R/kfuns.R
In innager/edgee: Edgeworth Expansions and Higher-Order Inference
Defines functions
calculateK2smp
K61two
K51two
K42two
K41two
K32two
K31two
K23two
K22two
K21two
K13two
K12two
calculateK1smp
K61one
K51one
K42one
K41one
K32one
K31one
K23one
K22one
K21one
K13one
K12one
He11
He10
He9
He8
He7
He6
He5
He4
He3
He2
He1
He0
Documented in
K12one
K12two
K13one
K13two
K21one
K21two
K22one
K22two
K23one
K23two
K31one
K31two
K32one
K32two
K41one
K41two
K42one
K42two
K51one
K51two
K61one
K61two
# Probabilists' Hermite polynomials
#
# @param x numeric vector.
#
# @return A vector of the same length as \code{x}.
#
# @examples
# He9(1:5)
#
# @name Hermite
He0 <- function(x) 1
# @rdname Hermite
He1 <- function(x) x
He2 <- function(x) x^2 - 1
He3 <- function(x) x^3 - 3*x
He4 <- function(x) x^4 - 6*x^2 + 3
He5 <- function(x) x^5 - 10*x^3 + 15*x
He6 <- function(x) x^6 - 15*x^4 + 45*x^2 - 15
He7 <- function(x) x^7 - 21*x^5 + 105*x^3 - 105*x
He8 <- function(x) x^8 - 28*x^6 + 210*x^4 - 420*x^2 + 105
He9 <- function(x) x^9 - 36*x^7 + 378*x^5 - 1260*x^3 + 945*x
He10 <- function(x) x^10 - 45*x^8 + 630*x^6 - 3150*x^4 + 4725*x^2 - 945
He11 <- function(x) x^11 - 55*x^9 + 990*x^7 - 6930*x^5 + 17325*x^3 - 10395*x
#' \code{k()} functions for Edgeworth expansions - one-sample
#'
#' Calculate \code{k}'s (cumulant components) for a general version of Edgeworth
#' expansions (EE) for one-sample t-statistic.
#'
#' Variance adjustment \eqn{r^2} is equal to the output of \code{K21one()},
#' unless different variance estimates are used for \code{A}, numerator of
#' \code{k}, and \code{r}.
#'
#' @name kfuns1
#' @family \code{k()} functions
#'
#' @param A value of \code{A} (depends on the type of the test).
#' @param B value of \code{B} (depends on the type of the test).
#' @param mu2,mu3,mu4,mu5,mu6 central moments (2 - 6) or their estimates.
#'
#' @return A calculated value for the respective component.
#' @examples
#' # moderated t-statistic
#' if (requireNamespace("limma")) {
#' # simulate high-dimensional data
#' n <- 10
#' m <- 1e4 # number of tests
#' ns <- 0.05*m # number of significant features
#' dat <- matrix(rgamma(m*n, shape = 3) - 3, nrow = m)
#' shifts <- runif(ns, 1, 5)
#' dat[1:ns, ] <- dat[1:ns, ] - shifts
#' # estimate prior information
#' fit <- limma::lmFit(dat, rep(1, n))
#' fbay <- limma::eBayes(fit)
#' # look at one feature (row of data)
#' i <- 625
#' stats <- smpStats(dat[i, ], moder = TRUE, d0 = fbay$df.prior,
#' s20 = fbay$s2.prior, varpost = fbay$s2.post[i])
#' vars <- names(stats) # if want to remove carryover names
#' names(stats) <- NULL
#' for (j in 1:length(stats)) {
#' assign(vars[j], stats[j])
#' }
#' K32one(A, B, mu2, mu3, mu4, mu5, mu6)
#' }
#' @rdname kfuns1
#' @export
K12one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/2*B*mu3/A^(3/2)
}
#' @rdname kfuns1
#' @export
K13one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/16*(6*(8*mu2*mu3 - mu5)*A*B^2 - 15*(mu2^2*mu3 - mu3*mu4)*B^3 -
8*A^2*B*mu3)/A^(7/2)
}
#' @rdname kfuns1
#' @export
K21one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
mu2/A
}
#' @rdname kfuns1
#' @export
K22one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
1/4*(4*(4*mu2^2 - mu4)*A*B - (4*mu2^3 - 7*mu3^2 - 4*mu2*mu4)*B^2)/A^3
}
#' @rdname kfuns1
#' @export
K23one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/16*(16*(3*mu2^2 - mu4)*A^3*B - 8*(58*mu2^3 - 19*mu3^2 - 30*mu2*mu4 +
2*mu6)*A^2*B^2 + 2*(112*mu2^4 - 360*mu2*mu3^2 - 144*mu2^2*mu4 + 24*mu4^2 +
45*mu3*mu5 + 8*mu2*mu6)*A*B^3 - 3*(16*mu2^5 - 59*mu2^2*mu3^2 +
16*mu2*mu4^2 - (32*mu2^3 - 59*mu3^2)*mu4)*B^4)/A^5
}
#' @rdname kfuns1
#' @export
K31one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-(3*B*mu2 - A)*mu3/A^(5/2)
}
#' @rdname kfuns1
#' @export
K32one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
1/8*(12*(13*mu2*mu3 - mu5)*A^2*B - 3*(175*mu2^2*mu3 - 31*mu3*mu4 -
12*mu2*mu5)*A*B^2 + (123*mu2^3*mu3 - 83*mu3^3 -
123*mu2*mu3*mu4)*B^3)/A^(9/2)
}
#' @rdname kfuns1
#' @export
K41one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-((3*mu2^2 - mu4)*A^2 - 6*(3*mu2^3 - mu3^2 - mu2*mu4)*A*B + 3*(mu2^4 -
6*mu2*mu3^2 - mu2^2*mu4)*B^2)/A^4
}
#' @rdname kfuns1
#' @export
K42one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/8*(16*(42*mu2^3 - 13*mu3^2 - 19*mu2*mu4 + mu6)*A^3*B - 12*(284*mu2^4 -
266*mu2*mu3^2 - 172*mu2^2*mu4 + 12*mu4^2 + 21*mu3*mu5 +
8*mu2*mu6)*A^2*B^2 + 12*(112*mu2^5 - 702*mu2^2*mu3^2 + 32*mu2*mu4^2 +
63*mu2*mu3*mu5 + 4*mu2^2*mu6 - 2*(74*mu2^3 - 51*mu3^2)*mu4)*A*B^3 -
3*(64*mu2^6 - 624*mu2^3*mu3^2 + 233*mu3^4 + 64*mu2^2*mu4^2 - 16*(8*mu2^4 -
39*mu2*mu3^2)*mu4)*B^4)/A^6
}
#' @rdname kfuns1
#' @export
K51one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/2*(2*(10*mu2*mu3 - mu5)*A^3 - 10*(35*mu2^2*mu3 - 5*mu3*mu4 -
2*mu2*mu5)*A^2*B + 15*(56*mu2^3*mu3 - 7*mu3^3 - 16*mu2*mu3*mu4 -
2*mu2^2*mu5)*A*B^2 - 15*(10*mu2^4*mu3 - 21*mu2*mu3^3 -
10*mu2^2*mu3*mu4)*B^3)/A^(11/2)
}
#' @rdname kfuns1
#' @export
K61one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
1/2*(2*(30*mu2^3 - 10*mu3^2 - 15*mu2*mu4 + mu6)*A^4 - 30*(48*mu2^4 -
40*mu2*mu3^2 - 27*mu2^2*mu4 + 2*mu4^2 + 3*mu3*mu5 + mu2*mu6)*A^3*B +
15*(336*mu2^5 - 623*mu2^2*mu3^2 + 28*mu2*mu4^2 + 44*mu2*mu3*mu5 +
6*mu2^2*mu6 - (226*mu2^3 - 67*mu3^2)*mu4)*A^2*B^2 - 30*(56*mu2^6 -
597*mu2^3*mu3^2 + 40*mu3^4 + 18*mu2^2*mu4^2 + 33*mu2^2*mu3*mu5 +
mu2^3*mu6 - 3*(25*mu2^4 - 51*mu2*mu3^2)*mu4)*A*B^3 + 45*(4*mu2^7 -
73*mu2^4*mu3^2 + 80*mu2*mu3^4 + 4*mu2^3*mu4^2 - (8*mu2^5 -
73*mu2^2*mu3^2)*mu4)*B^4)/A^7
}
# internal
calculateK1smp <- function(stats) {
c(K12one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K13one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K21one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K22one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K23one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K31one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K32one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K41one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K42one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K51one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K61one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']))
}
#' \code{k()} functions for Edgeworth expansions - two-sample
#'
#' Calculate \code{k}'s (cumulant components) for a general version of Edgeworth
#' expansions (EE) for two-sample t-statistic.
#'
#' Note that the test statistic for this Edgeworth expansion is defined as
#' \eqn{\sqrt{n}(\bar{X} - \bar{Y})/s}{sqrt(n)(X-bar - Y-bar)/s} and therefore
#' \code{X} would normally represent a treatment group and \code{Y} - control
#' group. Variance adjustment \eqn{r^2} is equal to the output of
#' \code{K21two()}, unless different variance estimates are used for \code{A},
#' numerator of \code{k}, and \code{r}.
#'
#' @name kfuns2
#' @family \code{k()} functions
#'
#' @param A value of \code{A}.
#' @param B_x value of \eqn{B_x}{B[x]} (depends on the type of the test).
#' @param B_y value of \eqn{B_y}{B[y]} (depends on the type of the test).
#' @param b_x value of \eqn{b_x}{b[x]} - equal to \eqn{n/n_x}{n/n[x]}, where
#' \eqn{n = (n_x + n_y)/2}{n = (n[x] + n[y])/2}.
#' @param b_y value of \eqn{b_y}{b[y]} - equal to \eqn{n/n_y}{n/n[y]}, where
#' \eqn{n = (n_x + n_y)/2}{n = (n[x] + n[y])/2}.
#' @param mu_x2,mu_x3,mu_x4,mu_x5,mu_x6 central moments (2 - 6) for a treatment
#' group or their estimates.
#' @param mu_y2,mu_y3,mu_y4,mu_y5,mu_y6 central moments (2 - 6) for a control
#' group or their estimates.
#'
#' @return A calculated value for a respective component.
#' @examples
#' n1 <- 10
#' n2 <- 8
#' smp <- c(rgamma(n1, shape = 3), rnorm(n2))
#' a <- rep(1:0, c(n1, n2))
#' stats <- smpStats(smp, a, type = "Welch")
#' vars <- names(stats) # if want to remove carryover names
#' names(stats) <- NULL
#' for (j in 1:length(stats)) {
#' assign(vars[j], stats[j])
#' }
#' K32two(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
#' mu_y2, mu_y3, mu_y4, mu_y5, mu_y6)
#' @rdname kfuns2
#' @export
K12two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-1/2*(B_x*b_x*mu_x3 - B_y*b_y*mu_y3)/A^(3/2)
}
#' @rdname kfuns2
#' @export
K13two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/16*(15*B_x^3*b_x^2*mu_x2^2*mu_x3 + 15*B_x*B_y^2*b_x*b_y*mu_x3*mu_y2^2
- 15*B_x^3*b_x^2*mu_x3*mu_x4 + 8*(B_x*b_x^2*mu_x3 - B_y*b_y^2*mu_y3)*A^2
- 6*(8*B_x^2*b_x^2*mu_x2*mu_x3 + 2*B_x*B_y*b_x*b_y*mu_x3*mu_y2 -
B_x^2*b_x^2*mu_x5 + B_y^2*b_y^2*mu_y5 - 2*(B_x*B_y*b_x*b_y*mu_x2 +
4*B_y^2*b_y^2*mu_y2)*mu_y3)*A - 15*(B_x^2*B_y*b_x*b_y*mu_x2^2 +
B_y^3*b_y^2*mu_y2^2 - B_x^2*B_y*b_x*b_y*mu_x4)*mu_y3 -
15*(B_x*B_y^2*b_x*b_y*mu_x3 - B_y^3*b_y^2*mu_y3)*mu_y4)/A^(7/2)
}
#' @rdname kfuns2
#' @export
K21two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
(b_x*mu_x2 + b_y*mu_y2)/A
}
#' @rdname kfuns2
#' @export
K22two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-1/4*(4*B_x^2*b_x^2*mu_x2^3 + 4*B_y^2*b_x*b_y*mu_x2*mu_y2^2 +
4*B_y^2*b_y^2*mu_y2^3 - 7*B_x^2*b_x^2*mu_x3^2 -
4*B_x^2*b_x^2*mu_x2*mu_x4 + 14*B_x*B_y*b_x*b_y*mu_x3*mu_y3 -
7*B_y^2*b_y^2*mu_y3^2 - 4*(4*B_x*b_x^2*mu_x2^2 + 4*B_y*b_y^2*mu_y2^2 -
B_x*b_x^2*mu_x4 - B_y*b_y^2*mu_y4 + (B_x*b_x*b_y +
B_y*b_x*b_y)*mu_x2*mu_y2)*A + 4*(B_x^2*b_x*b_y*mu_x2^2 -
