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Hyperbolic Class

Hyperbolic functions.
Inheritance Hierarchy
SystemObject
  Altaxo.CalcHyperbolic

Namespace: Altaxo.Calc
Assembly: AltaxoCore (in AltaxoCore.dll) Version: 4.8.3618.0 (4.8.3618.0)
Syntax
C#
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public static class Hyperbolic

The Hyperbolic type exposes the following members.

Methods
 NameDescription
Public methodStatic memberAcosh Returns the inverse hyperbolic cosine of a value.
Public methodStatic memberAsinh Returns the inverse hyperbolic sine of a value.
Public methodStatic memberAtanh Returns the inverse hyperbolic tangent of a value.
Public methodStatic memberCosh Hyperbolic cosine, i.e. (Exp(x)+Exp(-x))/2.
Public methodStatic memberCoth Hyperbolic cotangent, i.e. Cosh(x)/Sinh(x).
Public methodStatic memberCsch Hyperbolic cosecant, i.e. 1/Sinh(x) = 2/(Exp(x)-Exp(-x)).
Public methodStatic memberCschTimesX Hyperbolic cosecant, multiplied with the argument x, i.e. x*Csch(x) = x/Sinh(x).
Public methodStatic memberExpMinusOne Calculates Exp(x)-1 with better accuracy around x=0.
Public methodStatic memberInverseLangevin High-accuracy inverse Langevin function: starts from Kröger's rational approximation (max. relative error ~0.28%) and refines it with a fixed number of Newton-Raphson steps on the exact equation L(x) = y, where L is the true Langevin function (not an approximation). Each Newton step roughly doubles the number of correct digits, so starting from ~0.28% (about 2-3 correct digits), 3 steps reach full double precision (~15-16 digits) everywhere except in the immediate vicinity of y = +-1, where L'(x) -> 0 and the problem becomes ill-conditioned regardless of solver.
Public methodStatic memberInverseLangevinApproximationByCohen Padé approximation of the inverse Langevin function (Cohen, 1991). Accurate to within ~5% over the full domain (0,1).
Public methodStatic memberInverseLangevinApproximationByJedynak Inverse Langevin function approximation by Jedynak (2015).
Public methodStatic memberInverseLangevinApproximationByKroger Kröger's rational approximation of the inverse Langevin function (Kröger, M., J. Non-Newtonian Fluid Mech. 223 (2015) 77-87). Substantially more accurate than Cohen's Padé approximant (max. relative error ~0.28% vs. ~4.9%), at a similar computational cost, since it is still a single rational expression with only integer powers.
Public methodStatic memberInverseLangevinDerivative High-accuracy derivative of the inverse Langevin function, d(L^-1)/dy. Uses the inverse function theorem: if x = L^-1(y), then d(L^-1)/dy (y) = 1 / L'(x) evaluated with the EXACT Langevin derivative (not an approximant's derivative), at the highly accurate x produced by InverseLangevinAccurate. This is far more accurate than differentiating a closed-form approximation analytically (e.g. InverseLangevinDerivative, which inherits Cohen's ~5% approximation error) -- the error here is limited only by the Newton refinement in InverseLangevinAccurate, i.e. essentially full double precision away from y = +-1.
Public methodStatic memberInverseLangevinDerivativeApproxiationByCohen Closed-form derivative of Cohen's Padé approximant of the inverse Langevin function, dL^-1/dx = (3 + x^4) / (1 - x^2)^2.
Public methodStatic memberLangevin Langevin function, which is defined as Coth(x)-1/x.
Public methodStatic memberLangevinFirstDerivative Exact derivative of the Langevin function, L'(x) = 1/x^2 - csch^2(x), evaluated via its Taylor series near x = 0 to avoid cancellation error.
Public methodStatic memberLog1p Calculates the natural logarithm of 1+x with better accuracy for very small x.
Public methodStatic memberOneMinusExp Calculates 1-Exp(x) with better accuracy around x=0.
Public methodStatic memberSech Hyperbolic secant, i.e. 1/Cosh(x) = 2/(Exp(x)+Exp(-x)).
Public methodStatic memberSinh Hyperbolic sine, i.e. (Exp(x)-Exp(-x))/2.
Public methodStatic memberSinhAxBxTimesCschX Calculates [Exp(a x)-Exp(b x)]/[Exp(x)-Exp(-x)].
Public methodStatic memberTanh Hyperbolic tangent, i.e. Sinh(x)/Cosh(x).
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