dot-bf_lagrangian_project: Lagrangian Projection for the Spline-Only Subproblem

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Lagrangian Projection for the Spline-Only Subproblem

Description

Projects the adjusted cross-product \mathbf{X}_s^{\top} \tilde{\mathbf{y}} through \mathbf{G}_s^{1/2} and removes the component along \mathbf{G}_s^{1/2}\mathbf{A}_s to enforce the smoothness constraints in the spline block. When cv_spline contains nonzero values (e.g.\ from no_intercept), the corresponding contribution is added back after projection.

Usage

.bf_lagrangian_project(
  Xy_adj,
  Ghalf_cur,
  GhalfA_cur,
  cv_spline,
  K,
  nc_spline,
  A_spline,
  parallel_qr = FALSE,
  cl = NULL
)

Arguments

Xy_adj	List of K+1 adjusted cross-product vectors (nc_s \times 1).
Ghalf_cur	List of K+1 \mathbf{G}_s^{1/2} matrices (each nc_s \times nc_s).
GhalfA_cur	(nc_s \cdot (K+1)) \times nca_s matrix formed by row-stacking \mathbf{G}_s^{1/2,(k)} \mathbf{A}_s^{(k)}.
cv_spline	List of constraint right-hand-side vectors for the spline subproblem (may be empty).
K	Integer; number of interior knots.
nc_spline	Integer; number of spline columns.
A_spline	Spline-only constraint matrix.

Value

List of K+1 coefficient vectors (nc_s \times 1).

lgspline documentation built on May 8, 2026, 5:07 p.m.