B_x^2*b_x*b_y*mu_x4)*mu_y2 - 4*(B_y^2*b_x*b_y*mu_x2 +
B_y^2*b_y^2*mu_y2)*mu_y4)/A^3
}
#' @rdname kfuns2
#' @export
K23two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/16*(48*B_x^4*b_x^3*mu_x2^5 + 48*B_y^4*b_x*b_y^2*mu_x2*mu_y2^4 +
48*B_y^4*b_y^3*mu_y2^5 - 177*B_x^4*b_x^3*mu_x2^2*mu_x3^2 +
48*B_x^4*b_x^3*mu_x2*mu_x4^2 - 16*(3*B_x*b_x^3*mu_x2^2 +
3*B_y*b_y^3*mu_y2^2 - B_x*b_x^3*mu_x4 - B_y*b_y^3*mu_y4)*A^3 +
96*(B_x^2*B_y^2*b_x*b_y^2*mu_x2^2 - B_x^2*B_y^2*b_x*b_y^2*mu_x4)*mu_y2^3
+ 8*(58*B_x^2*b_x^3*mu_x2^3 + 58*B_y^2*b_y^3*mu_y2^3 -
19*B_x^2*b_x^3*mu_x3^2 - 30*B_x^2*b_x^3*mu_x2*mu_x4 -
19*B_y^2*b_y^3*mu_y3^2 + 2*B_x^2*b_x^3*mu_x6 + 2*B_y^2*b_y^3*mu_y6 +
7*(b_x^2*b_y + b_x*b_y^2)*B_x*B_y*mu_x3*mu_y3 + 2*(8*B_x*B_y*b_x*b_y^2 +
5*B_y^2*b_x*b_y^2)*mu_x2*mu_y2^2 + 2*((5*B_x^2*b_x^2*b_y +
8*B_x*B_y*b_x^2*b_y)*mu_x2^2 - 2*(B_x^2*b_x^2*b_y +
B_x*B_y*b_x^2*b_y)*mu_x4)*mu_y2 - 2*(15*B_y^2*b_y^3*mu_y2 +
2*(B_x*B_y*b_x*b_y^2 + B_y^2*b_x*b_y^2)*mu_x2)*mu_y4)*A^2 +
3*(32*B_x^2*B_y^2*b_x^2*b_y*mu_x2^3 - 59*B_x^2*B_y^2*b_x^2*b_y*mu_x3^2 -
32*B_x^2*B_y^2*b_x^2*b_y*mu_x2*mu_x4)*mu_y2^2 -
177*(B_x^2*B_y^2*b_x*b_y^2*mu_x2^2 + B_y^4*b_y^3*mu_y2^2 -
B_x^2*B_y^2*b_x*b_y^2*mu_x4)*mu_y3^2 + 48*(B_y^4*b_x*b_y^2*mu_x2 +
B_y^4*b_y^3*mu_y2)*mu_y4^2 - 2*(112*B_x^3*b_x^3*mu_x2^4 +
112*B_y^3*b_y^3*mu_y2^4 - 360*B_x^3*b_x^3*mu_x2*mu_x3^2 -
144*B_x^3*b_x^3*mu_x2^2*mu_x4 + 24*B_x^3*b_x^3*mu_x4^2 +
45*B_x^3*b_x^3*mu_x3*mu_x5 + 8*B_x^3*b_x^3*mu_x2*mu_x6 +
24*B_y^3*b_y^3*mu_y4^2 + 8*(3*B_x*B_y^2*b_x*b_y^2 +
5*B_y^3*b_x*b_y^2)*mu_x2*mu_y2^3 + 24*(4*(B_x*B_y^2*b_x^2*b_y +
B_x^2*B_y*b_x*b_y^2)*mu_x2^2 - (B_x*B_y^2*b_x^2*b_y +
4*B_x^2*B_y*b_x*b_y^2)*mu_x4)*mu_y2^2 - 6*(60*B_y^3*b_y^3*mu_y2 +
(7*B_x*B_y^2*b_x*b_y^2 + 8*B_y^3*b_x*b_y^2)*mu_x2)*mu_y3^2 +
2*(4*B_x^3*b_x^2*b_y*mu_x6 + 4*(5*B_x^3*b_x^2*b_y +
3*B_x^2*B_y*b_x^2*b_y)*mu_x2^3 - 3*(8*B_x^3*b_x^2*b_y +
7*B_x^2*B_y*b_x^2*b_y)*mu_x3^2 - 12*(2*B_x^3*b_x^2*b_y +
B_x^2*B_y*b_x^2*b_y)*mu_x2*mu_x4)*mu_y2 +
3*(118*B_x^2*B_y*b_x^2*b_y*mu_x2*mu_x3 +
118*B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y2 -
15*B_x^2*B_y*b_x^2*b_y*mu_x5)*mu_y3 - 24*(6*B_y^3*b_y^3*mu_y2^2 +
(4*B_x*B_y^2*b_x^2*b_y + B_x^2*B_y*b_x*b_y^2)*mu_x2^2 +
(B_x*B_y^2*b_x*b_y^2 + 2*B_y^3*b_x*b_y^2)*mu_x2*mu_y2 -
(B_x*B_y^2*b_x^2*b_y + B_x^2*B_y*b_x*b_y^2)*mu_x4)*mu_y4 -
45*(B_x*B_y^2*b_x*b_y^2*mu_x3 - B_y^3*b_y^3*mu_y3)*mu_y5 +
8*(B_y^3*b_x*b_y^2*mu_x2 + B_y^3*b_y^3*mu_y2)*mu_y6)*A -
3*(32*B_x^4*b_x^3*mu_x2^3 - 59*B_x^4*b_x^3*mu_x3^2)*mu_x4 +
48*(B_x^4*b_x^2*b_y*mu_x2^4 - 2*B_x^4*b_x^2*b_y*mu_x2^2*mu_x4 +
B_x^4*b_x^2*b_y*mu_x4^2)*mu_y2 + 354*(B_x^3*B_y*b_x^2*b_y*mu_x2^2*mu_x3
+ B_x*B_y^3*b_x*b_y^2*mu_x3*mu_y2^2 -
B_x^3*B_y*b_x^2*b_y*mu_x3*mu_x4)*mu_y3 -
3*(32*B_x^2*B_y^2*b_x^2*b_y*mu_x2^3 + 32*B_y^4*b_x*b_y^2*mu_x2*mu_y2^2 +
32*B_y^4*b_y^3*mu_y2^3 - 59*B_x^2*B_y^2*b_x^2*b_y*mu_x3^2 -
32*B_x^2*B_y^2*b_x^2*b_y*mu_x2*mu_x4 +
118*B_x*B_y^3*b_x*b_y^2*mu_x3*mu_y3 - 59*B_y^4*b_y^3*mu_y3^2 +
32*(B_x^2*B_y^2*b_x*b_y^2*mu_x2^2 -
B_x^2*B_y^2*b_x*b_y^2*mu_x4)*mu_y2)*mu_y4)/A^5
}
#' @rdname kfuns2
#' @export
K31two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-(3*B_x*b_x^2*mu_x2*mu_x3 + 3*B_x*b_x*b_y*mu_x3*mu_y2 - (b_x^2*mu_x3 -
b_y^2*mu_y3)*A - 3*(B_y*b_x*b_y*mu_x2 + B_y*b_y^2*mu_y2)*mu_y3)/A^(5/2)
}
#' @rdname kfuns2
#' @export
K32two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/8*(123*B_x^3*b_x^3*mu_x2^3*mu_x3 +
123*B_x*B_y^2*b_x^2*b_y*mu_x2*mu_x3*mu_y2^2 +
123*B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y2^3 - 83*B_x^3*b_x^3*mu_x3^3 -
123*B_x^3*b_x^3*mu_x2*mu_x3*mu_x4 -
249*B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y3^2 + 83*B_y^3*b_y^3*mu_y3^3 +
12*(13*B_x*b_x^3*mu_x2*mu_x3 - B_x*b_x^3*mu_x5 + B_y*b_y^3*mu_y5 +
(2*B_x*b_x^2*b_y + B_y*b_x^2*b_y)*mu_x3*mu_y2 - (13*B_y*b_y^3*mu_y2 +
(B_x*b_x*b_y^2 + 2*B_y*b_x*b_y^2)*mu_x2)*mu_y3)*A^2 -
3*(175*B_x^2*b_x^3*mu_x2^2*mu_x3 - 31*B_x^2*b_x^3*mu_x3*mu_x4 -
12*B_x^2*b_x^3*mu_x2*mu_x5 + (5*B_y^2*b_x^2*b_y +
98*B_x*B_y*b_x*b_y^2)*mu_x3*mu_y2^2 - 4*(3*B_x^2*b_x^2*b_y*mu_x5 -
(23*B_x^2*b_x^2*b_y + 5*B_x*B_y*b_x^2*b_y)*mu_x2*mu_x3)*mu_y2 -
(175*B_y^2*b_y^3*mu_y2^2 + (98*B_x*B_y*b_x^2*b_y +
5*B_x^2*b_x*b_y^2)*mu_x2^2 + 4*(5*B_x*B_y*b_x*b_y^2 +
23*B_y^2*b_x*b_y^2)*mu_x2*mu_y2 - (26*B_x*B_y*b_x^2*b_y +
5*B_x^2*b_x*b_y^2)*mu_x4)*mu_y3 + (31*B_y^2*b_y^3*mu_y3 -
(5*B_y^2*b_x^2*b_y + 26*B_x*B_y*b_x*b_y^2)*mu_x3)*mu_y4 +
12*(B_y^2*b_x*b_y^2*mu_x2 + B_y^2*b_y^3*mu_y2)*mu_y5)*A +
123*(B_x^3*b_x^2*b_y*mu_x2^2*mu_x3 - B_x^3*b_x^2*b_y*mu_x3*mu_x4)*mu_y2
- 3*(41*B_x^2*B_y*b_x^2*b_y*mu_x2^3 + 41*B_y^3*b_x*b_y^2*mu_x2*mu_y2^2 +
41*B_y^3*b_y^3*mu_y2^3 - 83*B_x^2*B_y*b_x^2*b_y*mu_x3^2 -
41*B_x^2*B_y*b_x^2*b_y*mu_x2*mu_x4 + 41*(B_x^2*B_y*b_x*b_y^2*mu_x2^2 -
B_x^2*B_y*b_x*b_y^2*mu_x4)*mu_y2)*mu_y3 -
123*(B_x*B_y^2*b_x^2*b_y*mu_x2*mu_x3 + B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y2 -
(B_y^3*b_x*b_y^2*mu_x2 + B_y^3*b_y^3*mu_y2)*mu_y3)*mu_y4)/A^(9/2)
}
#' @rdname kfuns2
#' @export
K41two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-(3*B_x^2*b_x^3*mu_x2^4 + 6*B_y^2*b_x*b_y^2*mu_x2*mu_y2^3 +
3*B_y^2*b_y^3*mu_y2^4 - 18*B_x^2*b_x^3*mu_x2*mu_x3^2 -
3*B_x^2*b_x^3*mu_x2^2*mu_x4 + (3*b_x^3*mu_x2^2 + 3*b_y^3*mu_y2^2 -
b_x^3*mu_x4 - b_y^3*mu_y4)*A^2 - 3*(B_x^2*b_x*b_y^2*mu_x4 -
(B_y^2*b_x^2*b_y + B_x^2*b_x*b_y^2)*mu_x2^2)*mu_y2^2 -
18*(B_y^2*b_x*b_y^2*mu_x2 + B_y^2*b_y^3*mu_y2)*mu_y3^2 -
6*(3*B_x*b_x^3*mu_x2^3 + 3*B_y*b_x*b_y^2*mu_x2*mu_y2^2 +
3*B_y*b_y^3*mu_y2^3 - B_x*b_x^3*mu_x3^2 - B_x*b_x^3*mu_x2*mu_x4 -
B_y*b_y^3*mu_y3^2 + (B_y*b_x^2*b_y + B_x*b_x*b_y^2)*mu_x3*mu_y3 +
(3*B_x*b_x^2*b_y*mu_x2^2 - B_x*b_x^2*b_y*mu_x4)*mu_y2 -
(B_y*b_x*b_y^2*mu_x2 + B_y*b_y^3*mu_y2)*mu_y4)*A +
6*(B_x^2*b_x^2*b_y*mu_x2^3 - 3*B_x^2*b_x^2*b_y*mu_x3^2 -
B_x^2*b_x^2*b_y*mu_x2*mu_x4)*mu_y2 + 36*(B_x*B_y*b_x^2*b_y*mu_x2*mu_x3 +
B_x*B_y*b_x*b_y^2*mu_x3*mu_y2)*mu_y3 - 3*(B_y^2*b_x^2*b_y*mu_x2^2 +
2*B_y^2*b_x*b_y^2*mu_x2*mu_y2 + B_y^2*b_y^3*mu_y2^2)*mu_y4)/A^4
}
#' @rdname kfuns2
#' @export
K42two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/8*(192*B_x^4*b_x^4*mu_x2^6 + 384*B_y^4*b_x*b_y^3*mu_x2*mu_y2^5 +
192*B_y^4*b_y^4*mu_y2^6 - 1872*B_x^4*b_x^4*mu_x2^3*mu_x3^2 +
699*B_x^4*b_x^4*mu_x3^4 + 192*B_x^4*b_x^4*mu_x2^2*mu_x4^2 -
2796*B_x*B_y^3*b_x*b_y^3*mu_x3*mu_y3^3 + 699*B_y^4*b_y^4*mu_y3^4 -
192*(2*B_x^2*B_y^2*b_x*b_y^3*mu_x4 - (B_y^4*b_x^2*b_y^2 +
2*B_x^2*B_y^2*b_x*b_y^3)*mu_x2^2)*mu_y2^4 - 16*(42*B_x*b_x^4*mu_x2^3 +
42*B_y*b_y^4*mu_y2^3 - 13*B_x*b_x^4*mu_x3^2 - 19*B_x*b_x^4*mu_x2*mu_x4 -
13*B_y*b_y^4*mu_y3^2 + B_x*b_x^4*mu_x6 + B_y*b_y^4*mu_y6 +
3*(B_x*b_x*b_y^3 + 3*B_y*b_x*b_y^3)*mu_x2*mu_y2^2 + 3*(B_x*b_x^2*b_y^2 +
B_y*b_x^2*b_y^2)*mu_x3*mu_y3 + (3*(3*B_x*b_x^3*b_y +
B_y*b_x^3*b_y)*mu_x2^2 - (3*B_x*b_x^3*b_y + B_y*b_x^3*b_y)*mu_x4)*mu_y2
- (19*B_y*b_y^4*mu_y2 + (B_x*b_x*b_y^3 +
3*B_y*b_x*b_y^3)*mu_x2)*mu_y4)*A^3 +
48*(16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2^3 -
39*B_x^2*B_y^2*b_x^2*b_y^2*mu_x3^2 -
16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2*mu_x4)*mu_y2^3 +
12*(284*B_x^2*b_x^4*mu_x2^4 + 284*B_y^2*b_y^4*mu_y2^4 -
266*B_x^2*b_x^4*mu_x2*mu_x3^2 - 172*B_x^2*b_x^4*mu_x2^2*mu_x4 +
12*B_x^2*b_x^4*mu_x4^2 + 21*B_x^2*b_x^4*mu_x3*mu_x5 +
8*B_x^2*b_x^4*mu_x2*mu_x6 + 12*B_y^2*b_y^4*mu_y4^2 +
4*(9*B_x*B_y*b_x*b_y^3 + 49*B_y^2*b_x*b_y^3)*mu_x2*mu_y2^3 +
2*((126*B_x*B_y*b_x^2*b_y^2 + (4*b_x^2*b_y^2 + 3*b_x*b_y^3)*B_x^2 +
(3*b_x^3*b_y + 4*b_x^2*b_y^2)*B_y^2)*mu_x2^2 - (B_y^2*b_x^3*b_y +
36*B_x*B_y*b_x^2*b_y^2 + (2*b_x^2*b_y^2 +
3*b_x*b_y^3)*B_x^2)*mu_x4)*mu_y2^2 - 2*(133*B_y^2*b_y^4*mu_y2 +
6*(B_x*B_y*b_x*b_y^3 + 6*B_y^2*b_x*b_y^3)*mu_x2)*mu_y3^2 +
4*(2*B_x^2*b_x^3*b_y*mu_x6 + (49*B_x^2*b_x^3*b_y +
9*B_x*B_y*b_x^3*b_y)*mu_x2^3 - 3*(6*B_x^2*b_x^3*b_y +
B_x*B_y*b_x^3*b_y)*mu_x3^2 - 3*(9*B_x^2*b_x^3*b_y +
B_x*B_y*b_x^3*b_y)*mu_x2*mu_x4)*mu_y2 + (2*(27*B_x^2*b_x^2*b_y^2 +
4*(22*b_x^3*b_y + 3*b_x^2*b_y^2)*B_x*B_y)*mu_x2*mu_x3 +
2*(27*B_y^2*b_x^2*b_y^2 + 4*(3*b_x^2*b_y^2 +
22*b_x*b_y^3)*B_x*B_y)*mu_x3*mu_y2 - 7*(2*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x5)*mu_y3 - 2*(86*B_y^2*b_y^4*mu_y2^2 +
(36*B_x*B_y*b_x^2*b_y^2 + B_x^2*b_x*b_y^3 + (3*b_x^3*b_y +
2*b_x^2*b_y^2)*B_y^2)*mu_x2^2 + 6*(B_x*B_y*b_x*b_y^3 +
9*B_y^2*b_x*b_y^3)*mu_x2*mu_y2 - (B_y^2*b_x^3*b_y +
10*B_x*B_y*b_x^2*b_y^2 + B_x^2*b_x*b_y^3)*mu_x4)*mu_y4 +
7*(3*B_y^2*b_y^4*mu_y3 - (B_y^2*b_x^2*b_y^2 +
2*B_x*B_y*b_x*b_y^3)*mu_x3)*mu_y5 + 8*(B_y^2*b_x*b_y^3*mu_x2 +
B_y^2*b_y^4*mu_y2)*mu_y6)*A^2 -
48*(39*B_x^2*B_y^2*b_x^3*b_y*mu_x2*mu_x3^2 - 4*B_x^4*b_x^2*b_y^2*mu_x4^2
- 4*(2*B_x^2*B_y^2*b_x^3*b_y + B_x^4*b_x^2*b_y^2)*mu_x2^4 +
8*(B_x^2*B_y^2*b_x^3*b_y + B_x^4*b_x^2*b_y^2)*mu_x2^2*mu_x4)*mu_y2^2 -
18*(104*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2^3 +
104*B_y^4*b_x*b_y^3*mu_x2*mu_y2^2 + 104*B_y^4*b_y^4*mu_y2^3 -
233*B_x^2*B_y^2*b_x^2*b_y^2*mu_x3^2 -
104*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2*mu_x4 +
104*(B_x^2*B_y^2*b_x*b_y^3*mu_x2^2 -
B_x^2*B_y^2*b_x*b_y^3*mu_x4)*mu_y2)*mu_y3^2 +
192*(B_y^4*b_x^2*b_y^2*mu_x2^2 + 2*B_y^4*b_x*b_y^3*mu_x2*mu_y2 +
B_y^4*b_y^4*mu_y2^2)*mu_y4^2 - 12*(112*B_x^3*b_x^4*mu_x2^5 +
112*B_y^3*b_y^4*mu_y2^5 - 702*B_x^3*b_x^4*mu_x2^2*mu_x3^2 +
32*B_x^3*b_x^4*mu_x2*mu_x4^2 + 63*B_x^3*b_x^4*mu_x2*mu_x3*mu_x5 +
4*B_x^3*b_x^4*mu_x2^2*mu_x6 + 8*(B_x*B_y^2*b_x*b_y^3 +
