dot-bf_constrained_flat_update: Constrained Flat Update Given Current Spline Solution

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Constrained Flat Update Given Current Spline Solution

Description

Solves for flat coefficients subject to the residual equality constraint imposed by the full constraint system, conditional on the current spline block solution.

Usage

.bf_constrained_flat_update(
  XfWXf_pen,
  Xfr,
  A_full,
  constraint_values,
  beta_spline,
  flat_rows_all,
  spline_rows_all,
  spline_cols,
  flat_cols,
  p_expansions,
  K,
  nc_flat
)

Arguments

XfWXf_pen	Penalized flat Gram matrix (nc_f \times nc_f).
Xfr	Right-hand side cross-product for flat update (nc_f \times 1).
A_full	Full constraint matrix (P \times R).
constraint_values	List of constraint RHS vectors.
beta_spline	List of K+1 current spline coefficient vectors.
flat_rows_all	Integer vector of flat row indices in the full P-space.
spline_rows_all	Integer vector of spline row indices in the full P-space.
spline_cols	Integer vector of spline column indices within each partition.
flat_cols	Integer vector of flat column indices within each partition.
p_expansions	Integer; columns per partition.
K	Integer; number of interior knots.
nc_flat	Integer; number of flat columns.

Details

When the full constraint system is \mathbf{A}^\top \boldsymbol{\beta} = \mathbf{c}, and we partition \boldsymbol{\beta} into spline and flat blocks, the flat update must satisfy:

\mathbf{A}_{\mathrm{flat}}^\top \boldsymbol{\beta}_{\mathrm{flat}} = \mathbf{c} - \mathbf{A}_{\mathrm{spline}}^\top \boldsymbol{\beta}_{\mathrm{spline}}

This function solves the penalized least-squares problem for flat coefficients subject to this residual equality constraint using a Lagrangian approach.

Value

Numeric vector of flat coefficients (nc_f \times 1).

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