us assume that we wish to evaluate the mapping f : D → R where D ⊂ R is the domain and R ⊂ R is the range (codomain). Now, we will let f̂ : D → R denote a computer implementation of this function. Generally, for x ∈ D it is the case that f(x) 6= f̂(x). Thus, the computed value is not “correct”. From the Notes on Conditioning, we know that it may not be the case that f̂(x) is “close to” f(x). After all, even if f̂ is an exact implementation of f , the mere act of storing x may introduce a small error δx and f(x+ δx) may be far from f(x) if f is ill-conditioned. The following defines a property that captures correctness in the presense of the kinds of errors that are introduced by computer arithmetic:
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تاریخ انتشار 2014