matinvsqrt: Calculate Matrix Inverse Square Root for Symmetric Matrices

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Calculate Matrix Inverse Square Root for Symmetric Matrices

Description

Calculate Matrix Inverse Square Root for Symmetric Matrices

Usage

matinvsqrt(mat)

Arguments

mat	A symmetric matrix \textbf{M}

Details

Uses an eigenvalue-decomposition-based approach.

Non-positive eigenvalues are set to 0 before taking inverse fourth roots.

This implementation is particularly useful for whitening procedures in GLMs with correlation structures and for computing variance-covariance matrices under constraints.

You can use this to help construct a custom VhalfInv_fxn for lgspline. When only VhalfInv is supplied there, the corresponding Vhalf is reconstructed internally by inversion for the GEE code paths.

Value

A matrix \textbf{B} such that \textbf{B}\textbf{B} equals the Moore-Penrose-style inverse on the positive-eigenvalue subspace, with non-positive components truncated to 0.

Examples

## Identity matrix
m1 <- diag(2)
matinvsqrt(m1)  # Returns identity matrix

## Compound symmetry correlation matrix
rho <- 0.5
m2 <- matrix(rho, 3, 3) + diag(1-rho, 3)
B <- matinvsqrt(m2)
# Verify: B %**% B approximately equals solve(m2)
all.equal(B %**% B, solve(m2))

## Example for GLM correlation structure
n_blocks <- 2  # Number of subjects
block_size <- 3  # Measurements per subject
rho <- 0.7  # Within-subject correlation
# Correlation matrix for one subject
R <- matrix(rho, block_size, block_size) +
     diag(1-rho, block_size)
## Full correlation matrix for all subjects
V <- kronecker(diag(n_blocks), R)
## Create whitening matrix
VhalfInv <- matinvsqrt(V)

# Example construction of VhalfInv_fxn for lgspline
VhalfInv_fxn <- function(par) {
  rho <- tanh(par)  # Transform parameter to (-1, 1)
  R <- matrix(rho, block_size, block_size) +
       diag(1-rho, block_size)
  kronecker(diag(n_blocks), matinvsqrt(R))
}

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