16*B_y^3*b_x*b_y^3)*mu_x2*mu_y2^4 + 8*((14*B_x*B_y^2*b_x^2*b_y^2 +
2*B_y^3*b_x^2*b_y^2 + 13*B_x^2*B_y*b_x*b_y^3)*mu_x2^2 -
(4*B_x*B_y^2*b_x^2*b_y^2 + 13*B_x^2*B_y*b_x*b_y^3)*mu_x4)*mu_y2^3 +
(4*B_x^3*b_x^2*b_y^2*mu_x6 + 8*(13*B_x*B_y^2*b_x^3*b_y +
2*B_x^3*b_x^2*b_y^2 + 14*B_x^2*B_y*b_x^2*b_y^2)*mu_x2^3 -
3*(9*B_x*B_y^2*b_x^3*b_y + 8*B_x^3*b_x^2*b_y^2 +
91*B_x^2*B_y*b_x^2*b_y^2)*mu_x3^2 - 4*(8*B_x*B_y^2*b_x^3*b_y +
5*B_x^3*b_x^2*b_y^2 + 28*B_x^2*B_y*b_x^2*b_y^2)*mu_x2*mu_x4)*mu_y2^2 -
3*(234*B_y^3*b_y^4*mu_y2^2 + (91*B_x*B_y^2*b_x^2*b_y^2 +
8*B_y^3*b_x^2*b_y^2 + 9*B_x^2*B_y*b_x*b_y^3)*mu_x2^2 +
2*(8*B_x*B_y^2*b_x*b_y^3 + 79*B_y^3*b_x*b_y^3)*mu_x2*mu_y2 -
(25*B_x*B_y^2*b_x^2*b_y^2 + 9*B_x^2*B_y*b_x*b_y^3)*mu_x4)*mu_y3^2 +
32*(B_y^3*b_x*b_y^3*mu_x2 + B_y^3*b_y^4*mu_y2)*mu_y4^2 -
2*(74*B_x^3*b_x^4*mu_x2^3 - 51*B_x^3*b_x^4*mu_x3^2)*mu_x4 +
(32*B_x^3*b_x^3*b_y*mu_x4^2 + 63*B_x^3*b_x^3*b_y*mu_x3*mu_x5 +
8*B_x^3*b_x^3*b_y*mu_x2*mu_x6 + 8*(16*B_x^3*b_x^3*b_y +
B_x^2*B_y*b_x^3*b_y)*mu_x2^4 - 6*(79*B_x^3*b_x^3*b_y +
8*B_x^2*B_y*b_x^3*b_y)*mu_x2*mu_x3^2 - 8*(21*B_x^3*b_x^3*b_y +
B_x^2*B_y*b_x^3*b_y)*mu_x2^2*mu_x4)*mu_y2 -
3*(21*B_x^2*B_y*b_x^3*b_y*mu_x2*mu_x5 - (317*B_x^2*B_y*b_x^3*b_y +
9*B_x^3*b_x^2*b_y^2)*mu_x2^2*mu_x3 - (9*B_y^3*b_x^2*b_y^2 +
317*B_x*B_y^2*b_x*b_y^3)*mu_x3*mu_y2^2 + (59*B_x^2*B_y*b_x^3*b_y +
9*B_x^3*b_x^2*b_y^2)*mu_x3*mu_x4 + (21*B_x^2*B_y*b_x^2*b_y^2*mu_x5 -
158*(B_x^2*B_y*b_x^2*b_y^2 +
B_x*B_y^2*b_x^2*b_y^2)*mu_x2*mu_x3)*mu_y2)*mu_y3 -
(148*B_y^3*b_y^4*mu_y2^3 - 102*B_y^3*b_y^4*mu_y3^2 +
8*(13*B_x*B_y^2*b_x^3*b_y + 4*B_x^2*B_y*b_x^2*b_y^2)*mu_x2^3 +
8*(B_x*B_y^2*b_x*b_y^3 + 21*B_y^3*b_x*b_y^3)*mu_x2*mu_y2^2 -
3*(9*B_x*B_y^2*b_x^3*b_y + 25*B_x^2*B_y*b_x^2*b_y^2)*mu_x3^2 -
32*(B_x*B_y^2*b_x^3*b_y + B_x^2*B_y*b_x^2*b_y^2)*mu_x2*mu_x4 +
3*(9*B_y^3*b_x^2*b_y^2 + 59*B_x*B_y^2*b_x*b_y^3)*mu_x3*mu_y3 +
4*((28*B_x*B_y^2*b_x^2*b_y^2 + 5*B_y^3*b_x^2*b_y^2 +
8*B_x^2*B_y*b_x*b_y^3)*mu_x2^2 - 8*(B_x*B_y^2*b_x^2*b_y^2 +
B_x^2*B_y*b_x*b_y^3)*mu_x4)*mu_y2)*mu_y4 -
63*(B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3 + B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2
- (B_y^3*b_x*b_y^3*mu_x2 + B_y^3*b_y^4*mu_y2)*mu_y3)*mu_y5 +
4*(B_y^3*b_x^2*b_y^2*mu_x2^2 + 2*B_y^3*b_x*b_y^3*mu_x2*mu_y2 +
B_y^3*b_y^4*mu_y2^2)*mu_y6)*A - 48*(8*B_x^4*b_x^4*mu_x2^4 -
39*B_x^4*b_x^4*mu_x2*mu_x3^2)*mu_x4 + 48*(8*B_x^4*b_x^3*b_y*mu_x2^5 -
39*B_x^4*b_x^3*b_y*mu_x2^2*mu_x3^2 + 8*B_x^4*b_x^3*b_y*mu_x2*mu_x4^2 -
(16*B_x^4*b_x^3*b_y*mu_x2^3 - 39*B_x^4*b_x^3*b_y*mu_x3^2)*mu_x4)*mu_y2 +
12*(312*B_x^3*B_y*b_x^3*b_y*mu_x2^3*mu_x3 +
312*B_x*B_y^3*b_x^2*b_y^2*mu_x2*mu_x3*mu_y2^2 +
312*B_x*B_y^3*b_x*b_y^3*mu_x3*mu_y2^3 - 233*B_x^3*B_y*b_x^3*b_y*mu_x3^3
- 312*B_x^3*B_y*b_x^3*b_y*mu_x2*mu_x3*mu_x4 +
312*(B_x^3*B_y*b_x^2*b_y^2*mu_x2^2*mu_x3 -
B_x^3*B_y*b_x^2*b_y^2*mu_x3*mu_x4)*mu_y2)*mu_y3 -
48*(8*B_x^2*B_y^2*b_x^3*b_y*mu_x2^4 + 16*B_y^4*b_x*b_y^3*mu_x2*mu_y2^3 +
8*B_y^4*b_y^4*mu_y2^4 - 39*B_x^2*B_y^2*b_x^3*b_y*mu_x2*mu_x3^2 -
8*B_x^2*B_y^2*b_x^3*b_y*mu_x2^2*mu_x4 - 8*(B_x^2*B_y^2*b_x*b_y^3*mu_x4 -
(B_y^4*b_x^2*b_y^2 + B_x^2*B_y^2*b_x*b_y^3)*mu_x2^2)*mu_y2^2 -
39*(B_y^4*b_x*b_y^3*mu_x2 + B_y^4*b_y^4*mu_y2)*mu_y3^2 +
(16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2^3 - 39*B_x^2*B_y^2*b_x^2*b_y^2*mu_x3^2
- 16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2*mu_x4)*mu_y2 +
78*(B_x*B_y^3*b_x^2*b_y^2*mu_x2*mu_x3 +
B_x*B_y^3*b_x*b_y^3*mu_x3*mu_y2)*mu_y3)*mu_y4)/A^6
}
#' @rdname kfuns2
#' @export
K51two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/2*(150*B_x^3*b_x^4*mu_x2^4*mu_x3 +
300*B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3*mu_y2^3 +
150*B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2^4 - 315*B_x^3*b_x^4*mu_x2*mu_x3^3 -
150*B_x^3*b_x^4*mu_x2^2*mu_x3*mu_x4 - 2*(10*b_x^4*mu_x2*mu_x3 -
10*b_y^4*mu_y2*mu_y3 - b_x^4*mu_x5 + b_y^4*mu_y5)*A^3 +
315*(B_y^3*b_x*b_y^3*mu_x2 + B_y^3*b_y^4*mu_y2)*mu_y3^3 +
10*(35*B_x*b_x^4*mu_x2^2*mu_x3 - 5*B_x*b_x^4*mu_x3*mu_x4 -
2*B_x*b_x^4*mu_x2*mu_x5 + 3*(3*B_y*b_x^2*b_y^2 +
2*B_x*b_x*b_y^3)*mu_x3*mu_y2^2 + 2*(10*B_x*b_x^3*b_y*mu_x2*mu_x3 -
B_x*b_x^3*b_y*mu_x5)*mu_y2 - (20*B_y*b_x*b_y^3*mu_x2*mu_y2 +
35*B_y*b_y^4*mu_y2^2 + 3*(2*B_y*b_x^3*b_y + 3*B_x*b_x^2*b_y^2)*mu_x2^2 -
(2*B_y*b_x^3*b_y + 3*B_x*b_x^2*b_y^2)*mu_x4)*mu_y3 + (5*B_y*b_y^4*mu_y3
- (3*B_y*b_x^2*b_y^2 + 2*B_x*b_x*b_y^3)*mu_x3)*mu_y4 +
2*(B_y*b_x*b_y^3*mu_x2 + B_y*b_y^4*mu_y2)*mu_y5)*A^2 -
150*(B_x^3*b_x^2*b_y^2*mu_x3*mu_x4 - (B_x*B_y^2*b_x^3*b_y +
B_x^3*b_x^2*b_y^2)*mu_x2^2*mu_x3)*mu_y2^2 -
945*(B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3 +
B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2)*mu_y3^2 -
15*(56*B_x^2*b_x^4*mu_x2^3*mu_x3 - 7*B_x^2*b_x^4*mu_x3^3 -
16*B_x^2*b_x^4*mu_x2*mu_x3*mu_x4 - 2*B_x^2*b_x^4*mu_x2^2*mu_x5 +
7*B_y^2*b_y^4*mu_y3^3 + 2*(B_y^2*b_x^2*b_y^2 +
21*B_x*B_y*b_x*b_y^3)*mu_x3*mu_y2^3 - 7*(B_y^2*b_x^2*b_y^2 +
2*B_x*B_y*b_x*b_y^3)*mu_x3*mu_y3^2 - 2*(B_x^2*b_x^2*b_y^2*mu_x5 -
(B_y^2*b_x^3*b_y + 6*B_x^2*b_x^2*b_y^2 +
21*B_x*B_y*b_x^2*b_y^2)*mu_x2*mu_x3)*mu_y2^2 +
4*(17*B_x^2*b_x^3*b_y*mu_x2^2*mu_x3 - 4*B_x^2*b_x^3*b_y*mu_x3*mu_x4 -
B_x^2*b_x^3*b_y*mu_x2*mu_x5)*mu_y2 - (68*B_y^2*b_x*b_y^3*mu_x2*mu_y2^2 +
56*B_y^2*b_y^4*mu_y2^3 + 2*(21*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x2^3 - 7*(2*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x3^2 - 2*(7*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x2*mu_x4 + 2*((21*B_x*B_y*b_x^2*b_y^2 +
6*B_y^2*b_x^2*b_y^2 + B_x^2*b_x*b_y^3)*mu_x2^2 - (7*B_x*B_y*b_x^2*b_y^2
+ B_x^2*b_x*b_y^3)*mu_x4)*mu_y2)*mu_y3 - 2*((B_y^2*b_x^3*b_y +
7*B_x*B_y*b_x^2*b_y^2)*mu_x2*mu_x3 + (B_y^2*b_x^2*b_y^2 +
7*B_x*B_y*b_x*b_y^3)*mu_x3*mu_y2 - 8*(B_y^2*b_x*b_y^3*mu_x2 +
B_y^2*b_y^4*mu_y2)*mu_y3)*mu_y4 + 2*(B_y^2*b_x^2*b_y^2*mu_x2^2 +
2*B_y^2*b_x*b_y^3*mu_x2*mu_y2 + B_y^2*b_y^4*mu_y2^2)*mu_y5)*A +
15*(20*B_x^3*b_x^3*b_y*mu_x2^3*mu_x3 - 21*B_x^3*b_x^3*b_y*mu_x3^3 -
20*B_x^3*b_x^3*b_y*mu_x2*mu_x3*mu_x4)*mu_y2 -
15*(10*B_x^2*B_y*b_x^3*b_y*mu_x2^4 + 20*B_y^3*b_x*b_y^3*mu_x2*mu_y2^3 +
10*B_y^3*b_y^4*mu_y2^4 - 63*B_x^2*B_y*b_x^3*b_y*mu_x2*mu_x3^2 -
10*B_x^2*B_y*b_x^3*b_y*mu_x2^2*mu_x4 - 10*(B_x^2*B_y*b_x*b_y^3*mu_x4 -
(B_y^3*b_x^2*b_y^2 + B_x^2*B_y*b_x*b_y^3)*mu_x2^2)*mu_y2^2 +
(20*B_x^2*B_y*b_x^2*b_y^2*mu_x2^3 - 63*B_x^2*B_y*b_x^2*b_y^2*mu_x3^2 -
20*B_x^2*B_y*b_x^2*b_y^2*mu_x2*mu_x4)*mu_y2)*mu_y3 -
150*(B_x*B_y^2*b_x^3*b_y*mu_x2^2*mu_x3 +
2*B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3*mu_y2 +
B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2^2 - (B_y^3*b_x^2*b_y^2*mu_x2^2 +
2*B_y^3*b_x*b_y^3*mu_x2*mu_y2 +
B_y^3*b_y^4*mu_y2^2)*mu_y3)*mu_y4)/A^(11/2)
}
#' @rdname kfuns2
#' @export
K61two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/2*(180*B_x^4*b_x^5*mu_x2^7 + 540*B_y^4*b_x*b_y^4*mu_x2*mu_y2^6 +
180*B_y^4*b_y^5*mu_y2^7 - 3285*B_x^4*b_x^5*mu_x2^4*mu_x3^2 +
3600*B_x^4*b_x^5*mu_x2*mu_x3^4 + 180*B_x^4*b_x^5*mu_x2^3*mu_x4^2 -
180*(2*B_x^2*B_y^2*b_x*b_y^4*mu_x4 - (3*B_y^4*b_x^2*b_y^3 +
2*B_x^2*B_y^2*b_x*b_y^4)*mu_x2^2)*mu_y2^5 + 2*(30*b_x^5*mu_x2^3 +
30*b_y^5*mu_y2^3 - 10*b_x^5*mu_x3^2 - 15*b_x^5*mu_x2*mu_x4 -
10*b_y^5*mu_y3^2 - 15*b_y^5*mu_y2*mu_y4 + b_x^5*mu_x6 + b_y^5*mu_y6)*A^4
- 45*(73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x3^2 +
24*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2*mu_x4 - 4*(B_y^4*b_x^3*b_y^2 +
6*B_x^2*B_y^2*b_x^2*b_y^3)*mu_x2^3)*mu_y2^4 +
3600*(B_y^4*b_x*b_y^4*mu_x2 + B_y^4*b_y^5*mu_y2)*mu_y3^4 -
30*(48*B_x*b_x^5*mu_x2^4 + 30*B_y*b_x*b_y^4*mu_x2*mu_y2^3 +
48*B_y*b_y^5*mu_y2^4 - 40*B_x*b_x^5*mu_x2*mu_x3^2 -
27*B_x*b_x^5*mu_x2^2*mu_x4 + 2*B_x*b_x^5*mu_x4^2 +
3*B_x*b_x^5*mu_x3*mu_x5 + B_x*b_x^5*mu_x2*mu_x6 + 2*B_y*b_y^5*mu_y4^2 +
6*(3*(B_y*b_x^3*b_y^2 + B_x*b_x^2*b_y^3)*mu_x2^2 - (B_y*b_x^3*b_y^2 +
B_x*b_x^2*b_y^3)*mu_x4)*mu_y2^2 - 10*(B_y*b_x*b_y^4*mu_x2 +
4*B_y*b_y^5*mu_y2)*mu_y3^2 + (30*B_x*b_x^4*b_y*mu_x2^3 -
10*B_x*b_x^4*b_y*mu_x3^2 - 15*B_x*b_x^4*b_y*mu_x2*mu_x4 +
B_x*b_x^4*b_y*mu_x6)*mu_y2 + (10*(B_y*b_x^4*b_y +
2*B_x*b_x^3*b_y^2)*mu_x2*mu_x3 + 10*(2*B_y*b_x^2*b_y^3 +
B_x*b_x*b_y^4)*mu_x3*mu_y2 - (B_y*b_x^4*b_y +
2*B_x*b_x^3*b_y^2)*mu_x5)*mu_y3 - (15*B_y*b_x*b_y^4*mu_x2*mu_y2 +
27*B_y*b_y^5*mu_y2^2 + 6*(B_y*b_x^3*b_y^2 + B_x*b_x^2*b_y^3)*mu_x2^2 -
2*(B_y*b_x^3*b_y^2 + B_x*b_x^2*b_y^3)*mu_x4)*mu_y4 + (3*B_y*b_y^5*mu_y3
- (2*B_y*b_x^2*b_y^3 + B_x*b_x*b_y^4)*mu_x3)*mu_y5 +
(B_y*b_x*b_y^4*mu_x2 + B_y*b_y^5*mu_y2)*mu_y6)*A^3 -
90*(73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2*mu_x3^2 -
2*B_x^4*b_x^2*b_y^3*mu_x4^2 - 2*(6*B_x^2*B_y^2*b_x^3*b_y^2 +
B_x^4*b_x^2*b_y^3)*mu_x2^4 + 4*(3*B_x^2*B_y^2*b_x^3*b_y^2 +
B_x^4*b_x^2*b_y^3)*mu_x2^2*mu_x4)*mu_y2^3 -
14400*(B_x*B_y^3*b_x^2*b_y^3*mu_x2*mu_x3 +
B_x*B_y^3*b_x*b_y^4*mu_x3*mu_y2)*mu_y3^3 + 15*(336*B_x^2*b_x^5*mu_x2^5 +
444*B_y^2*b_x*b_y^4*mu_x2*mu_y2^4 + 336*B_y^2*b_y^5*mu_y2^5 -
623*B_x^2*b_x^5*mu_x2^2*mu_x3^2 + 28*B_x^2*b_x^5*mu_x2*mu_x4^2 +
44*B_x^2*b_x^5*mu_x2*mu_x3*mu_x5 + 6*B_x^2*b_x^5*mu_x2^2*mu_x6 +
4*(3*(36*B_x*B_y*b_x^2*b_y^3 + B_x^2*b_x*b_y^4 + (b_x^3*b_y^2 +
9*b_x^2*b_y^3)*B_y^2)*mu_x2^2 - (B_y^2*b_x^3*b_y^2 +
36*B_x*B_y*b_x^2*b_y^3 + 3*B_x^2*b_x*b_y^4)*mu_x4)*mu_y2^3 +
(6*B_x^2*b_x^3*b_y^2*mu_x6 + 12*(B_y^2*b_x^4*b_y +
36*B_x*B_y*b_x^3*b_y^2 + (9*b_x^3*b_y^2 + b_x^2*b_y^3)*B_x^2)*mu_x2^3 -
3*(B_y^2*b_x^4*b_y + 48*B_x*B_y*b_x^3*b_y^2 + 4*(3*b_x^3*b_y^2 +
4*b_x^2*b_y^3)*B_x^2)*mu_x3^2 - 2*(2*B_y^2*b_x^4*b_y +
72*B_x*B_y*b_x^3*b_y^2 + 3*(11*b_x^3*b_y^2 +
2*b_x^2*b_y^3)*B_x^2)*mu_x2*mu_x4)*mu_y2^2 -
(464*B_y^2*b_x*b_y^4*mu_x2*mu_y2 + 623*B_y^2*b_y^5*mu_y2^2 +
3*(48*B_x*B_y*b_x^2*b_y^3 + B_x^2*b_x*b_y^4 + 4*(4*b_x^3*b_y^2 +
3*b_x^2*b_y^3)*B_y^2)*mu_x2^2 - (16*B_y^2*b_x^3*b_y^2 +
48*B_x*B_y*b_x^2*b_y^3 + 3*B_x^2*b_x*b_y^4)*mu_x4)*mu_y3^2 +
28*(B_y^2*b_x*b_y^4*mu_x2 + B_y^2*b_y^5*mu_y2)*mu_y4^2 -
(226*B_x^2*b_x^5*mu_x2^3 - 67*B_x^2*b_x^5*mu_x3^2)*mu_x4 +
4*(111*B_x^2*b_x^4*b_y*mu_x2^4 - 116*B_x^2*b_x^4*b_y*mu_x2*mu_x3^2 -
73*B_x^2*b_x^4*b_y*mu_x2^2*mu_x4 + 7*B_x^2*b_x^4*b_y*mu_x4^2 +
11*B_x^2*b_x^4*b_y*mu_x3*mu_x5 + 3*B_x^2*b_x^4*b_y*mu_x2*mu_x6)*mu_y2 +
2*((280*B_x*B_y*b_x^4*b_y + 111*B_x^2*b_x^3*b_y^2)*mu_x2^2*mu_x3 +
(111*B_y^2*b_x^2*b_y^3 + 280*B_x*B_y*b_x*b_y^4)*mu_x3*mu_y2^2 -
(40*B_x*B_y*b_x^4*b_y + 27*B_x^2*b_x^3*b_y^2)*mu_x3*mu_x4 -
2*(8*B_x*B_y*b_x^4*b_y + 3*B_x^2*b_x^3*b_y^2)*mu_x2*mu_x5 +
2*(2*(9*B_y^2*b_x^3*b_y^2 + 9*B_x^2*b_x^2*b_y^3 + 40*(b_x^3*b_y^2 +
b_x^2*b_y^3)*B_x*B_y)*mu_x2*mu_x3 - (8*B_x*B_y*b_x^3*b_y^2 +
3*B_x^2*b_x^2*b_y^3)*mu_x5)*mu_y2)*mu_y3 -
(292*B_y^2*b_x*b_y^4*mu_x2*mu_y2^2 + 226*B_y^2*b_y^5*mu_y2^3 -
67*B_y^2*b_y^5*mu_y3^2 + 4*(3*B_y^2*b_x^4*b_y + 36*B_x*B_y*b_x^3*b_y^2 +
B_x^2*b_x^2*b_y^3)*mu_x2^3 - (3*B_y^2*b_x^4*b_y + 48*B_x*B_y*b_x^3*b_y^2
+ 16*B_x^2*b_x^2*b_y^3)*mu_x3^2 - 4*(B_y^2*b_x^4*b_y +
12*B_x*B_y*b_x^3*b_y^2 + B_x^2*b_x^2*b_y^3)*mu_x2*mu_x4 +
2*(27*B_y^2*b_x^2*b_y^3 + 40*B_x*B_y*b_x*b_y^4)*mu_x3*mu_y3 +
2*((72*B_x*B_y*b_x^2*b_y^3 + 2*B_x^2*b_x*b_y^4 + 3*(2*b_x^3*b_y^2 +
11*b_x^2*b_y^3)*B_y^2)*mu_x2^2 - 2*(B_y^2*b_x^3*b_y^2 +
12*B_x*B_y*b_x^2*b_y^3 + B_x^2*b_x*b_y^4)*mu_x4)*mu_y2)*mu_y4 -
4*((3*B_y^2*b_x^3*b_y^2 + 8*B_x*B_y*b_x^2*b_y^3)*mu_x2*mu_x3 +
(3*B_y^2*b_x^2*b_y^3 + 8*B_x*B_y*b_x*b_y^4)*mu_x3*mu_y2 -
11*(B_y^2*b_x*b_y^4*mu_x2 + B_y^2*b_y^5*mu_y2)*mu_y3)*mu_y5 +
6*(B_y^2*b_x^2*b_y^3*mu_x2^2 + 2*B_y^2*b_x*b_y^4*mu_x2*mu_y2 +
B_y^2*b_y^5*mu_y2^2)*mu_y6)*A^2 + 45*(12*B_x^4*b_x^3*b_y^2*mu_x2*mu_x4^2
+ 4*(2*B_x^2*B_y^2*b_x^4*b_y + 3*B_x^4*b_x^3*b_y^2)*mu_x2^5 -
73*(B_x^2*B_y^2*b_x^4*b_y + B_x^4*b_x^3*b_y^2)*mu_x2^2*mu_x3^2 +
(73*B_x^4*b_x^3*b_y^2*mu_x3^2 - 8*(B_x^2*B_y^2*b_x^4*b_y +
3*B_x^4*b_x^3*b_y^2)*mu_x2^3)*mu_x4)*mu_y2^2 -
45*(73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^4 +
146*B_y^4*b_x*b_y^4*mu_x2*mu_y2^3 + 73*B_y^4*b_y^5*mu_y2^4 -
480*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2*mu_x3^2 -
73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^2*mu_x4 -
73*(B_x^2*B_y^2*b_x*b_y^4*mu_x4 - (B_y^4*b_x^2*b_y^3 +
B_x^2*B_y^2*b_x*b_y^4)*mu_x2^2)*mu_y2^2 +
2*(73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2^3 -
240*B_x^2*B_y^2*b_x^2*b_y^3*mu_x3^2 -
73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2*mu_x4)*mu_y2)*mu_y3^2 +
180*(B_y^4*b_x^3*b_y^2*mu_x2^3 + 3*B_y^4*b_x^2*b_y^3*mu_x2^2*mu_y2 +
3*B_y^4*b_x*b_y^4*mu_x2*mu_y2^2 + B_y^4*b_y^5*mu_y2^3)*mu_y4^2 -
30*(56*B_x^3*b_x^5*mu_x2^6 + 114*B_y^3*b_x*b_y^4*mu_x2*mu_y2^5 +
56*B_y^3*b_y^5*mu_y2^6 - 597*B_x^3*b_x^5*mu_x2^3*mu_x3^2 +
40*B_x^3*b_x^5*mu_x3^4 + 18*B_x^3*b_x^5*mu_x2^2*mu_x4^2 +
33*B_x^3*b_x^5*mu_x2^2*mu_x3*mu_x5 + B_x^3*b_x^5*mu_x2^3*mu_x6 +
40*B_y^3*b_y^5*mu_y3^4 + 6*((9*B_x*B_y^2*b_x^2*b_y^3 +
10*B_y^3*b_x^2*b_y^3 + 9*B_x^2*B_y*b_x*b_y^4)*mu_x2^2 -
3*(B_x*B_y^2*b_x^2*b_y^3 + 3*B_x^2*B_y*b_x*b_y^4)*mu_x4)*mu_y2^4 -
40*(B_y^3*b_x^2*b_y^3 + 3*B_x*B_y^2*b_x*b_y^4)*mu_x3*mu_y3^3 +
(B_x^3*b_x^2*b_y^3*mu_x6 + 2*(54*B_x*B_y^2*b_x^3*b_y^2 +
B_y^3*b_x^3*b_y^2 + B_x^3*b_x^2*b_y^3 +
54*B_x^2*B_y*b_x^2*b_y^3)*mu_x2^3 - 3*(11*B_x*B_y^2*b_x^3*b_y^2 +
2*B_x^3*b_x^2*b_y^3 + 120*B_x^2*B_y*b_x^2*b_y^3)*mu_x3^2 -
3*(12*B_x*B_y^2*b_x^3*b_y^2 + B_x^3*b_x^2*b_y^3 +
36*B_x^2*B_y*b_x^2*b_y^3)*mu_x2*mu_x4)*mu_y2^3 +
3*(6*B_x^3*b_x^3*b_y^2*mu_x4^2 + 11*B_x^3*b_x^3*b_y^2*mu_x3*mu_x5 +
B_x^3*b_x^3*b_y^2*mu_x2*mu_x6 + 2*(9*B_x*B_y^2*b_x^4*b_y +
10*B_x^3*b_x^3*b_y^2 + 9*B_x^2*B_y*b_x^3*b_y^2)*mu_x2^4 -
(11*B_x*B_y^2*b_x^4*b_y + 72*B_x^3*b_x^3*b_y^2 +
120*B_x^2*B_y*b_x^3*b_y^2)*mu_x2*mu_x3^2 - 3*(2*B_x*B_y^2*b_x^4*b_y +
9*B_x^3*b_x^3*b_y^2 + 6*B_x^2*B_y*b_x^3*b_y^2)*mu_x2^2*mu_x4)*mu_y2^2 -
3*(269*B_y^3*b_x*b_y^4*mu_x2*mu_y2^2 + 199*B_y^3*b_y^5*mu_y2^3 +
(120*B_x*B_y^2*b_x^3*b_y^2 + 2*B_y^3*b_x^3*b_y^2 +
11*B_x^2*B_y*b_x^2*b_y^3)*mu_x2^3 - 40*(B_x*B_y^2*b_x^3*b_y^2 +
B_x^2*B_y*b_x^2*b_y^3)*mu_x3^2 - (40*B_x*B_y^2*b_x^3*b_y^2 +
11*B_x^2*B_y*b_x^2*b_y^3)*mu_x2*mu_x4 + ((120*B_x*B_y^2*b_x^2*b_y^3 +
72*B_y^3*b_x^2*b_y^3 + 11*B_x^2*B_y*b_x*b_y^4)*mu_x2^2 -
(40*B_x*B_y^2*b_x^2*b_y^3 +
11*B_x^2*B_y*b_x*b_y^4)*mu_x4)*mu_y2)*mu_y3^2 +
18*(B_y^3*b_x^2*b_y^3*mu_x2^2 + 2*B_y^3*b_x*b_y^4*mu_x2*mu_y2 +
B_y^3*b_y^5*mu_y2^2)*mu_y4^2 - 3*(25*B_x^3*b_x^5*mu_x2^4 -
51*B_x^3*b_x^5*mu_x2*mu_x3^2)*mu_x4 + 3*(38*B_x^3*b_x^4*b_y*mu_x2^5 -
269*B_x^3*b_x^4*b_y*mu_x2^2*mu_x3^2 + 12*B_x^3*b_x^4*b_y*mu_x2*mu_x4^2 +
22*B_x^3*b_x^4*b_y*mu_x2*mu_x3*mu_x5 + B_x^3*b_x^4*b_y*mu_x2^2*mu_x6 -
51*(B_x^3*b_x^4*b_y*mu_x2^3 - B_x^3*b_x^4*b_y*mu_x3^2)*mu_x4)*mu_y2 -
(33*B_x^2*B_y*b_x^4*b_y*mu_x2^2*mu_x5 - 3*(317*B_x^2*B_y*b_x^4*b_y +
11*B_x^3*b_x^3*b_y^2)*mu_x2^3*mu_x3 - 3*(11*B_y^3*b_x^2*b_y^3 +
317*B_x*B_y^2*b_x*b_y^4)*mu_x3*mu_y2^3 + 40*(3*B_x^2*B_y*b_x^4*b_y +
B_x^3*b_x^3*b_y^2)*mu_x3^3 + 3*(91*B_x^2*B_y*b_x^4*b_y +
11*B_x^3*b_x^3*b_y^2)*mu_x2*mu_x3*mu_x4 +
3*(11*B_x^2*B_y*b_x^2*b_y^3*mu_x5 - (11*B_y^3*b_x^3*b_y^2 +
66*B_x^2*B_y*b_x^2*b_y^3 +
383*B_x*B_y^2*b_x^2*b_y^3)*mu_x2*mu_x3)*mu_y2^2 +
3*(22*B_x^2*B_y*b_x^3*b_y^2*mu_x2*mu_x5 - (383*B_x^2*B_y*b_x^3*b_y^2 +
66*B_x*B_y^2*b_x^3*b_y^2 + 11*B_x^3*b_x^2*b_y^3)*mu_x2^2*mu_x3 +
(91*B_x^2*B_y*b_x^3*b_y^2 +
11*B_x^3*b_x^2*b_y^3)*mu_x3*mu_x4)*mu_y2)*mu_y3 -
3*(51*B_y^3*b_x*b_y^4*mu_x2*mu_y2^3 + 25*B_y^3*b_y^5*mu_y2^4 +
6*(3*B_x*B_y^2*b_x^4*b_y + B_x^2*B_y*b_x^3*b_y^2)*mu_x2^4 -
(11*B_x*B_y^2*b_x^4*b_y + 40*B_x^2*B_y*b_x^3*b_y^2)*mu_x2*mu_x3^2 -
6*(B_x*B_y^2*b_x^4*b_y + B_x^2*B_y*b_x^3*b_y^2)*mu_x2^2*mu_x4 +
3*((6*B_x*B_y^2*b_x^2*b_y^3 + 9*B_y^3*b_x^2*b_y^3 +
2*B_x^2*B_y*b_x*b_y^4)*mu_x2^2 - 2*(B_x*B_y^2*b_x^2*b_y^3 +
B_x^2*B_y*b_x*b_y^4)*mu_x4)*mu_y2^2 - 51*(B_y^3*b_x*b_y^4*mu_x2 +
B_y^3*b_y^5*mu_y2)*mu_y3^2 + ((36*B_x*B_y^2*b_x^3*b_y^2 +
B_y^3*b_x^3*b_y^2 + 12*B_x^2*B_y*b_x^2*b_y^3)*mu_x2^3 -
(11*B_x*B_y^2*b_x^3*b_y^2 + 40*B_x^2*B_y*b_x^2*b_y^3)*mu_x3^2 -
12*(B_x*B_y^2*b_x^3*b_y^2 + B_x^2*B_y*b_x^2*b_y^3)*mu_x2*mu_x4)*mu_y2 +
((11*B_y^3*b_x^3*b_y^2 + 91*B_x*B_y^2*b_x^2*b_y^3)*mu_x2*mu_x3 +
(11*B_y^3*b_x^2*b_y^3 +
91*B_x*B_y^2*b_x*b_y^4)*mu_x3*mu_y2)*mu_y3)*mu_y4 -
33*(B_x*B_y^2*b_x^3*b_y^2*mu_x2^2*mu_x3 +
2*B_x*B_y^2*b_x^2*b_y^3*mu_x2*mu_x3*mu_y2 +
B_x*B_y^2*b_x*b_y^4*mu_x3*mu_y2^2 - (B_y^3*b_x^2*b_y^3*mu_x2^2 +
2*B_y^3*b_x*b_y^4*mu_x2*mu_y2 + B_y^3*b_y^5*mu_y2^2)*mu_y3)*mu_y5 +
(B_y^3*b_x^3*b_y^2*mu_x2^3 + 3*B_y^3*b_x^2*b_y^3*mu_x2^2*mu_y2 +
3*B_y^3*b_x*b_y^4*mu_x2*mu_y2^2 + B_y^3*b_y^5*mu_y2^3)*mu_y6)*A -
45*(8*B_x^4*b_x^5*mu_x2^5 - 73*B_x^4*b_x^5*mu_x2^2*mu_x3^2)*mu_x4 +
90*(6*B_x^4*b_x^4*b_y*mu_x2^6 - 73*B_x^4*b_x^4*b_y*mu_x2^3*mu_x3^2 +
40*B_x^4*b_x^4*b_y*mu_x3^4 + 6*B_x^4*b_x^4*b_y*mu_x2^2*mu_x4^2 -
(12*B_x^4*b_x^4*b_y*mu_x2^4 -
73*B_x^4*b_x^4*b_y*mu_x2*mu_x3^2)*mu_x4)*mu_y2 +
90*(73*B_x^3*B_y*b_x^4*b_y*mu_x2^4*mu_x3 +
146*B_x*B_y^3*b_x^2*b_y^3*mu_x2*mu_x3*mu_y2^3 +
73*B_x*B_y^3*b_x*b_y^4*mu_x3*mu_y2^4 -
160*B_x^3*B_y*b_x^4*b_y*mu_x2*mu_x3^3 -
73*B_x^3*B_y*b_x^4*b_y*mu_x2^2*mu_x3*mu_x4 -
73*(B_x^3*B_y*b_x^2*b_y^3*mu_x3*mu_x4 - (B_x*B_y^3*b_x^3*b_y^2 +
B_x^3*B_y*b_x^2*b_y^3)*mu_x2^2*mu_x3)*mu_y2^2 +
2*(73*B_x^3*B_y*b_x^3*b_y^2*mu_x2^3*mu_x3 -
80*B_x^3*B_y*b_x^3*b_y^2*mu_x3^3 -
73*B_x^3*B_y*b_x^3*b_y^2*mu_x2*mu_x3*mu_x4)*mu_y2)*mu_y3 -
45*(8*B_x^2*B_y^2*b_x^4*b_y*mu_x2^5 + 24*B_y^4*b_x*b_y^4*mu_x2*mu_y2^4 +
8*B_y^4*b_y^5*mu_y2^5 - 73*B_x^2*B_y^2*b_x^4*b_y*mu_x2^2*mu_x3^2 -
8*B_x^2*B_y^2*b_x^4*b_y*mu_x2^3*mu_x4 - 8*(B_x^2*B_y^2*b_x*b_y^4*mu_x4 -
(3*B_y^4*b_x^2*b_y^3 + B_x^2*B_y^2*b_x*b_y^4)*mu_x2^2)*mu_y2^3 -
(73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x3^2 +
24*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2*mu_x4 - 8*(B_y^4*b_x^3*b_y^2 +
3*B_x^2*B_y^2*b_x^2*b_y^3)*mu_x2^3)*mu_y2^2 -
73*(B_y^4*b_x^2*b_y^3*mu_x2^2 + 2*B_y^4*b_x*b_y^4*mu_x2*mu_y2 +
B_y^4*b_y^5*mu_y2^2)*mu_y3^2 + 2*(12*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^4 -
73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2*mu_x3^2 -
12*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^2*mu_x4)*mu_y2 +
146*(B_x*B_y^3*b_x^3*b_y^2*mu_x2^2*mu_x3 +
2*B_x*B_y^3*b_x^2*b_y^3*mu_x2*mu_x3*mu_y2 +
B_x*B_y^3*b_x*b_y^4*mu_x3*mu_y2^2)*mu_y3)*mu_y4)/A^7
}
# internal
calculateK2smp <- function(stats) {
c(K12two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K13two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K21two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K22two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K23two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K31two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K32two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K41two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K42two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K51two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K61two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']))
}
innager/edgee documentation built on April 24, 2024, 8:14 p.m.
