relative error

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1—10 of 18 matching pages

1: 33.25 Approximations

… ►Maximum relative errors range from 1. …

2: 3.1 Arithmetics and Error Measures

… ►Then rounding by chopping or rounding down of

x

gives

x_{-}

, with maximum relative error

\epsilon_{M}

. Symmetric rounding or rounding to nearest of

x

gives

x_{-}

x_{+}

, whichever is nearer to

x

, with maximum relative error equal to the machine precision

\frac{1}{2}\epsilon_{M}=2^{-p}

. … ►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. … ►If

x\neq 0

, the relative error is …The relative precision is …

Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ and $2\leq x<\infty$ , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ . For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$ .

4: DLMF Project News

error generating summary

5: 11.6 Asymptotic Expansions

… ►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).

6: 19.36 Methods of Computation

… ►If the iteration of (19.36.6) and (19.36.12) is stopped when

c_{s}<\sqrt{r}t_{s}

(

M

and

T

being approximated by

a_{s}

and

t_{s}

, and the infinite series being truncated), then the relative error in

R_{F}

and

R_{G}

is less than

r

if we neglect terms of order

r^{2}

. …

7: 7.24 Approximations

… ►

•

Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

… ►

•

Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

…

8: 3.3 Interpolation

… ►With an error term the Lagrange interpolation formula for

f

is given by …If

f

x

(

=z

), and the nodes

x_{k}

are real, and

f^{(n+1)}

is continuous on the smallest closed interval

I

containing

x,x_{0},x_{1},\dots,x_{n}

, then the error can be expressed … ►The divided differences of

f

relative to a sequence of distinct points

z_{0},z_{1},z_{2},\dots

are defined by … ►The interpolation error

R_{n}(z)

is as in §3.3(i). …

9: Bibliography W

… ►

E. J. Weniger (2007) Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

ⓘ

External Links:: ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

… ►

R. Wong and J.-M. Zhang (1994a) Asymptotic monotonicity of the relative extrema of Jacobi polynomials. Canad. J. Math. 46 (6), pp. 1318–1337.

ⓘ

External Links:: ISSN 0008-414X, MathReview (S. I. Drobnies), ZentralBlatt
Referenced by:: §18.14(iii).
Encodings:: BibTeX

►

R. Wong and J.-M. Zhang (1994b) On the relative extrema of the Jacobi polynomials

P_{n}^{(0,-1)}(x)

. SIAM J. Math. Anal. 25 (2), pp. 776–811.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.14(iii).
Encodings:: BibTeX

… ►

R. Wong and Y.-Q. Zhao (2003) Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.15(i), §18.15(i).
Encodings:: BibTeX

… ►

R. Wong (1976) Error bounds for asymptotic expansions of Hankel transforms. SIAM J. Math. Anal. 7 (6), pp. 799–808.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (John S. Lew), ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

…

10: 3.6 Linear Difference Equations

… ►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a). … ►Suppose again that

f_{0}\neq 0

w_{0}

is given, and we wish to calculate

w_{1},w_{2},\dots,w_{M}

to a prescribed relative accuracy

\epsilon

for a given value of

M

. … ►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …