Common quadratic forms

Description

Common quadratic forms

Usage

xTAx(x, A)

xAxT(x, A)

xTAx_solve(x, A, ...)

xTAx_qrsolve(x, A, tol = 1e-07, ...)

sandwich_solve(A, B, ...)

xTAx_eigen(x, A, tol = sqrt(.Machine$double.eps), ...)

Arguments

Details

These are somewhat inspired by emulator::quad.form.inv() and others.

Functions

• xTAx(): Evaluate x'Ax for vector x and square matrix A.

• xAxT(): Evaluate xAx' for vector x and square matrix A.

• xTAx_solve(): Evaluate x'A^{-1}x for vector x and invertible matrix A using solve().

• xTAx_qrsolve(): Evaluate x'A^{-1}x for vector x and matrix A using QR decomposition and confirming that x is in the span of A if A is singular; returns rank and nullity as attributes just in case subsequent calculations (e.g., hypothesis test degrees of freedom) are affected.

• sandwich_solve(): Evaluate A^{-1}B(A')^{-1} for B a square matrix and A invertible.

• xTAx_eigen(): Evaluate x' A^{-1} x for vector x and matrix A (symmetric, nonnegative-definite) via eigendecomposition; returns rank and nullity as attributes just in case subsequent calculations (e.g., hypothesis test degrees of freedom) are affected.

  Decompose A = P L P' for L diagonal matrix of eigenvalues and P orthogonal. Then A^{-1} = P L^{-1} P'.

  Substituting,

  x' A^{-1} x = x' P L^{-1} P' x = h' L^{-1} h

  for h = P' x.