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Defines functions calculateK2smp K61two K51two K42two K41two K32two K31two K23two K22two K21two K13two K12two calculateK1smp K61one K51one K42one K41one K32one K31one K23one K22one K21one K13one K12one He11 He10 He9 He8 He7 He6 He5 He4 He3 He2 He1 He0
Documented in K12one K12two K13one K13two K21one K21two K22one K22two K23one K23two K31one K31two K32one K32two K41one K41two K42one K42two K51one K51two K61one K61two
# Probabilists' Hermite polynomials
#
# @param x numeric vector.
#
# @return A vector of the same length as \code{x}.
#
# @examples
# He9(1:5)
#
# @name Hermite
He0 <- function(x) 1
# @rdname Hermite
He1 <- function(x) x
He2 <- function(x) x^2 - 1
He3 <- function(x) x^3 - 3*x
He4 <- function(x) x^4 - 6*x^2 + 3
He5 <- function(x) x^5 - 10*x^3 + 15*x
He6 <- function(x) x^6 - 15*x^4 + 45*x^2 - 15
He7 <- function(x) x^7 - 21*x^5 + 105*x^3 - 105*x
He8 <- function(x) x^8 - 28*x^6 + 210*x^4 - 420*x^2 + 105
He9 <- function(x) x^9 - 36*x^7 + 378*x^5 - 1260*x^3 + 945*x
He10 <- function(x) x^10 - 45*x^8 + 630*x^6 - 3150*x^4 + 4725*x^2 - 945
He11 <- function(x) x^11 - 55*x^9 + 990*x^7 - 6930*x^5 + 17325*x^3 - 10395*x
#' \code{k()} functions for Edgeworth expansions - one-sample
#'
#' Calculate \code{k}'s (cumulant components) for a general version of Edgeworth
#' expansions (EE) for one-sample t-statistic.
#'
#' Variance adjustment \eqn{r^2} is equal to the output of \code{K21one()},
#' unless different variance estimates are used for \code{A}, numerator of
#' \code{k}, and \code{r}.
#'
#' @name kfuns1
#' @family \code{k()} functions
#'
#' @param A value of \code{A} (depends on the type of the test).
#' @param B value of \code{B} (depends on the type of the test).
#' @param mu2,mu3,mu4,mu5,mu6 central moments (2 - 6) or their estimates.
#'
#' @return A calculated value for the respective component.
#' @examples
#' # moderated t-statistic
#' if (requireNamespace("limma")) {
#' # simulate high-dimensional data
#' n <- 10
#' m <- 1e4 # number of tests
#' ns <- 0.05*m # number of significant features
#' dat <- matrix(rgamma(m*n, shape = 3) - 3, nrow = m)
#' shifts <- runif(ns, 1, 5)
#' dat[1:ns, ] <- dat[1:ns, ] - shifts
#' # estimate prior information
#' fit <- limma::lmFit(dat, rep(1, n))
#' fbay <- limma::eBayes(fit)
#' # look at one feature (row of data)
#' i <- 625
#' stats <- smpStats(dat[i, ], moder = TRUE, d0 = fbay$df.prior,
#' s20 = fbay$s2.prior, varpost = fbay$s2.post[i])
#' vars <- names(stats) # if want to remove carryover names
#' names(stats) <- NULL
#' for (j in 1:length(stats)) {
#' assign(vars[j], stats[j])
#' }
#' K32one(A, B, mu2, mu3, mu4, mu5, mu6)
#' }
#' @rdname kfuns1
#' @export
K12one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/2*B*mu3/A^(3/2)
}
#' @rdname kfuns1
#' @export
K13one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/16*(6*(8*mu2*mu3 - mu5)*A*B^2 - 15*(mu2^2*mu3 - mu3*mu4)*B^3 -
8*A^2*B*mu3)/A^(7/2)
}
#' @rdname kfuns1
#' @export
K21one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
mu2/A
}
#' @rdname kfuns1
#' @export
K22one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
1/4*(4*(4*mu2^2 - mu4)*A*B - (4*mu2^3 - 7*mu3^2 - 4*mu2*mu4)*B^2)/A^3
}
#' @rdname kfuns1
#' @export
K23one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/16*(16*(3*mu2^2 - mu4)*A^3*B - 8*(58*mu2^3 - 19*mu3^2 - 30*mu2*mu4 +
2*mu6)*A^2*B^2 + 2*(112*mu2^4 - 360*mu2*mu3^2 - 144*mu2^2*mu4 + 24*mu4^2 +
45*mu3*mu5 + 8*mu2*mu6)*A*B^3 - 3*(16*mu2^5 - 59*mu2^2*mu3^2 +
16*mu2*mu4^2 - (32*mu2^3 - 59*mu3^2)*mu4)*B^4)/A^5
}
#' @rdname kfuns1
#' @export
K31one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-(3*B*mu2 - A)*mu3/A^(5/2)
}
#' @rdname kfuns1
#' @export
K32one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
1/8*(12*(13*mu2*mu3 - mu5)*A^2*B - 3*(175*mu2^2*mu3 - 31*mu3*mu4 -
12*mu2*mu5)*A*B^2 + (123*mu2^3*mu3 - 83*mu3^3 -
123*mu2*mu3*mu4)*B^3)/A^(9/2)
}
#' @rdname kfuns1
#' @export
K41one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-((3*mu2^2 - mu4)*A^2 - 6*(3*mu2^3 - mu3^2 - mu2*mu4)*A*B + 3*(mu2^4 -
6*mu2*mu3^2 - mu2^2*mu4)*B^2)/A^4
}
#' @rdname kfuns1
#' @export
K42one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/8*(16*(42*mu2^3 - 13*mu3^2 - 19*mu2*mu4 + mu6)*A^3*B - 12*(284*mu2^4 -
266*mu2*mu3^2 - 172*mu2^2*mu4 + 12*mu4^2 + 21*mu3*mu5 +
8*mu2*mu6)*A^2*B^2 + 12*(112*mu2^5 - 702*mu2^2*mu3^2 + 32*mu2*mu4^2 +
63*mu2*mu3*mu5 + 4*mu2^2*mu6 - 2*(74*mu2^3 - 51*mu3^2)*mu4)*A*B^3 -
3*(64*mu2^6 - 624*mu2^3*mu3^2 + 233*mu3^4 + 64*mu2^2*mu4^2 - 16*(8*mu2^4 -
39*mu2*mu3^2)*mu4)*B^4)/A^6
}
#' @rdname kfuns1
#' @export
K51one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
-1/2*(2*(10*mu2*mu3 - mu5)*A^3 - 10*(35*mu2^2*mu3 - 5*mu3*mu4 -
2*mu2*mu5)*A^2*B + 15*(56*mu2^3*mu3 - 7*mu3^3 - 16*mu2*mu3*mu4 -
2*mu2^2*mu5)*A*B^2 - 15*(10*mu2^4*mu3 - 21*mu2*mu3^3 -
10*mu2^2*mu3*mu4)*B^3)/A^(11/2)
}
#' @rdname kfuns1
#' @export
K61one <- function(A, B, mu2, mu3, mu4, mu5, mu6) {
1/2*(2*(30*mu2^3 - 10*mu3^2 - 15*mu2*mu4 + mu6)*A^4 - 30*(48*mu2^4 -
40*mu2*mu3^2 - 27*mu2^2*mu4 + 2*mu4^2 + 3*mu3*mu5 + mu2*mu6)*A^3*B +
15*(336*mu2^5 - 623*mu2^2*mu3^2 + 28*mu2*mu4^2 + 44*mu2*mu3*mu5 +
6*mu2^2*mu6 - (226*mu2^3 - 67*mu3^2)*mu4)*A^2*B^2 - 30*(56*mu2^6 -
597*mu2^3*mu3^2 + 40*mu3^4 + 18*mu2^2*mu4^2 + 33*mu2^2*mu3*mu5 +
mu2^3*mu6 - 3*(25*mu2^4 - 51*mu2*mu3^2)*mu4)*A*B^3 + 45*(4*mu2^7 -
73*mu2^4*mu3^2 + 80*mu2*mu3^4 + 4*mu2^3*mu4^2 - (8*mu2^5 -
73*mu2^2*mu3^2)*mu4)*B^4)/A^7
}
# internal
calculateK1smp <- function(stats) {
c(K12one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K13one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K21one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K22one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K23one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K31one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K32one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K41one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K42one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K51one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']),
K61one(stats['A'], stats['B'], stats['mu2'], stats['mu3'], stats['mu4'],
stats['mu5'], stats['mu6']))
}
#' \code{k()} functions for Edgeworth expansions - two-sample
#'
#' Calculate \code{k}'s (cumulant components) for a general version of Edgeworth
#' expansions (EE) for two-sample t-statistic.
#'
#' Note that the test statistic for this Edgeworth expansion is defined as
#' \eqn{\sqrt{n}(\bar{X} - \bar{Y})/s}{sqrt(n)(X-bar - Y-bar)/s} and therefore
#' \code{X} would normally represent a treatment group and \code{Y} - control
#' group. Variance adjustment \eqn{r^2} is equal to the output of
#' \code{K21two()}, unless different variance estimates are used for \code{A},
#' numerator of \code{k}, and \code{r}.
#'
#' @name kfuns2
#' @family \code{k()} functions
#'
#' @param A value of \code{A}.
#' @param B_x value of \eqn{B_x}{B[x]} (depends on the type of the test).
#' @param B_y value of \eqn{B_y}{B[y]} (depends on the type of the test).
#' @param b_x value of \eqn{b_x}{b[x]} - equal to \eqn{n/n_x}{n/n[x]}, where
#' \eqn{n = (n_x + n_y)/2}{n = (n[x] + n[y])/2}.
#' @param b_y value of \eqn{b_y}{b[y]} - equal to \eqn{n/n_y}{n/n[y]}, where
#' \eqn{n = (n_x + n_y)/2}{n = (n[x] + n[y])/2}.
#' @param mu_x2,mu_x3,mu_x4,mu_x5,mu_x6 central moments (2 - 6) for a treatment
#' group or their estimates.
#' @param mu_y2,mu_y3,mu_y4,mu_y5,mu_y6 central moments (2 - 6) for a control
#' group or their estimates.
#'
#' @return A calculated value for a respective component.
#' @examples
#' n1 <- 10
#' n2 <- 8
#' smp <- c(rgamma(n1, shape = 3), rnorm(n2))
#' a <- rep(1:0, c(n1, n2))
#' stats <- smpStats(smp, a, type = "Welch")
#' vars <- names(stats) # if want to remove carryover names
#' names(stats) <- NULL
#' for (j in 1:length(stats)) {
#' assign(vars[j], stats[j])
#' }
#' K32two(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
#' mu_y2, mu_y3, mu_y4, mu_y5, mu_y6)
#' @rdname kfuns2
#' @export
K12two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-1/2*(B_x*b_x*mu_x3 - B_y*b_y*mu_y3)/A^(3/2)
}
#' @rdname kfuns2
#' @export
K13two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/16*(15*B_x^3*b_x^2*mu_x2^2*mu_x3 + 15*B_x*B_y^2*b_x*b_y*mu_x3*mu_y2^2
- 15*B_x^3*b_x^2*mu_x3*mu_x4 + 8*(B_x*b_x^2*mu_x3 - B_y*b_y^2*mu_y3)*A^2
- 6*(8*B_x^2*b_x^2*mu_x2*mu_x3 + 2*B_x*B_y*b_x*b_y*mu_x3*mu_y2 -
B_x^2*b_x^2*mu_x5 + B_y^2*b_y^2*mu_y5 - 2*(B_x*B_y*b_x*b_y*mu_x2 +
4*B_y^2*b_y^2*mu_y2)*mu_y3)*A - 15*(B_x^2*B_y*b_x*b_y*mu_x2^2 +
B_y^3*b_y^2*mu_y2^2 - B_x^2*B_y*b_x*b_y*mu_x4)*mu_y3 -
15*(B_x*B_y^2*b_x*b_y*mu_x3 - B_y^3*b_y^2*mu_y3)*mu_y4)/A^(7/2)
}
#' @rdname kfuns2
#' @export
K21two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
(b_x*mu_x2 + b_y*mu_y2)/A
}
#' @rdname kfuns2
#' @export
K22two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-1/4*(4*B_x^2*b_x^2*mu_x2^3 + 4*B_y^2*b_x*b_y*mu_x2*mu_y2^2 +
4*B_y^2*b_y^2*mu_y2^3 - 7*B_x^2*b_x^2*mu_x3^2 -
4*B_x^2*b_x^2*mu_x2*mu_x4 + 14*B_x*B_y*b_x*b_y*mu_x3*mu_y3 -
7*B_y^2*b_y^2*mu_y3^2 - 4*(4*B_x*b_x^2*mu_x2^2 + 4*B_y*b_y^2*mu_y2^2 -
B_x*b_x^2*mu_x4 - B_y*b_y^2*mu_y4 + (B_x*b_x*b_y +
B_y*b_x*b_y)*mu_x2*mu_y2)*A + 4*(B_x^2*b_x*b_y*mu_x2^2 -
