# maximum

(0.001 seconds)

## 1—10 of 76 matching pages

##### 1: 19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^{2}$ with $0\leq m<1$ can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for $0 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. …
##### 2: 3.1 Arithmetics and Error Measures
Let $E_{{\rm min}}\leq E\leq E_{{\rm max}}$ with $E_{{\rm min}}<0$ and $E_{{\rm max}}>0$. …The integers $p$, $E_{{\rm min}}$, and $E_{{\rm max}}$ are characteristics of the machine. … In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ($N=32$, $p=24$, $E_{{\rm min}}=-126$, $E_{{\rm max}}=127$), binary64 (previously double precision) ($N=64$, $p=53$, $E_{{\rm min}}=-1022$, $E_{{\rm max}}=1023$) and binary128 (previously quad precision) ($N=128$, $p=113$, $E_{{\rm min}}=-16382$, $E_{{\rm max}}=16383$) are as in Figure 3.1.1. … $N_{{\rm min}}\leq x\leq N_{{\rm max}}$, and …Then rounding by chopping or rounding down of $x$ gives $x_{-}$, with maximum relative error $\epsilon_{M}$. …
##### 3: 25.20 Approximations
• Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$, $5\leq s\leq 11$, $11\leq s\leq 25$, $25\leq s\leq 55$. Precision is varied, with a maximum of 20S.

• Morris (1979) gives rational approximations for $\operatorname{Li}_{2}\left(x\right)$25.12(i)) for $0.5\leq x\leq 1$. Precision is varied with a maximum of 24S.

• 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 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: 33.25 Approximations
Maximum relative errors range from 1. …
##### 5: 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 (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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

• ##### 6: 14.16 Zeros
The number of zeros of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ in the interval $(-1,1)$ is $\max(\lceil\nu-|\mu|\rceil,0)$ if any of the following sets of conditions hold: … The number of zeros of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ in the interval $(-1,1)$ is $\max(\lceil\nu-|\mu|\rceil,0)+1$ if either of the following sets of conditions holds: … $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ has $\max(\lceil\nu-|\mu|\rceil,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max(\lceil\nu-|\mu|\rceil,0)+k$ being even or odd according as $\cos\left(\nu\pi\right)$ and $\cos\left(\mu\pi\right)$ have opposite signs or the same sign. …
##### 7: 15.15 Sums
Here $z_{0}$ (${\neq 0}$) is an arbitrary complex constant and the expansion converges when $|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)$. …
##### 8: 29.20 Methods of Computation
Alternatively, the zeros can be found by locating the maximum of function $g$ in (29.12.11).
##### 9: 1.10 Functions of a Complex Variable
###### Analytic Functions
If $f(z)$ is continuous on $\overline{D}$ and analytic in $D$, then $\left|f(z)\right|$ attains its maximum on $\partial D$.
##### 10: Bibliography P
• K. A. Paciorek (1970) Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.
• W. F. Perger, A. Bhalla, and M. Nardin (1993) A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.
• R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
• G. P. M. Poppe and C. M. J. Wijers (1990) Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.
• M. J. D. Powell (1967) On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.