range_qfr {qfratio}

Get range of ratio of quadratic forms

Description

range_qfr(): internal function to obtain the possible range of a ratio of quadratic forms, \frac{ \mathbf{x^{\mathit{T}} A x} }{ \mathbf{x^{\mathit{T}} B x} } .

gen_eig() is an internal function to obtain generalized eigenvalues, i.e., roots of \det{\mathbf{A} - \lambda \mathbf{B}} = 0, which are the eigenvalues of \mathbf{B}^{-1} \mathbf{A} if \mathbf{B} is nonsingular.

Usage

range_qfr(
  A,
  B,
  eigB = eigen(B, symmetric = TRUE),
  tol = .Machine$double.eps * 100,
  t = 0.001
)

gen_eig(
  A,
  B,
  eigB = eigen(B, symmetric = TRUE),
  Ad = with(eigB, crossprod(crossprod(A, vectors), vectors)),
  tol = .Machine$double.eps * 100,
  t = 0.001
)

Arguments

A, B

Symmetric matrices. No check is done.

eigB

Result of eigen(B) can be passed when already computed

tol

Tolerance to determine numerical zero

t

Tolerance used to determine whether estimates are numerically stable; t in Jennings et al. (1978).

Ad

A rotated with eigenvectors of B can be passed when already computed

Details

gen_eig() solves the generalized eigenvalue problem with Jennings et al.'s (1978) algorithm. The sign of infinite eigenvalue (when present) cannot be determined from this algorithm, so is deduced as follows: (1) \mathbf{A} and \mathbf{B} are rotated by the eigenvectors of \mathbf{B}; (2) the submatrix of rotated \mathbf{A} corresponding to the null space of \mathbf{B} is examined; (3) if this is nonnegative (nonpositive) definite, the result must have positive (negative, resp.) infinity; if this is indefinite, the result must have both positive and negative infinities; if this is (numerically) zero, the result must have NaN. The last case is expeted to happen very rarely, as in this case Jennings algorithm would fail. This is where the null space of \mathbf{B} is a subspace of that of \mathbf{A}, so that the range of ratio of quadratic forms can be well-behaved. range_qfr() tries to detect this case and handle the range accordingly, but if that is infeasible it returns c(-Inf, Inf).

References

Jennings, A., Halliday, J. and Cole, M. J. (1978) Solution of linear generalized eigenvalue problems containing singular matrices. Journal of the Institute of Mathematics and Its Applications, 22, 401–410. doi:10.1093/imamat/22.4.401.