B_x^2*b_x*b_y*mu_x4)*mu_y2 - 4*(B_y^2*b_x*b_y*mu_x2 +
B_y^2*b_y^2*mu_y2)*mu_y4)/A^3
}
#' @rdname kfuns2
#' @export
K23two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/16*(48*B_x^4*b_x^3*mu_x2^5 + 48*B_y^4*b_x*b_y^2*mu_x2*mu_y2^4 +
48*B_y^4*b_y^3*mu_y2^5 - 177*B_x^4*b_x^3*mu_x2^2*mu_x3^2 +
48*B_x^4*b_x^3*mu_x2*mu_x4^2 - 16*(3*B_x*b_x^3*mu_x2^2 +
3*B_y*b_y^3*mu_y2^2 - B_x*b_x^3*mu_x4 - B_y*b_y^3*mu_y4)*A^3 +
96*(B_x^2*B_y^2*b_x*b_y^2*mu_x2^2 - B_x^2*B_y^2*b_x*b_y^2*mu_x4)*mu_y2^3
+ 8*(58*B_x^2*b_x^3*mu_x2^3 + 58*B_y^2*b_y^3*mu_y2^3 -
19*B_x^2*b_x^3*mu_x3^2 - 30*B_x^2*b_x^3*mu_x2*mu_x4 -
19*B_y^2*b_y^3*mu_y3^2 + 2*B_x^2*b_x^3*mu_x6 + 2*B_y^2*b_y^3*mu_y6 +
7*(b_x^2*b_y + b_x*b_y^2)*B_x*B_y*mu_x3*mu_y3 + 2*(8*B_x*B_y*b_x*b_y^2 +
5*B_y^2*b_x*b_y^2)*mu_x2*mu_y2^2 + 2*((5*B_x^2*b_x^2*b_y +
8*B_x*B_y*b_x^2*b_y)*mu_x2^2 - 2*(B_x^2*b_x^2*b_y +
B_x*B_y*b_x^2*b_y)*mu_x4)*mu_y2 - 2*(15*B_y^2*b_y^3*mu_y2 +
2*(B_x*B_y*b_x*b_y^2 + B_y^2*b_x*b_y^2)*mu_x2)*mu_y4)*A^2 +
3*(32*B_x^2*B_y^2*b_x^2*b_y*mu_x2^3 - 59*B_x^2*B_y^2*b_x^2*b_y*mu_x3^2 -
32*B_x^2*B_y^2*b_x^2*b_y*mu_x2*mu_x4)*mu_y2^2 -
177*(B_x^2*B_y^2*b_x*b_y^2*mu_x2^2 + B_y^4*b_y^3*mu_y2^2 -
B_x^2*B_y^2*b_x*b_y^2*mu_x4)*mu_y3^2 + 48*(B_y^4*b_x*b_y^2*mu_x2 +
B_y^4*b_y^3*mu_y2)*mu_y4^2 - 2*(112*B_x^3*b_x^3*mu_x2^4 +
112*B_y^3*b_y^3*mu_y2^4 - 360*B_x^3*b_x^3*mu_x2*mu_x3^2 -
144*B_x^3*b_x^3*mu_x2^2*mu_x4 + 24*B_x^3*b_x^3*mu_x4^2 +
45*B_x^3*b_x^3*mu_x3*mu_x5 + 8*B_x^3*b_x^3*mu_x2*mu_x6 +
24*B_y^3*b_y^3*mu_y4^2 + 8*(3*B_x*B_y^2*b_x*b_y^2 +
5*B_y^3*b_x*b_y^2)*mu_x2*mu_y2^3 + 24*(4*(B_x*B_y^2*b_x^2*b_y +
B_x^2*B_y*b_x*b_y^2)*mu_x2^2 - (B_x*B_y^2*b_x^2*b_y +
4*B_x^2*B_y*b_x*b_y^2)*mu_x4)*mu_y2^2 - 6*(60*B_y^3*b_y^3*mu_y2 +
(7*B_x*B_y^2*b_x*b_y^2 + 8*B_y^3*b_x*b_y^2)*mu_x2)*mu_y3^2 +
2*(4*B_x^3*b_x^2*b_y*mu_x6 + 4*(5*B_x^3*b_x^2*b_y +
3*B_x^2*B_y*b_x^2*b_y)*mu_x2^3 - 3*(8*B_x^3*b_x^2*b_y +
7*B_x^2*B_y*b_x^2*b_y)*mu_x3^2 - 12*(2*B_x^3*b_x^2*b_y +
B_x^2*B_y*b_x^2*b_y)*mu_x2*mu_x4)*mu_y2 +
3*(118*B_x^2*B_y*b_x^2*b_y*mu_x2*mu_x3 +
118*B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y2 -
15*B_x^2*B_y*b_x^2*b_y*mu_x5)*mu_y3 - 24*(6*B_y^3*b_y^3*mu_y2^2 +
(4*B_x*B_y^2*b_x^2*b_y + B_x^2*B_y*b_x*b_y^2)*mu_x2^2 +
(B_x*B_y^2*b_x*b_y^2 + 2*B_y^3*b_x*b_y^2)*mu_x2*mu_y2 -
(B_x*B_y^2*b_x^2*b_y + B_x^2*B_y*b_x*b_y^2)*mu_x4)*mu_y4 -
45*(B_x*B_y^2*b_x*b_y^2*mu_x3 - B_y^3*b_y^3*mu_y3)*mu_y5 +
8*(B_y^3*b_x*b_y^2*mu_x2 + B_y^3*b_y^3*mu_y2)*mu_y6)*A -
3*(32*B_x^4*b_x^3*mu_x2^3 - 59*B_x^4*b_x^3*mu_x3^2)*mu_x4 +
48*(B_x^4*b_x^2*b_y*mu_x2^4 - 2*B_x^4*b_x^2*b_y*mu_x2^2*mu_x4 +
B_x^4*b_x^2*b_y*mu_x4^2)*mu_y2 + 354*(B_x^3*B_y*b_x^2*b_y*mu_x2^2*mu_x3
+ B_x*B_y^3*b_x*b_y^2*mu_x3*mu_y2^2 -
B_x^3*B_y*b_x^2*b_y*mu_x3*mu_x4)*mu_y3 -
3*(32*B_x^2*B_y^2*b_x^2*b_y*mu_x2^3 + 32*B_y^4*b_x*b_y^2*mu_x2*mu_y2^2 +
32*B_y^4*b_y^3*mu_y2^3 - 59*B_x^2*B_y^2*b_x^2*b_y*mu_x3^2 -
32*B_x^2*B_y^2*b_x^2*b_y*mu_x2*mu_x4 +
118*B_x*B_y^3*b_x*b_y^2*mu_x3*mu_y3 - 59*B_y^4*b_y^3*mu_y3^2 +
32*(B_x^2*B_y^2*b_x*b_y^2*mu_x2^2 -
B_x^2*B_y^2*b_x*b_y^2*mu_x4)*mu_y2)*mu_y4)/A^5
}
#' @rdname kfuns2
#' @export
K31two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-(3*B_x*b_x^2*mu_x2*mu_x3 + 3*B_x*b_x*b_y*mu_x3*mu_y2 - (b_x^2*mu_x3 -
b_y^2*mu_y3)*A - 3*(B_y*b_x*b_y*mu_x2 + B_y*b_y^2*mu_y2)*mu_y3)/A^(5/2)
}
#' @rdname kfuns2
#' @export
K32two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/8*(123*B_x^3*b_x^3*mu_x2^3*mu_x3 +
123*B_x*B_y^2*b_x^2*b_y*mu_x2*mu_x3*mu_y2^2 +
123*B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y2^3 - 83*B_x^3*b_x^3*mu_x3^3 -
123*B_x^3*b_x^3*mu_x2*mu_x3*mu_x4 -
249*B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y3^2 + 83*B_y^3*b_y^3*mu_y3^3 +
12*(13*B_x*b_x^3*mu_x2*mu_x3 - B_x*b_x^3*mu_x5 + B_y*b_y^3*mu_y5 +
(2*B_x*b_x^2*b_y + B_y*b_x^2*b_y)*mu_x3*mu_y2 - (13*B_y*b_y^3*mu_y2 +
(B_x*b_x*b_y^2 + 2*B_y*b_x*b_y^2)*mu_x2)*mu_y3)*A^2 -
3*(175*B_x^2*b_x^3*mu_x2^2*mu_x3 - 31*B_x^2*b_x^3*mu_x3*mu_x4 -
12*B_x^2*b_x^3*mu_x2*mu_x5 + (5*B_y^2*b_x^2*b_y +
98*B_x*B_y*b_x*b_y^2)*mu_x3*mu_y2^2 - 4*(3*B_x^2*b_x^2*b_y*mu_x5 -
(23*B_x^2*b_x^2*b_y + 5*B_x*B_y*b_x^2*b_y)*mu_x2*mu_x3)*mu_y2 -
(175*B_y^2*b_y^3*mu_y2^2 + (98*B_x*B_y*b_x^2*b_y +
5*B_x^2*b_x*b_y^2)*mu_x2^2 + 4*(5*B_x*B_y*b_x*b_y^2 +
23*B_y^2*b_x*b_y^2)*mu_x2*mu_y2 - (26*B_x*B_y*b_x^2*b_y +
5*B_x^2*b_x*b_y^2)*mu_x4)*mu_y3 + (31*B_y^2*b_y^3*mu_y3 -
(5*B_y^2*b_x^2*b_y + 26*B_x*B_y*b_x*b_y^2)*mu_x3)*mu_y4 +
12*(B_y^2*b_x*b_y^2*mu_x2 + B_y^2*b_y^3*mu_y2)*mu_y5)*A +
123*(B_x^3*b_x^2*b_y*mu_x2^2*mu_x3 - B_x^3*b_x^2*b_y*mu_x3*mu_x4)*mu_y2
- 3*(41*B_x^2*B_y*b_x^2*b_y*mu_x2^3 + 41*B_y^3*b_x*b_y^2*mu_x2*mu_y2^2 +
41*B_y^3*b_y^3*mu_y2^3 - 83*B_x^2*B_y*b_x^2*b_y*mu_x3^2 -
41*B_x^2*B_y*b_x^2*b_y*mu_x2*mu_x4 + 41*(B_x^2*B_y*b_x*b_y^2*mu_x2^2 -
B_x^2*B_y*b_x*b_y^2*mu_x4)*mu_y2)*mu_y3 -
123*(B_x*B_y^2*b_x^2*b_y*mu_x2*mu_x3 + B_x*B_y^2*b_x*b_y^2*mu_x3*mu_y2 -
(B_y^3*b_x*b_y^2*mu_x2 + B_y^3*b_y^3*mu_y2)*mu_y3)*mu_y4)/A^(9/2)
}
#' @rdname kfuns2
#' @export
K41two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
-(3*B_x^2*b_x^3*mu_x2^4 + 6*B_y^2*b_x*b_y^2*mu_x2*mu_y2^3 +
3*B_y^2*b_y^3*mu_y2^4 - 18*B_x^2*b_x^3*mu_x2*mu_x3^2 -
3*B_x^2*b_x^3*mu_x2^2*mu_x4 + (3*b_x^3*mu_x2^2 + 3*b_y^3*mu_y2^2 -
b_x^3*mu_x4 - b_y^3*mu_y4)*A^2 - 3*(B_x^2*b_x*b_y^2*mu_x4 -
(B_y^2*b_x^2*b_y + B_x^2*b_x*b_y^2)*mu_x2^2)*mu_y2^2 -
18*(B_y^2*b_x*b_y^2*mu_x2 + B_y^2*b_y^3*mu_y2)*mu_y3^2 -
6*(3*B_x*b_x^3*mu_x2^3 + 3*B_y*b_x*b_y^2*mu_x2*mu_y2^2 +
3*B_y*b_y^3*mu_y2^3 - B_x*b_x^3*mu_x3^2 - B_x*b_x^3*mu_x2*mu_x4 -
B_y*b_y^3*mu_y3^2 + (B_y*b_x^2*b_y + B_x*b_x*b_y^2)*mu_x3*mu_y3 +
(3*B_x*b_x^2*b_y*mu_x2^2 - B_x*b_x^2*b_y*mu_x4)*mu_y2 -
(B_y*b_x*b_y^2*mu_x2 + B_y*b_y^3*mu_y2)*mu_y4)*A +
6*(B_x^2*b_x^2*b_y*mu_x2^3 - 3*B_x^2*b_x^2*b_y*mu_x3^2 -
B_x^2*b_x^2*b_y*mu_x2*mu_x4)*mu_y2 + 36*(B_x*B_y*b_x^2*b_y*mu_x2*mu_x3 +
B_x*B_y*b_x*b_y^2*mu_x3*mu_y2)*mu_y3 - 3*(B_y^2*b_x^2*b_y*mu_x2^2 +
2*B_y^2*b_x*b_y^2*mu_x2*mu_y2 + B_y^2*b_y^3*mu_y2^2)*mu_y4)/A^4
}
#' @rdname kfuns2
#' @export
K42two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/8*(192*B_x^4*b_x^4*mu_x2^6 + 384*B_y^4*b_x*b_y^3*mu_x2*mu_y2^5 +
192*B_y^4*b_y^4*mu_y2^6 - 1872*B_x^4*b_x^4*mu_x2^3*mu_x3^2 +
699*B_x^4*b_x^4*mu_x3^4 + 192*B_x^4*b_x^4*mu_x2^2*mu_x4^2 -
2796*B_x*B_y^3*b_x*b_y^3*mu_x3*mu_y3^3 + 699*B_y^4*b_y^4*mu_y3^4 -
192*(2*B_x^2*B_y^2*b_x*b_y^3*mu_x4 - (B_y^4*b_x^2*b_y^2 +
2*B_x^2*B_y^2*b_x*b_y^3)*mu_x2^2)*mu_y2^4 - 16*(42*B_x*b_x^4*mu_x2^3 +
42*B_y*b_y^4*mu_y2^3 - 13*B_x*b_x^4*mu_x3^2 - 19*B_x*b_x^4*mu_x2*mu_x4 -
13*B_y*b_y^4*mu_y3^2 + B_x*b_x^4*mu_x6 + B_y*b_y^4*mu_y6 +
3*(B_x*b_x*b_y^3 + 3*B_y*b_x*b_y^3)*mu_x2*mu_y2^2 + 3*(B_x*b_x^2*b_y^2 +
B_y*b_x^2*b_y^2)*mu_x3*mu_y3 + (3*(3*B_x*b_x^3*b_y +
B_y*b_x^3*b_y)*mu_x2^2 - (3*B_x*b_x^3*b_y + B_y*b_x^3*b_y)*mu_x4)*mu_y2
- (19*B_y*b_y^4*mu_y2 + (B_x*b_x*b_y^3 +
3*B_y*b_x*b_y^3)*mu_x2)*mu_y4)*A^3 +
48*(16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2^3 -
39*B_x^2*B_y^2*b_x^2*b_y^2*mu_x3^2 -
16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2*mu_x4)*mu_y2^3 +
12*(284*B_x^2*b_x^4*mu_x2^4 + 284*B_y^2*b_y^4*mu_y2^4 -
266*B_x^2*b_x^4*mu_x2*mu_x3^2 - 172*B_x^2*b_x^4*mu_x2^2*mu_x4 +
12*B_x^2*b_x^4*mu_x4^2 + 21*B_x^2*b_x^4*mu_x3*mu_x5 +
8*B_x^2*b_x^4*mu_x2*mu_x6 + 12*B_y^2*b_y^4*mu_y4^2 +
4*(9*B_x*B_y*b_x*b_y^3 + 49*B_y^2*b_x*b_y^3)*mu_x2*mu_y2^3 +
2*((126*B_x*B_y*b_x^2*b_y^2 + (4*b_x^2*b_y^2 + 3*b_x*b_y^3)*B_x^2 +
(3*b_x^3*b_y + 4*b_x^2*b_y^2)*B_y^2)*mu_x2^2 - (B_y^2*b_x^3*b_y +
36*B_x*B_y*b_x^2*b_y^2 + (2*b_x^2*b_y^2 +
3*b_x*b_y^3)*B_x^2)*mu_x4)*mu_y2^2 - 2*(133*B_y^2*b_y^4*mu_y2 +
6*(B_x*B_y*b_x*b_y^3 + 6*B_y^2*b_x*b_y^3)*mu_x2)*mu_y3^2 +
4*(2*B_x^2*b_x^3*b_y*mu_x6 + (49*B_x^2*b_x^3*b_y +
9*B_x*B_y*b_x^3*b_y)*mu_x2^3 - 3*(6*B_x^2*b_x^3*b_y +
B_x*B_y*b_x^3*b_y)*mu_x3^2 - 3*(9*B_x^2*b_x^3*b_y +
B_x*B_y*b_x^3*b_y)*mu_x2*mu_x4)*mu_y2 + (2*(27*B_x^2*b_x^2*b_y^2 +
4*(22*b_x^3*b_y + 3*b_x^2*b_y^2)*B_x*B_y)*mu_x2*mu_x3 +
2*(27*B_y^2*b_x^2*b_y^2 + 4*(3*b_x^2*b_y^2 +
22*b_x*b_y^3)*B_x*B_y)*mu_x3*mu_y2 - 7*(2*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x5)*mu_y3 - 2*(86*B_y^2*b_y^4*mu_y2^2 +
(36*B_x*B_y*b_x^2*b_y^2 + B_x^2*b_x*b_y^3 + (3*b_x^3*b_y +
2*b_x^2*b_y^2)*B_y^2)*mu_x2^2 + 6*(B_x*B_y*b_x*b_y^3 +
9*B_y^2*b_x*b_y^3)*mu_x2*mu_y2 - (B_y^2*b_x^3*b_y +
10*B_x*B_y*b_x^2*b_y^2 + B_x^2*b_x*b_y^3)*mu_x4)*mu_y4 +
7*(3*B_y^2*b_y^4*mu_y3 - (B_y^2*b_x^2*b_y^2 +
2*B_x*B_y*b_x*b_y^3)*mu_x3)*mu_y5 + 8*(B_y^2*b_x*b_y^3*mu_x2 +
B_y^2*b_y^4*mu_y2)*mu_y6)*A^2 -
48*(39*B_x^2*B_y^2*b_x^3*b_y*mu_x2*mu_x3^2 - 4*B_x^4*b_x^2*b_y^2*mu_x4^2
- 4*(2*B_x^2*B_y^2*b_x^3*b_y + B_x^4*b_x^2*b_y^2)*mu_x2^4 +
8*(B_x^2*B_y^2*b_x^3*b_y + B_x^4*b_x^2*b_y^2)*mu_x2^2*mu_x4)*mu_y2^2 -
18*(104*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2^3 +
104*B_y^4*b_x*b_y^3*mu_x2*mu_y2^2 + 104*B_y^4*b_y^4*mu_y2^3 -
233*B_x^2*B_y^2*b_x^2*b_y^2*mu_x3^2 -
104*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2*mu_x4 +
104*(B_x^2*B_y^2*b_x*b_y^3*mu_x2^2 -
B_x^2*B_y^2*b_x*b_y^3*mu_x4)*mu_y2)*mu_y3^2 +
192*(B_y^4*b_x^2*b_y^2*mu_x2^2 + 2*B_y^4*b_x*b_y^3*mu_x2*mu_y2 +
B_y^4*b_y^4*mu_y2^2)*mu_y4^2 - 12*(112*B_x^3*b_x^4*mu_x2^5 +
112*B_y^3*b_y^4*mu_y2^5 - 702*B_x^3*b_x^4*mu_x2^2*mu_x3^2 +
32*B_x^3*b_x^4*mu_x2*mu_x4^2 + 63*B_x^3*b_x^4*mu_x2*mu_x3*mu_x5 +
4*B_x^3*b_x^4*mu_x2^2*mu_x6 + 8*(B_x*B_y^2*b_x*b_y^3 +
16*B_y^3*b_x*b_y^3)*mu_x2*mu_y2^4 + 8*((14*B_x*B_y^2*b_x^2*b_y^2 +
