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## Algorithm as in Davidson 1970
Davidson <- function(i, j, r_ij, w_ij, w_ji, t_ij, maxit = 500){
n <- max(c(i, j))
R <- length(i)
nt <- sum(t_ij)
theta <- (2*nt)/(sum(w_ij + w_ji + t_ij) - nt)
s <- numeric(n)
for (obj in 1:n){
s[obj] <- c(2*w_ij %*% (i == obj) + 2*w_ji %*% (j == obj) +
t_ij %*% (i == obj | j == obj))
}
gamma <- rep(1/n, n)
for (k in 1:maxit){
gamma2 <- numeric(n)
resi <- r_ij*(2 + theta*sqrt(gamma[j]/gamma[i]))/(
gamma[i] + gamma[j] + theta*sqrt(gamma[i]*gamma[j]))
resj <- r_ij*(2 + theta*sqrt(gamma[i]/gamma[j]))/(
gamma[i] + gamma[j] + theta*sqrt(gamma[i]*gamma[j]))
for (obj in 1:n){
gamma2[obj] <- s[obj]/(sum(resi[i == obj]) + sum(resj[j == obj]))
}
gamma2 <- gamma2/sum(gamma2)
theta2 <- nt/sum(r_ij*sqrt(gamma2[i]*gamma2[j])/(
gamma2[i] + gamma2[j] + theta*sqrt(gamma2[i]*gamma2[j])))
if (isTRUE(all.equal(log(gamma), log(gamma2)))) break
gamma <- gamma2
theta <- theta2
}
structure(c(theta2, gamma2), iter = k)
}
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