By W. J. Thron

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**Extra resources for Analytic Theory of Continued Fractions II**

**Example text**

Xp−1 × β−k−1, where k ≥ 0. That is, y is at least one magnitude smaller than x. To perform the subtraction the digits of y are shifted to the right in the following way. β−kβ−k−1⋯β−p︸p+1 digitsyp−k⋯yp−1↑↑β−p−1⋯β−p−k The lower row gives the power of β associated with the position of the digit. Let ŷ be y truncated to p + 1 digits. +β−p−k). From the definition of guard digit, the computed value of x − y is x − ŷ rounded to be a floating-point number, that is, (x − ŷ) + α, where the rounding error α satisfies | α |≤β2β−p.

Next we multiply by 8. 92 × 101. 0 ulps. 8u. The error measured in ulps has grown 8 times larger. The relative error, however, is the same, because the scaling factor to obtain the relative error has also been multiplied by 8. 4 Forward and Backward Error Analysis Forward error analysis examines how perturbations of the input propagate. For example, consider the function f(x) = x2. Let x* = x(1 + δ) be the representation of x, then squaring both sides gives (x*)2=x2(1+δ)2=x2(1+2δ+δ2)≈x2(1+2δ), because δ2 is small.

005. The quantities u and ulp can be viewed as measuring units. The absolute error is measured in ulps and the relative error in u. 35. 24 × 101. 8u. Next we multiply by 8. 92 × 101. 0 ulps. 8u. The error measured in ulps has grown 8 times larger. The relative error, however, is the same, because the scaling factor to obtain the relative error has also been multiplied by 8. 4 Forward and Backward Error Analysis Forward error analysis examines how perturbations of the input propagate. For example, consider the function f(x) = x2.