2*B_y^3*b_x^2*b_y^2 + 13*B_x^2*B_y*b_x*b_y^3)*mu_x2^2 -
(4*B_x*B_y^2*b_x^2*b_y^2 + 13*B_x^2*B_y*b_x*b_y^3)*mu_x4)*mu_y2^3 +
(4*B_x^3*b_x^2*b_y^2*mu_x6 + 8*(13*B_x*B_y^2*b_x^3*b_y +
2*B_x^3*b_x^2*b_y^2 + 14*B_x^2*B_y*b_x^2*b_y^2)*mu_x2^3 -
3*(9*B_x*B_y^2*b_x^3*b_y + 8*B_x^3*b_x^2*b_y^2 +
91*B_x^2*B_y*b_x^2*b_y^2)*mu_x3^2 - 4*(8*B_x*B_y^2*b_x^3*b_y +
5*B_x^3*b_x^2*b_y^2 + 28*B_x^2*B_y*b_x^2*b_y^2)*mu_x2*mu_x4)*mu_y2^2 -
3*(234*B_y^3*b_y^4*mu_y2^2 + (91*B_x*B_y^2*b_x^2*b_y^2 +
8*B_y^3*b_x^2*b_y^2 + 9*B_x^2*B_y*b_x*b_y^3)*mu_x2^2 +
2*(8*B_x*B_y^2*b_x*b_y^3 + 79*B_y^3*b_x*b_y^3)*mu_x2*mu_y2 -
(25*B_x*B_y^2*b_x^2*b_y^2 + 9*B_x^2*B_y*b_x*b_y^3)*mu_x4)*mu_y3^2 +
32*(B_y^3*b_x*b_y^3*mu_x2 + B_y^3*b_y^4*mu_y2)*mu_y4^2 -
2*(74*B_x^3*b_x^4*mu_x2^3 - 51*B_x^3*b_x^4*mu_x3^2)*mu_x4 +
(32*B_x^3*b_x^3*b_y*mu_x4^2 + 63*B_x^3*b_x^3*b_y*mu_x3*mu_x5 +
8*B_x^3*b_x^3*b_y*mu_x2*mu_x6 + 8*(16*B_x^3*b_x^3*b_y +
B_x^2*B_y*b_x^3*b_y)*mu_x2^4 - 6*(79*B_x^3*b_x^3*b_y +
8*B_x^2*B_y*b_x^3*b_y)*mu_x2*mu_x3^2 - 8*(21*B_x^3*b_x^3*b_y +
B_x^2*B_y*b_x^3*b_y)*mu_x2^2*mu_x4)*mu_y2 -
3*(21*B_x^2*B_y*b_x^3*b_y*mu_x2*mu_x5 - (317*B_x^2*B_y*b_x^3*b_y +
9*B_x^3*b_x^2*b_y^2)*mu_x2^2*mu_x3 - (9*B_y^3*b_x^2*b_y^2 +
317*B_x*B_y^2*b_x*b_y^3)*mu_x3*mu_y2^2 + (59*B_x^2*B_y*b_x^3*b_y +
9*B_x^3*b_x^2*b_y^2)*mu_x3*mu_x4 + (21*B_x^2*B_y*b_x^2*b_y^2*mu_x5 -
158*(B_x^2*B_y*b_x^2*b_y^2 +
B_x*B_y^2*b_x^2*b_y^2)*mu_x2*mu_x3)*mu_y2)*mu_y3 -
(148*B_y^3*b_y^4*mu_y2^3 - 102*B_y^3*b_y^4*mu_y3^2 +
8*(13*B_x*B_y^2*b_x^3*b_y + 4*B_x^2*B_y*b_x^2*b_y^2)*mu_x2^3 +
8*(B_x*B_y^2*b_x*b_y^3 + 21*B_y^3*b_x*b_y^3)*mu_x2*mu_y2^2 -
3*(9*B_x*B_y^2*b_x^3*b_y + 25*B_x^2*B_y*b_x^2*b_y^2)*mu_x3^2 -
32*(B_x*B_y^2*b_x^3*b_y + B_x^2*B_y*b_x^2*b_y^2)*mu_x2*mu_x4 +
3*(9*B_y^3*b_x^2*b_y^2 + 59*B_x*B_y^2*b_x*b_y^3)*mu_x3*mu_y3 +
4*((28*B_x*B_y^2*b_x^2*b_y^2 + 5*B_y^3*b_x^2*b_y^2 +
8*B_x^2*B_y*b_x*b_y^3)*mu_x2^2 - 8*(B_x*B_y^2*b_x^2*b_y^2 +
B_x^2*B_y*b_x*b_y^3)*mu_x4)*mu_y2)*mu_y4 -
63*(B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3 + B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2
- (B_y^3*b_x*b_y^3*mu_x2 + B_y^3*b_y^4*mu_y2)*mu_y3)*mu_y5 +
4*(B_y^3*b_x^2*b_y^2*mu_x2^2 + 2*B_y^3*b_x*b_y^3*mu_x2*mu_y2 +
B_y^3*b_y^4*mu_y2^2)*mu_y6)*A - 48*(8*B_x^4*b_x^4*mu_x2^4 -
39*B_x^4*b_x^4*mu_x2*mu_x3^2)*mu_x4 + 48*(8*B_x^4*b_x^3*b_y*mu_x2^5 -
39*B_x^4*b_x^3*b_y*mu_x2^2*mu_x3^2 + 8*B_x^4*b_x^3*b_y*mu_x2*mu_x4^2 -
(16*B_x^4*b_x^3*b_y*mu_x2^3 - 39*B_x^4*b_x^3*b_y*mu_x3^2)*mu_x4)*mu_y2 +
12*(312*B_x^3*B_y*b_x^3*b_y*mu_x2^3*mu_x3 +
312*B_x*B_y^3*b_x^2*b_y^2*mu_x2*mu_x3*mu_y2^2 +
312*B_x*B_y^3*b_x*b_y^3*mu_x3*mu_y2^3 - 233*B_x^3*B_y*b_x^3*b_y*mu_x3^3
- 312*B_x^3*B_y*b_x^3*b_y*mu_x2*mu_x3*mu_x4 +
312*(B_x^3*B_y*b_x^2*b_y^2*mu_x2^2*mu_x3 -
B_x^3*B_y*b_x^2*b_y^2*mu_x3*mu_x4)*mu_y2)*mu_y3 -
48*(8*B_x^2*B_y^2*b_x^3*b_y*mu_x2^4 + 16*B_y^4*b_x*b_y^3*mu_x2*mu_y2^3 +
8*B_y^4*b_y^4*mu_y2^4 - 39*B_x^2*B_y^2*b_x^3*b_y*mu_x2*mu_x3^2 -
8*B_x^2*B_y^2*b_x^3*b_y*mu_x2^2*mu_x4 - 8*(B_x^2*B_y^2*b_x*b_y^3*mu_x4 -
(B_y^4*b_x^2*b_y^2 + B_x^2*B_y^2*b_x*b_y^3)*mu_x2^2)*mu_y2^2 -
39*(B_y^4*b_x*b_y^3*mu_x2 + B_y^4*b_y^4*mu_y2)*mu_y3^2 +
(16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2^3 - 39*B_x^2*B_y^2*b_x^2*b_y^2*mu_x3^2
- 16*B_x^2*B_y^2*b_x^2*b_y^2*mu_x2*mu_x4)*mu_y2 +
78*(B_x*B_y^3*b_x^2*b_y^2*mu_x2*mu_x3 +
B_x*B_y^3*b_x*b_y^3*mu_x3*mu_y2)*mu_y3)*mu_y4)/A^6
}
#' @rdname kfuns2
#' @export
K51two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/2*(150*B_x^3*b_x^4*mu_x2^4*mu_x3 +
300*B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3*mu_y2^3 +
150*B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2^4 - 315*B_x^3*b_x^4*mu_x2*mu_x3^3 -
150*B_x^3*b_x^4*mu_x2^2*mu_x3*mu_x4 - 2*(10*b_x^4*mu_x2*mu_x3 -
10*b_y^4*mu_y2*mu_y3 - b_x^4*mu_x5 + b_y^4*mu_y5)*A^3 +
315*(B_y^3*b_x*b_y^3*mu_x2 + B_y^3*b_y^4*mu_y2)*mu_y3^3 +
10*(35*B_x*b_x^4*mu_x2^2*mu_x3 - 5*B_x*b_x^4*mu_x3*mu_x4 -
2*B_x*b_x^4*mu_x2*mu_x5 + 3*(3*B_y*b_x^2*b_y^2 +
2*B_x*b_x*b_y^3)*mu_x3*mu_y2^2 + 2*(10*B_x*b_x^3*b_y*mu_x2*mu_x3 -
B_x*b_x^3*b_y*mu_x5)*mu_y2 - (20*B_y*b_x*b_y^3*mu_x2*mu_y2 +
35*B_y*b_y^4*mu_y2^2 + 3*(2*B_y*b_x^3*b_y + 3*B_x*b_x^2*b_y^2)*mu_x2^2 -
(2*B_y*b_x^3*b_y + 3*B_x*b_x^2*b_y^2)*mu_x4)*mu_y3 + (5*B_y*b_y^4*mu_y3
- (3*B_y*b_x^2*b_y^2 + 2*B_x*b_x*b_y^3)*mu_x3)*mu_y4 +
2*(B_y*b_x*b_y^3*mu_x2 + B_y*b_y^4*mu_y2)*mu_y5)*A^2 -
150*(B_x^3*b_x^2*b_y^2*mu_x3*mu_x4 - (B_x*B_y^2*b_x^3*b_y +
B_x^3*b_x^2*b_y^2)*mu_x2^2*mu_x3)*mu_y2^2 -
945*(B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3 +
B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2)*mu_y3^2 -
15*(56*B_x^2*b_x^4*mu_x2^3*mu_x3 - 7*B_x^2*b_x^4*mu_x3^3 -
16*B_x^2*b_x^4*mu_x2*mu_x3*mu_x4 - 2*B_x^2*b_x^4*mu_x2^2*mu_x5 +
7*B_y^2*b_y^4*mu_y3^3 + 2*(B_y^2*b_x^2*b_y^2 +
21*B_x*B_y*b_x*b_y^3)*mu_x3*mu_y2^3 - 7*(B_y^2*b_x^2*b_y^2 +
2*B_x*B_y*b_x*b_y^3)*mu_x3*mu_y3^2 - 2*(B_x^2*b_x^2*b_y^2*mu_x5 -
(B_y^2*b_x^3*b_y + 6*B_x^2*b_x^2*b_y^2 +
21*B_x*B_y*b_x^2*b_y^2)*mu_x2*mu_x3)*mu_y2^2 +
4*(17*B_x^2*b_x^3*b_y*mu_x2^2*mu_x3 - 4*B_x^2*b_x^3*b_y*mu_x3*mu_x4 -
B_x^2*b_x^3*b_y*mu_x2*mu_x5)*mu_y2 - (68*B_y^2*b_x*b_y^3*mu_x2*mu_y2^2 +
56*B_y^2*b_y^4*mu_y2^3 + 2*(21*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x2^3 - 7*(2*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x3^2 - 2*(7*B_x*B_y*b_x^3*b_y +
B_x^2*b_x^2*b_y^2)*mu_x2*mu_x4 + 2*((21*B_x*B_y*b_x^2*b_y^2 +
6*B_y^2*b_x^2*b_y^2 + B_x^2*b_x*b_y^3)*mu_x2^2 - (7*B_x*B_y*b_x^2*b_y^2
+ B_x^2*b_x*b_y^3)*mu_x4)*mu_y2)*mu_y3 - 2*((B_y^2*b_x^3*b_y +
7*B_x*B_y*b_x^2*b_y^2)*mu_x2*mu_x3 + (B_y^2*b_x^2*b_y^2 +
7*B_x*B_y*b_x*b_y^3)*mu_x3*mu_y2 - 8*(B_y^2*b_x*b_y^3*mu_x2 +
B_y^2*b_y^4*mu_y2)*mu_y3)*mu_y4 + 2*(B_y^2*b_x^2*b_y^2*mu_x2^2 +
2*B_y^2*b_x*b_y^3*mu_x2*mu_y2 + B_y^2*b_y^4*mu_y2^2)*mu_y5)*A +
15*(20*B_x^3*b_x^3*b_y*mu_x2^3*mu_x3 - 21*B_x^3*b_x^3*b_y*mu_x3^3 -
20*B_x^3*b_x^3*b_y*mu_x2*mu_x3*mu_x4)*mu_y2 -
15*(10*B_x^2*B_y*b_x^3*b_y*mu_x2^4 + 20*B_y^3*b_x*b_y^3*mu_x2*mu_y2^3 +
10*B_y^3*b_y^4*mu_y2^4 - 63*B_x^2*B_y*b_x^3*b_y*mu_x2*mu_x3^2 -
10*B_x^2*B_y*b_x^3*b_y*mu_x2^2*mu_x4 - 10*(B_x^2*B_y*b_x*b_y^3*mu_x4 -
(B_y^3*b_x^2*b_y^2 + B_x^2*B_y*b_x*b_y^3)*mu_x2^2)*mu_y2^2 +
(20*B_x^2*B_y*b_x^2*b_y^2*mu_x2^3 - 63*B_x^2*B_y*b_x^2*b_y^2*mu_x3^2 -
20*B_x^2*B_y*b_x^2*b_y^2*mu_x2*mu_x4)*mu_y2)*mu_y3 -
150*(B_x*B_y^2*b_x^3*b_y*mu_x2^2*mu_x3 +
2*B_x*B_y^2*b_x^2*b_y^2*mu_x2*mu_x3*mu_y2 +
B_x*B_y^2*b_x*b_y^3*mu_x3*mu_y2^2 - (B_y^3*b_x^2*b_y^2*mu_x2^2 +
2*B_y^3*b_x*b_y^3*mu_x2*mu_y2 +
B_y^3*b_y^4*mu_y2^2)*mu_y3)*mu_y4)/A^(11/2)
}
#' @rdname kfuns2
#' @export
K61two <- function(A, B_x, B_y, b_x, b_y, mu_x2, mu_x3, mu_x4, mu_x5, mu_x6,
mu_y2, mu_y3, mu_y4, mu_y5, mu_y6) {
1/2*(180*B_x^4*b_x^5*mu_x2^7 + 540*B_y^4*b_x*b_y^4*mu_x2*mu_y2^6 +
180*B_y^4*b_y^5*mu_y2^7 - 3285*B_x^4*b_x^5*mu_x2^4*mu_x3^2 +
3600*B_x^4*b_x^5*mu_x2*mu_x3^4 + 180*B_x^4*b_x^5*mu_x2^3*mu_x4^2 -
180*(2*B_x^2*B_y^2*b_x*b_y^4*mu_x4 - (3*B_y^4*b_x^2*b_y^3 +
2*B_x^2*B_y^2*b_x*b_y^4)*mu_x2^2)*mu_y2^5 + 2*(30*b_x^5*mu_x2^3 +
30*b_y^5*mu_y2^3 - 10*b_x^5*mu_x3^2 - 15*b_x^5*mu_x2*mu_x4 -
10*b_y^5*mu_y3^2 - 15*b_y^5*mu_y2*mu_y4 + b_x^5*mu_x6 + b_y^5*mu_y6)*A^4
- 45*(73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x3^2 +
24*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2*mu_x4 - 4*(B_y^4*b_x^3*b_y^2 +
6*B_x^2*B_y^2*b_x^2*b_y^3)*mu_x2^3)*mu_y2^4 +
3600*(B_y^4*b_x*b_y^4*mu_x2 + B_y^4*b_y^5*mu_y2)*mu_y3^4 -
30*(48*B_x*b_x^5*mu_x2^4 + 30*B_y*b_x*b_y^4*mu_x2*mu_y2^3 +
48*B_y*b_y^5*mu_y2^4 - 40*B_x*b_x^5*mu_x2*mu_x3^2 -
27*B_x*b_x^5*mu_x2^2*mu_x4 + 2*B_x*b_x^5*mu_x4^2 +
3*B_x*b_x^5*mu_x3*mu_x5 + B_x*b_x^5*mu_x2*mu_x6 + 2*B_y*b_y^5*mu_y4^2 +
6*(3*(B_y*b_x^3*b_y^2 + B_x*b_x^2*b_y^3)*mu_x2^2 - (B_y*b_x^3*b_y^2 +
B_x*b_x^2*b_y^3)*mu_x4)*mu_y2^2 - 10*(B_y*b_x*b_y^4*mu_x2 +
4*B_y*b_y^5*mu_y2)*mu_y3^2 + (30*B_x*b_x^4*b_y*mu_x2^3 -
10*B_x*b_x^4*b_y*mu_x3^2 - 15*B_x*b_x^4*b_y*mu_x2*mu_x4 +
B_x*b_x^4*b_y*mu_x6)*mu_y2 + (10*(B_y*b_x^4*b_y +
2*B_x*b_x^3*b_y^2)*mu_x2*mu_x3 + 10*(2*B_y*b_x^2*b_y^3 +
B_x*b_x*b_y^4)*mu_x3*mu_y2 - (B_y*b_x^4*b_y +
2*B_x*b_x^3*b_y^2)*mu_x5)*mu_y3 - (15*B_y*b_x*b_y^4*mu_x2*mu_y2 +
27*B_y*b_y^5*mu_y2^2 + 6*(B_y*b_x^3*b_y^2 + B_x*b_x^2*b_y^3)*mu_x2^2 -
2*(B_y*b_x^3*b_y^2 + B_x*b_x^2*b_y^3)*mu_x4)*mu_y4 + (3*B_y*b_y^5*mu_y3
- (2*B_y*b_x^2*b_y^3 + B_x*b_x*b_y^4)*mu_x3)*mu_y5 +
(B_y*b_x*b_y^4*mu_x2 + B_y*b_y^5*mu_y2)*mu_y6)*A^3 -
90*(73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2*mu_x3^2 -
2*B_x^4*b_x^2*b_y^3*mu_x4^2 - 2*(6*B_x^2*B_y^2*b_x^3*b_y^2 +
B_x^4*b_x^2*b_y^3)*mu_x2^4 + 4*(3*B_x^2*B_y^2*b_x^3*b_y^2 +
B_x^4*b_x^2*b_y^3)*mu_x2^2*mu_x4)*mu_y2^3 -
14400*(B_x*B_y^3*b_x^2*b_y^3*mu_x2*mu_x3 +
B_x*B_y^3*b_x*b_y^4*mu_x3*mu_y2)*mu_y3^3 + 15*(336*B_x^2*b_x^5*mu_x2^5 +
444*B_y^2*b_x*b_y^4*mu_x2*mu_y2^4 + 336*B_y^2*b_y^5*mu_y2^5 -
623*B_x^2*b_x^5*mu_x2^2*mu_x3^2 + 28*B_x^2*b_x^5*mu_x2*mu_x4^2 +
44*B_x^2*b_x^5*mu_x2*mu_x3*mu_x5 + 6*B_x^2*b_x^5*mu_x2^2*mu_x6 +
4*(3*(36*B_x*B_y*b_x^2*b_y^3 + B_x^2*b_x*b_y^4 + (b_x^3*b_y^2 +
9*b_x^2*b_y^3)*B_y^2)*mu_x2^2 - (B_y^2*b_x^3*b_y^2 +
36*B_x*B_y*b_x^2*b_y^3 + 3*B_x^2*b_x*b_y^4)*mu_x4)*mu_y2^3 +
(6*B_x^2*b_x^3*b_y^2*mu_x6 + 12*(B_y^2*b_x^4*b_y +
36*B_x*B_y*b_x^3*b_y^2 + (9*b_x^3*b_y^2 + b_x^2*b_y^3)*B_x^2)*mu_x2^3 -
3*(B_y^2*b_x^4*b_y + 48*B_x*B_y*b_x^3*b_y^2 + 4*(3*b_x^3*b_y^2 +
4*b_x^2*b_y^3)*B_x^2)*mu_x3^2 - 2*(2*B_y^2*b_x^4*b_y +
72*B_x*B_y*b_x^3*b_y^2 + 3*(11*b_x^3*b_y^2 +
