naginterfaces.library.specfun.log_​shifted

naginterfaces.library.specfun.log_shifted(x)

log_shifted returns a value of the shifted logarithmic function, $ln(1+x)$.

For full information please refer to the NAG Library document for s01ba

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s01baf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

lnx1float

The value of the shifted logarithmic function, $ln(1+x)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x>-1.0$.

Notes

log_shifted computes values of $ln(1+x)$, retaining full relative precision even when $\left|x\right|$ is small. The function is based on the Chebyshev expansion

$$ln\left(\frac{1+p^2+2p\overline{x}}{1+p^2-2p\overline{x}}\right)=4\sum _{k=0}^{\infty }\frac{p^{2k+1}}{2k+1}{T}_{2k+1}\left(\overline{x}\right)\text{.}$$

Setting $\overline{x}=\frac{x(1+p^2)}{2p(x+2)}$, and choosing $p=\frac{q-1}{q+1}$, $q=\sqrt[4]{42}$ the expansion is valid in the domain $x\in [\frac{1}{\sqrt{2}}-1,\sqrt{2}-1]$.

Outside this domain, $ln(1+x)$ is computed by the standard logarithmic function.

References

Lyusternik, L A, Chervonenkis, O A and Yanpolskii, A R, 1965, Handbook for Computing Elementary Functions, p. 57, Pergamon Press