J1 {kader}R Documentation

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 (5/3)\sqrt(5/3).

Details

Using, for brevity's sake, J1a(u,c):=qc(u)J_{1a}(u, c) := -q_c(u) and J1b(u,c):=J1a(u,c)2+pc3J_{1b}(u, c) := J_{1a}(u, c)^2 + p_c^3, the definition of J1J_1 reads:

J1(u,c):=[J1a(u,c)+(J1b(u,c))]1/3+[J1a(u,c)(J1b(u,c))]1/3J_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 qc(u)q_c(u) and pcp_c see qc and pc, respectively.

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

Value

Vector of same length and mode as u.

Note

Eq. (15.16) in Eichner (2017), and hence J1(u,c)J_1(u, c), requires cc to be in [(5/3),3)[\sqrt(5/3), 3). If cc does not satisfy this requirement a warning (only) is issued.

See Also

J_admissible.

Examples


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)



[Package kader version 0.0.8 Index]