kader (version 0.0.8)

J1: J1

Description

Eq. (15.16) in Eichner (2017) as a result of Cardano's formula.

Usage

J1(u, cc = sqrt(5/3))

Arguments

u

Numeric vector.

cc

Numeric constant, defaults to \(\sqrt(5/3)\).

Value

Vector of same length and mode as u.

Details

Using, for brevity's sake, \(J_{1a}(u, c) := -q_c(u)\) and \(J_{1b}(u, c) := J_{1a}(u, c)^2 + p_c^3\), the definition of \(J_1\) reads:

\(J_1(u, c) := [J_{1a}(u, c) + \sqrt(J_{1b}(u, c))]^{1/3} + [J_{1a}(u, c) - \sqrt(J_{1b}(u, c))]^{1/3}\).

For implementation details of \(q_c(u)\) and \(p_c\) see qc and pc, respectively.

For further mathematical details see Eichner (2017) and/or Eichner & Stute (2013).

See Also

J_admissible.

Examples

# NOT RUN {
u <- seq(0, 1, by = 0.01)
c0 <- expression(sqrt(5/3))
c1 <- expression(sqrt(3) - 0.01)
cgrid <- c(1.35, seq(1.4, 1.7, by = 0.1))
cvals <- c(eval(c0), cgrid, eval(c1))

Y <- sapply(cvals, function(cc, u) J1(u, cc = cc), u = u)
cols <- rainbow(ncol(Y), end = 9/12)
matplot(u, Y, type = "l", lty = "solid", col = cols,
  ylab = expression(J[1](u, c)))
abline(h = 0)
legend("topleft", title = "c", legend = c(c0, cgrid, c1),
  lty = 1, col = cols, cex = 0.8)
# }
# NOT RUN {
# }