Rayleigh-Ritz approximation for the linear response eigenvalue problem (Q2923370)

One important question in random phase approximation is to compute a few eigenpairs associated with the smallest positive eigenvalues of the following eigenvalue problem: NEWLINE\[NEWLINE \mathcal{H}w:= \begin{bmatrix} A & B \\ -B & -A \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix}= \lambda \begin{bmatrix} u \\ v \end{bmatrix}, NEWLINE\]NEWLINE where \( A, B \in \mathbb{R}^{n \times n}\) are both symmetric matrices and \( \begin{bmatrix} A & B \\ B & A \end{bmatrix} \) is positive definite. This paper is concerned with approximation accuracy relationships between a pair of approximate deflating subspaces and approximate eigenvalues extracted by the pair. The lower and upper bounds on eigenvalue approximation errors are obtained. They are useful in analyzing numerical solutions to linear response eigenvalue problem.