By Riaz A Usmani

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For this purpose, we rely on the axiomatization of floating-point arithmetic described in Section 2 . 4. Chapter 2. 32) where lOl l is bounded by a small number E M that indicates the machine precision. 33) where 6 1 arises from multiplication, "'22 from addition, and 02 from division. The magnitudes of 6 1 , "'22 , and 02 are all bounded above by E M . 822 1 are bounded above by a constant multiple of E M . 8TJ I is bounded above by E M times a linear function of r. 34) where each element in the r-th row of oL is bounded in magnitude by an expression involving machine precision, r, and the corresponding element of the original L.

1 . 1 ). e. I (J I � EA. W hen f is non-zero, the error in fl(l) may sometimes be expressed in terms of relative accuracy. Relative errors are important because they are inherent in the nature of floating-point arithmetic and in the usual methods for computing standard functions. example, on most machines the computed value of For y'X contains an error of no more than one unit in the last place of the mantissa. W hen using relative errors, we write fl(l) as fl(l) = f (1 + 8 ) . 1 8 1 � ER • = 8 f 1 l I (J I .

25) is given by x = A- l b. 25) is perturbed to b + 6b, and that the exact solution of the perturbed system is x + 6x, i . e . , A(x + 6x) = b + 6b, where "6" denotes a small change in a vector or matrix. 3. Perturbation theory; condition number and since x = A-1 b, To measure ox , we invoke the properties of compatible vector and matrix norms: (2. 26) with equality possible for some vector o b . The perturbation in the exact solution is thus bounded above by I IA- 1 1 1 times the perturbation in the right-hand side.