2*b_x^2*b_y^3)*B_x^2)*mu_x2*mu_x4)*mu_y2^2 -
(464*B_y^2*b_x*b_y^4*mu_x2*mu_y2 + 623*B_y^2*b_y^5*mu_y2^2 +
3*(48*B_x*B_y*b_x^2*b_y^3 + B_x^2*b_x*b_y^4 + 4*(4*b_x^3*b_y^2 +
3*b_x^2*b_y^3)*B_y^2)*mu_x2^2 - (16*B_y^2*b_x^3*b_y^2 +
48*B_x*B_y*b_x^2*b_y^3 + 3*B_x^2*b_x*b_y^4)*mu_x4)*mu_y3^2 +
28*(B_y^2*b_x*b_y^4*mu_x2 + B_y^2*b_y^5*mu_y2)*mu_y4^2 -
(226*B_x^2*b_x^5*mu_x2^3 - 67*B_x^2*b_x^5*mu_x3^2)*mu_x4 +
4*(111*B_x^2*b_x^4*b_y*mu_x2^4 - 116*B_x^2*b_x^4*b_y*mu_x2*mu_x3^2 -
73*B_x^2*b_x^4*b_y*mu_x2^2*mu_x4 + 7*B_x^2*b_x^4*b_y*mu_x4^2 +
11*B_x^2*b_x^4*b_y*mu_x3*mu_x5 + 3*B_x^2*b_x^4*b_y*mu_x2*mu_x6)*mu_y2 +
2*((280*B_x*B_y*b_x^4*b_y + 111*B_x^2*b_x^3*b_y^2)*mu_x2^2*mu_x3 +
(111*B_y^2*b_x^2*b_y^3 + 280*B_x*B_y*b_x*b_y^4)*mu_x3*mu_y2^2 -
(40*B_x*B_y*b_x^4*b_y + 27*B_x^2*b_x^3*b_y^2)*mu_x3*mu_x4 -
2*(8*B_x*B_y*b_x^4*b_y + 3*B_x^2*b_x^3*b_y^2)*mu_x2*mu_x5 +
2*(2*(9*B_y^2*b_x^3*b_y^2 + 9*B_x^2*b_x^2*b_y^3 + 40*(b_x^3*b_y^2 +
b_x^2*b_y^3)*B_x*B_y)*mu_x2*mu_x3 - (8*B_x*B_y*b_x^3*b_y^2 +
3*B_x^2*b_x^2*b_y^3)*mu_x5)*mu_y2)*mu_y3 -
(292*B_y^2*b_x*b_y^4*mu_x2*mu_y2^2 + 226*B_y^2*b_y^5*mu_y2^3 -
67*B_y^2*b_y^5*mu_y3^2 + 4*(3*B_y^2*b_x^4*b_y + 36*B_x*B_y*b_x^3*b_y^2 +
B_x^2*b_x^2*b_y^3)*mu_x2^3 - (3*B_y^2*b_x^4*b_y + 48*B_x*B_y*b_x^3*b_y^2
+ 16*B_x^2*b_x^2*b_y^3)*mu_x3^2 - 4*(B_y^2*b_x^4*b_y +
12*B_x*B_y*b_x^3*b_y^2 + B_x^2*b_x^2*b_y^3)*mu_x2*mu_x4 +
2*(27*B_y^2*b_x^2*b_y^3 + 40*B_x*B_y*b_x*b_y^4)*mu_x3*mu_y3 +
2*((72*B_x*B_y*b_x^2*b_y^3 + 2*B_x^2*b_x*b_y^4 + 3*(2*b_x^3*b_y^2 +
11*b_x^2*b_y^3)*B_y^2)*mu_x2^2 - 2*(B_y^2*b_x^3*b_y^2 +
12*B_x*B_y*b_x^2*b_y^3 + B_x^2*b_x*b_y^4)*mu_x4)*mu_y2)*mu_y4 -
4*((3*B_y^2*b_x^3*b_y^2 + 8*B_x*B_y*b_x^2*b_y^3)*mu_x2*mu_x3 +
(3*B_y^2*b_x^2*b_y^3 + 8*B_x*B_y*b_x*b_y^4)*mu_x3*mu_y2 -
11*(B_y^2*b_x*b_y^4*mu_x2 + B_y^2*b_y^5*mu_y2)*mu_y3)*mu_y5 +
6*(B_y^2*b_x^2*b_y^3*mu_x2^2 + 2*B_y^2*b_x*b_y^4*mu_x2*mu_y2 +
B_y^2*b_y^5*mu_y2^2)*mu_y6)*A^2 + 45*(12*B_x^4*b_x^3*b_y^2*mu_x2*mu_x4^2
+ 4*(2*B_x^2*B_y^2*b_x^4*b_y + 3*B_x^4*b_x^3*b_y^2)*mu_x2^5 -
73*(B_x^2*B_y^2*b_x^4*b_y + B_x^4*b_x^3*b_y^2)*mu_x2^2*mu_x3^2 +
(73*B_x^4*b_x^3*b_y^2*mu_x3^2 - 8*(B_x^2*B_y^2*b_x^4*b_y +
3*B_x^4*b_x^3*b_y^2)*mu_x2^3)*mu_x4)*mu_y2^2 -
45*(73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^4 +
146*B_y^4*b_x*b_y^4*mu_x2*mu_y2^3 + 73*B_y^4*b_y^5*mu_y2^4 -
480*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2*mu_x3^2 -
73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^2*mu_x4 -
73*(B_x^2*B_y^2*b_x*b_y^4*mu_x4 - (B_y^4*b_x^2*b_y^3 +
B_x^2*B_y^2*b_x*b_y^4)*mu_x2^2)*mu_y2^2 +
2*(73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2^3 -
240*B_x^2*B_y^2*b_x^2*b_y^3*mu_x3^2 -
73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2*mu_x4)*mu_y2)*mu_y3^2 +
180*(B_y^4*b_x^3*b_y^2*mu_x2^3 + 3*B_y^4*b_x^2*b_y^3*mu_x2^2*mu_y2 +
3*B_y^4*b_x*b_y^4*mu_x2*mu_y2^2 + B_y^4*b_y^5*mu_y2^3)*mu_y4^2 -
30*(56*B_x^3*b_x^5*mu_x2^6 + 114*B_y^3*b_x*b_y^4*mu_x2*mu_y2^5 +
56*B_y^3*b_y^5*mu_y2^6 - 597*B_x^3*b_x^5*mu_x2^3*mu_x3^2 +
40*B_x^3*b_x^5*mu_x3^4 + 18*B_x^3*b_x^5*mu_x2^2*mu_x4^2 +
33*B_x^3*b_x^5*mu_x2^2*mu_x3*mu_x5 + B_x^3*b_x^5*mu_x2^3*mu_x6 +
40*B_y^3*b_y^5*mu_y3^4 + 6*((9*B_x*B_y^2*b_x^2*b_y^3 +
10*B_y^3*b_x^2*b_y^3 + 9*B_x^2*B_y*b_x*b_y^4)*mu_x2^2 -
3*(B_x*B_y^2*b_x^2*b_y^3 + 3*B_x^2*B_y*b_x*b_y^4)*mu_x4)*mu_y2^4 -
40*(B_y^3*b_x^2*b_y^3 + 3*B_x*B_y^2*b_x*b_y^4)*mu_x3*mu_y3^3 +
(B_x^3*b_x^2*b_y^3*mu_x6 + 2*(54*B_x*B_y^2*b_x^3*b_y^2 +
B_y^3*b_x^3*b_y^2 + B_x^3*b_x^2*b_y^3 +
54*B_x^2*B_y*b_x^2*b_y^3)*mu_x2^3 - 3*(11*B_x*B_y^2*b_x^3*b_y^2 +
2*B_x^3*b_x^2*b_y^3 + 120*B_x^2*B_y*b_x^2*b_y^3)*mu_x3^2 -
3*(12*B_x*B_y^2*b_x^3*b_y^2 + B_x^3*b_x^2*b_y^3 +
36*B_x^2*B_y*b_x^2*b_y^3)*mu_x2*mu_x4)*mu_y2^3 +
3*(6*B_x^3*b_x^3*b_y^2*mu_x4^2 + 11*B_x^3*b_x^3*b_y^2*mu_x3*mu_x5 +
B_x^3*b_x^3*b_y^2*mu_x2*mu_x6 + 2*(9*B_x*B_y^2*b_x^4*b_y +
10*B_x^3*b_x^3*b_y^2 + 9*B_x^2*B_y*b_x^3*b_y^2)*mu_x2^4 -
(11*B_x*B_y^2*b_x^4*b_y + 72*B_x^3*b_x^3*b_y^2 +
120*B_x^2*B_y*b_x^3*b_y^2)*mu_x2*mu_x3^2 - 3*(2*B_x*B_y^2*b_x^4*b_y +
9*B_x^3*b_x^3*b_y^2 + 6*B_x^2*B_y*b_x^3*b_y^2)*mu_x2^2*mu_x4)*mu_y2^2 -
3*(269*B_y^3*b_x*b_y^4*mu_x2*mu_y2^2 + 199*B_y^3*b_y^5*mu_y2^3 +
(120*B_x*B_y^2*b_x^3*b_y^2 + 2*B_y^3*b_x^3*b_y^2 +
11*B_x^2*B_y*b_x^2*b_y^3)*mu_x2^3 - 40*(B_x*B_y^2*b_x^3*b_y^2 +
B_x^2*B_y*b_x^2*b_y^3)*mu_x3^2 - (40*B_x*B_y^2*b_x^3*b_y^2 +
11*B_x^2*B_y*b_x^2*b_y^3)*mu_x2*mu_x4 + ((120*B_x*B_y^2*b_x^2*b_y^3 +
72*B_y^3*b_x^2*b_y^3 + 11*B_x^2*B_y*b_x*b_y^4)*mu_x2^2 -
(40*B_x*B_y^2*b_x^2*b_y^3 +
11*B_x^2*B_y*b_x*b_y^4)*mu_x4)*mu_y2)*mu_y3^2 +
18*(B_y^3*b_x^2*b_y^3*mu_x2^2 + 2*B_y^3*b_x*b_y^4*mu_x2*mu_y2 +
B_y^3*b_y^5*mu_y2^2)*mu_y4^2 - 3*(25*B_x^3*b_x^5*mu_x2^4 -
51*B_x^3*b_x^5*mu_x2*mu_x3^2)*mu_x4 + 3*(38*B_x^3*b_x^4*b_y*mu_x2^5 -
269*B_x^3*b_x^4*b_y*mu_x2^2*mu_x3^2 + 12*B_x^3*b_x^4*b_y*mu_x2*mu_x4^2 +
22*B_x^3*b_x^4*b_y*mu_x2*mu_x3*mu_x5 + B_x^3*b_x^4*b_y*mu_x2^2*mu_x6 -
51*(B_x^3*b_x^4*b_y*mu_x2^3 - B_x^3*b_x^4*b_y*mu_x3^2)*mu_x4)*mu_y2 -
(33*B_x^2*B_y*b_x^4*b_y*mu_x2^2*mu_x5 - 3*(317*B_x^2*B_y*b_x^4*b_y +
11*B_x^3*b_x^3*b_y^2)*mu_x2^3*mu_x3 - 3*(11*B_y^3*b_x^2*b_y^3 +
317*B_x*B_y^2*b_x*b_y^4)*mu_x3*mu_y2^3 + 40*(3*B_x^2*B_y*b_x^4*b_y +
B_x^3*b_x^3*b_y^2)*mu_x3^3 + 3*(91*B_x^2*B_y*b_x^4*b_y +
11*B_x^3*b_x^3*b_y^2)*mu_x2*mu_x3*mu_x4 +
3*(11*B_x^2*B_y*b_x^2*b_y^3*mu_x5 - (11*B_y^3*b_x^3*b_y^2 +
66*B_x^2*B_y*b_x^2*b_y^3 +
383*B_x*B_y^2*b_x^2*b_y^3)*mu_x2*mu_x3)*mu_y2^2 +
3*(22*B_x^2*B_y*b_x^3*b_y^2*mu_x2*mu_x5 - (383*B_x^2*B_y*b_x^3*b_y^2 +
66*B_x*B_y^2*b_x^3*b_y^2 + 11*B_x^3*b_x^2*b_y^3)*mu_x2^2*mu_x3 +
(91*B_x^2*B_y*b_x^3*b_y^2 +
11*B_x^3*b_x^2*b_y^3)*mu_x3*mu_x4)*mu_y2)*mu_y3 -
3*(51*B_y^3*b_x*b_y^4*mu_x2*mu_y2^3 + 25*B_y^3*b_y^5*mu_y2^4 +
6*(3*B_x*B_y^2*b_x^4*b_y + B_x^2*B_y*b_x^3*b_y^2)*mu_x2^4 -
(11*B_x*B_y^2*b_x^4*b_y + 40*B_x^2*B_y*b_x^3*b_y^2)*mu_x2*mu_x3^2 -
6*(B_x*B_y^2*b_x^4*b_y + B_x^2*B_y*b_x^3*b_y^2)*mu_x2^2*mu_x4 +
3*((6*B_x*B_y^2*b_x^2*b_y^3 + 9*B_y^3*b_x^2*b_y^3 +
2*B_x^2*B_y*b_x*b_y^4)*mu_x2^2 - 2*(B_x*B_y^2*b_x^2*b_y^3 +
B_x^2*B_y*b_x*b_y^4)*mu_x4)*mu_y2^2 - 51*(B_y^3*b_x*b_y^4*mu_x2 +
B_y^3*b_y^5*mu_y2)*mu_y3^2 + ((36*B_x*B_y^2*b_x^3*b_y^2 +
B_y^3*b_x^3*b_y^2 + 12*B_x^2*B_y*b_x^2*b_y^3)*mu_x2^3 -
(11*B_x*B_y^2*b_x^3*b_y^2 + 40*B_x^2*B_y*b_x^2*b_y^3)*mu_x3^2 -
12*(B_x*B_y^2*b_x^3*b_y^2 + B_x^2*B_y*b_x^2*b_y^3)*mu_x2*mu_x4)*mu_y2 +
((11*B_y^3*b_x^3*b_y^2 + 91*B_x*B_y^2*b_x^2*b_y^3)*mu_x2*mu_x3 +
(11*B_y^3*b_x^2*b_y^3 +
91*B_x*B_y^2*b_x*b_y^4)*mu_x3*mu_y2)*mu_y3)*mu_y4 -
33*(B_x*B_y^2*b_x^3*b_y^2*mu_x2^2*mu_x3 +
2*B_x*B_y^2*b_x^2*b_y^3*mu_x2*mu_x3*mu_y2 +
B_x*B_y^2*b_x*b_y^4*mu_x3*mu_y2^2 - (B_y^3*b_x^2*b_y^3*mu_x2^2 +
2*B_y^3*b_x*b_y^4*mu_x2*mu_y2 + B_y^3*b_y^5*mu_y2^2)*mu_y3)*mu_y5 +
(B_y^3*b_x^3*b_y^2*mu_x2^3 + 3*B_y^3*b_x^2*b_y^3*mu_x2^2*mu_y2 +
3*B_y^3*b_x*b_y^4*mu_x2*mu_y2^2 + B_y^3*b_y^5*mu_y2^3)*mu_y6)*A -
45*(8*B_x^4*b_x^5*mu_x2^5 - 73*B_x^4*b_x^5*mu_x2^2*mu_x3^2)*mu_x4 +
90*(6*B_x^4*b_x^4*b_y*mu_x2^6 - 73*B_x^4*b_x^4*b_y*mu_x2^3*mu_x3^2 +
40*B_x^4*b_x^4*b_y*mu_x3^4 + 6*B_x^4*b_x^4*b_y*mu_x2^2*mu_x4^2 -
(12*B_x^4*b_x^4*b_y*mu_x2^4 -
73*B_x^4*b_x^4*b_y*mu_x2*mu_x3^2)*mu_x4)*mu_y2 +
90*(73*B_x^3*B_y*b_x^4*b_y*mu_x2^4*mu_x3 +
146*B_x*B_y^3*b_x^2*b_y^3*mu_x2*mu_x3*mu_y2^3 +
73*B_x*B_y^3*b_x*b_y^4*mu_x3*mu_y2^4 -
160*B_x^3*B_y*b_x^4*b_y*mu_x2*mu_x3^3 -
73*B_x^3*B_y*b_x^4*b_y*mu_x2^2*mu_x3*mu_x4 -
73*(B_x^3*B_y*b_x^2*b_y^3*mu_x3*mu_x4 - (B_x*B_y^3*b_x^3*b_y^2 +
B_x^3*B_y*b_x^2*b_y^3)*mu_x2^2*mu_x3)*mu_y2^2 +
2*(73*B_x^3*B_y*b_x^3*b_y^2*mu_x2^3*mu_x3 -
80*B_x^3*B_y*b_x^3*b_y^2*mu_x3^3 -
73*B_x^3*B_y*b_x^3*b_y^2*mu_x2*mu_x3*mu_x4)*mu_y2)*mu_y3 -
45*(8*B_x^2*B_y^2*b_x^4*b_y*mu_x2^5 + 24*B_y^4*b_x*b_y^4*mu_x2*mu_y2^4 +
8*B_y^4*b_y^5*mu_y2^5 - 73*B_x^2*B_y^2*b_x^4*b_y*mu_x2^2*mu_x3^2 -
8*B_x^2*B_y^2*b_x^4*b_y*mu_x2^3*mu_x4 - 8*(B_x^2*B_y^2*b_x*b_y^4*mu_x4 -
(3*B_y^4*b_x^2*b_y^3 + B_x^2*B_y^2*b_x*b_y^4)*mu_x2^2)*mu_y2^3 -
(73*B_x^2*B_y^2*b_x^2*b_y^3*mu_x3^2 +
24*B_x^2*B_y^2*b_x^2*b_y^3*mu_x2*mu_x4 - 8*(B_y^4*b_x^3*b_y^2 +
3*B_x^2*B_y^2*b_x^2*b_y^3)*mu_x2^3)*mu_y2^2 -
73*(B_y^4*b_x^2*b_y^3*mu_x2^2 + 2*B_y^4*b_x*b_y^4*mu_x2*mu_y2 +
B_y^4*b_y^5*mu_y2^2)*mu_y3^2 + 2*(12*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^4 -
73*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2*mu_x3^2 -
12*B_x^2*B_y^2*b_x^3*b_y^2*mu_x2^2*mu_x4)*mu_y2 +
146*(B_x*B_y^3*b_x^3*b_y^2*mu_x2^2*mu_x3 +
2*B_x*B_y^3*b_x^2*b_y^3*mu_x2*mu_x3*mu_y2 +
B_x*B_y^3*b_x*b_y^4*mu_x3*mu_y2^2)*mu_y3)*mu_y4)/A^7
}
# internal
calculateK2smp <- function(stats) {
c(K12two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K13two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K21two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K22two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K23two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K31two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K32two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K41two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K42two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K51two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']),
K61two(stats['A'], stats['B_x'], stats['B_y'], stats['b_x'], stats['b_y'],
stats['mu_x2'], stats['mu_x3'], stats['mu_x4'], stats['mu_x5'],
stats['mu_x6'], stats['mu_y2'], stats['mu_y3'], stats['mu_y4'],
stats['mu_y5'], stats['mu_y6']))
}
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