Jfunctions {logcondens}R Documentation

Numerical Routine J and Some Derivatives

Description

J00 represents the function J(x, y, v), where for real numbers x, y and v in [0, 1],

J(x, y, v) = int_0^v exp((1-t)x + ty) d t = frac{exp(x + v(y - x)) - exp(x)}{y - x}.

The functions Jab give the respective derivatives J_{ab} for v = 1, i.e.

J_{ab}(x, y) = frac{partial^{a+b}}{partial x^a partial y^b} J(x, y).

Specifically,

J_{10}(x, y) = frac{exp(y) - exp(x) - (y - x) exp(x)}{(y - x)^2};

J_{11}(x, y) = frac{(y - x)(exp(x) + exp(y)) + 2 (exp(y) - exp(x))}{(y - x)^3};

J_{20}(x, y) = 2frac{exp(y) - exp(x) - (y - x)exp(x)-(y - x)^2 exp(x)}{(y - x)^3}.

Usage

J00(x, y, v)
J10(x, y)
J11(x, y)
J20(x, y)

Arguments

x Vector of length d with real entries.
y Vector of length d with real entries.
v Number in [0, 1]^d.

Value

Value of the respective function.

Note

Taylor approximations are used if y-x is small.

Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.staff.unibe.ch/duembgen


[Package logcondens version 1.3.3 Index]