% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/blockfit_solve.R
\name{.bf_case_gauss_gee}
\alias{.bf_case_gauss_gee}
\title{Gaussian Identity + GEE: Closed-Form Solve (Case a)}
\usage{
.bf_case_gauss_gee(
  X,
  y,
  K,
  p_expansions,
  order_list,
  VhalfInv,
  Lambda,
  L_partition_list,
  unique_penalty_per_partition,
  A,
  constraint_values,
  spline_cols,
  flat_cols,
  observation_weights
)
}
\arguments{
\item{X}{List of \eqn{K+1} unwhitened design matrices.}

\item{y}{List of \eqn{K+1} response vectors.}

\item{K}{Integer; number of interior knots.}

\item{p_expansions}{Integer; columns per partition.}

\item{order_list}{List of observation-index vectors by partition.}

\item{VhalfInv}{\eqn{\mathbf{V}^{-1/2}} matrix in the original
observation ordering.}

\item{Lambda}{Shared penalty matrix.}

\item{L_partition_list}{Partition-specific penalty matrices.}

\item{unique_penalty_per_partition}{Logical.}

\item{A}{Full constraint matrix.}

\item{constraint_values}{List of constraint RHS vectors.}

\item{spline_cols, flat_cols}{Integer vectors of column indices.}
}
\value{
A named list:
\describe{
  \item{beta_spline}{List of \eqn{K+1} spline coefficient vectors.}
  \item{beta_flat}{Numeric vector of flat coefficients.}
  \item{result}{List of \eqn{K+1} full per-partition coefficient
    vectors.}
}
}
\description{
When the response is Gaussian with identity link and a working
correlation structure is present, the whitened system
\eqn{\mathbf{V}^{-1/2}\mathbf{X}} is not block-diagonal, so
partition-wise backfitting does not apply.
This function performs a single closed-form solve that replicates
\code{get_B} Path 1a but in the backfitting context.
}
\details{
In other words, this is the dense correlated solve on the pooled
coefficient space. The supplement's Woodbury factorization
\eqn{\mathbf{G}^{-1} =
\mathbf{G}_{\mathrm{on}}^{-1} + \mathbf{G}_{\mathrm{off}}^{-1}}
is handled in \code{get_B()}, not here, so this helper always uses the
dense correlated solve.
}
\keyword{internal}
