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#' #### OurICsurvPHgrad calculates the gradient of the survival likleihood under the I-spline model of Wang et al. (2016)
#' # The code heavily uses code from fast.PH.ICsurv.EM function in ICsurv. See also OurICsurvPH
#' #' @title FUNCTION_TITLE
#' #' @description FUNCTION_DESCRIPTION
#' #' @param Li PARAM_DESCRIPTION
#' #' @param Ri PARAM_DESCRIPTION
#' #' @param Xp PARAM_DESCRIPTION
#' #' @param n.int PARAM_DESCRIPTION
#' #' @param order PARAM_DESCRIPTION
#' #' @param g1 PARAM_DESCRIPTION
#' #' @param b1 PARAM_DESCRIPTION
#' #' @param tol PARAM_DESCRIPTION
#' #' @param t.seq PARAM_DESCRIPTION
#' #' @param equal PARAM_DESCRIPTION, Default: TRUE
#' #' @return OUTPUT_DESCRIPTION
#' #' @details DETAILS
#' #' @examples
#' #' \dontrun{
#' #' if(interactive()){
#' #' #EXAMPLE1
#' #' }
#' #' }
#' # @rdname OurICsurvPHgrad
#' OurICsurvPHgrad <- function (Li, Ri, Xp, n.int, order, g1, b1, tol, t.seq, equal = TRUE)
#' {
#' P <- length(b0)
#' L <- length(g0)
#' N <- length(Li)
#' ti <- c(Li, Ri)
#' if (equal == TRUE) {
#' ti.max <- max(ti) + 1e-05
#' ti.min <- min(ti) - 1e-05
#' knots <- seq(ti.min, ti.max, length.out = (n.int + 2))
#' }
#' if (equal == FALSE) {
#' id <- seq(0, 1, length.out = (n.int + 2))
#' id <- id[-c(1, (n.int + 2))]
#' ti.max <- max(ti) + 1e-05
#' ti.min <- min(ti) - 1e-05
#' knots <- c(ti.min, quantile(ti, id), ti.max)
#' }
#' bRi <- t(Ispline(x = Ri, order = order, knots = knots))
#' bLi <- t(Ispline(x = Li, order = order, knots = knots))
#' bt <- t(Ispline(x = t.seq, order = order, knots = knots))
#'
#' GRi <- bRi %*% g1
#' GLi <- bLi %*% g1
#' xb <- Xp %*% b1
#' LL <- exp(-GLi * exp(xb)) - exp(-GRi * exp(xb))
#' ll <- log(LL)
#'
#' u.b <- as.vector((1/LL)*(exp(-GLi * exp(xb)) * ( -GLi * exp(xb)) - exp(-GRi * exp(xb)) * (-GRi * exp(xb)))) * Xp
#' u.g <- as.vector((1/LL)*(exp(-GLi * exp(xb)) * ( -GLi * exp(xb)) ))* bLi -
#' as.vector((1/LL)*(exp(-GRi * exp(xb)) * (-GRi * exp(xb))))*bRi
#' u <- cbind(u.b, u.g)
#' return(u)
#' }
#'
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