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/*
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) 2008-2008 - INRIA - Bruno JOFRET
* Copyright (C) Bruno Pincon
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
*
*/
#include "lnp1m1.h"
#include "abs.h"
/*
PURPOSE : Compute v = log ( (1 + s)/(1 - s) )
for small s, this is for |s| < SLIM = 0.20
ALGORITHM :
1/ if |s| is "very small" we use a truncated
taylor dvp (by keeping 3 terms) from :
2 4 6
t = 2 * s * ( 1 + 1/3 s + 1/5 s + [ 1/7 s + ....] )
2 4
t = 2 * s * ( 1 + 1/3 s + 1/5 s + er)
The limit E until we use this formula may be simply
gotten so that the negliged part er is such that :
2 4
(#) er <= epsm * ( 1 + 1/3 s + 1/5 s ) for all |s|<= E
As er = 1/7 s^6 + 1/9 s^8 + ...
er <= 1/7 * s^6 ( 1 + s^2 + s^4 + ...) = 1/7 s^6/(1-s^2)
the inequality (#) is forced if :
1/7 s^6 / (1-s^2) <= epsm * ( 1 + 1/3 s^2 + 1/5 s^4 )
s^6 <= 7 epsm * (1 - 2/3 s^2 - 3/15 s^4 - 1/5 s^6)
So that E is very near (7 epsm)^(1/6) (approximately 3.032d-3):
2/ For larger |s| we used a minimax polynome :
yi = s * (2 + d3 s^3 + d5 s^5 .... + d13 s^13 + d15 s^15)
This polynome was computed (by some remes algorithm) following
(*) the sin(x) example (p 39) of the book :
"ELEMENTARY FUNCTIONS"
"Algorithms and implementation"
J.M. Muller (Birkhauser)
(*) without the additionnal raffinement to get the first coefs
very near floating point numbers)
*/
double dlnp1m1s(double Var)
{
static double D3 = 0.66666666666672679472;
static double D5 = 0.39999999996176889299;
static double D7 = 0.28571429392829380980;
static double D9 = 0.22222138684562683797;
static double D11 = 0.18186349187499222459;
static double D13 = 0.15250315884469364710;
static double D15 = 0.15367270224757008114;
static double E = 3.032E-3;
static double C3 = 2.0/3.0;
static double C5 = 2.0/5.0;
double S2 = Var * Var;
if( dabss(Var) <= E)
return Var * (2 + S2 * (C3 + C5 * S2));
else
return Var * (2 + S2 * (D3 + S2 * (D5 + S2 * (D7 + S2 * (D9 + S2 * (D11 + S2 * (D13 + S2 * D15)))))));
}
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