TABLE OF CONTENTS
- 1. /besselK
- 1.1. besselK/BesselI0
- 1.2. besselK/BesselI1
- 1.3. besselK/BesselK0
- 1.4. besselK/BesselK1
- 1.5. besselK/BesselK2
- 1.6. besselK/BesselKexp0
- 1.7. besselK/BesselKexp1
- 1.8. besselK/BesselKexp2
/besselK [ Modules ]
NAME
module besselK
PURPOSE
provide the 'modified Bessel functions of first kind', I_n, and the 'modified Bessel functions of second kind', K_n.
NOTES
- The expansions by Pavel Holoborodko are used, https://www.advanpix.com/2015/11/11/rational-approximations-for-the-modified-bessel-function-of-the-first-kind-i0-computations-double-precision/ (see for further references below at the definition of the fuction.) These expressions claim to be exact within one unit of machine precision, i.e. ~2e-16
- The series expansions given by Abramowitz&Stegun (chapter 9.8) only have single precission (~1e-7)
- K0 and K1 are imlemented explicitely, using own parametrization of I0 and I1. K2 is implemented by recursion.
- One should catch the case of x > 400, since then the exp results as NaN.
- One could think of implementing a second order Horner scheme in order to speed up the calculations of the polynomials. Here one splits the polynomial p(x) into the odd and even polynomials p_odd(y) and p_even(y) (with y=x^2) and returns p(x) = p_even + x*p_odd. see e.g.: J.-M. Muller, Elementary Functions. Algorithms and Implementation Birkhäuser / Springer, 2016
besselK/BesselI0 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselI0(x)
PURPOSE
return the value I(0,x)
NOTES
besselK/BesselI1 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselI1(x)
PURPOSE
return the value I(1,x)
NOTES
besselK/BesselK0 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselK0(x)
PURPOSE
return the value K(0,x)
NOTES
besselK/BesselK1 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselK1(x)
PURPOSE
return the value K(1,x)
NOTES
besselK/BesselK2 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselK2(x)
PURPOSE
return the value K(2,x)
besselK/BesselKexp0 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselKexp0(x)
PURPOSE
return the value exp(x)*K(0,x)
NOTES
besselK/BesselKexp1 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselKexp1(x)
PURPOSE
return the value exp(x)*K(1,x)
NOTES
besselK/BesselKexp2 [ Functions ]
[ Top ] [ besselK ] [ Functions ]
NAME
real function BesselKexp2(x)
PURPOSE
return the value exp(x)*K(2,x)