Conforming Rayleigh–Ritz procedure

Let $H$ be a nonzero complex Hilbert space whose inner product is linear in its first argument. Let $Q\subseteq H$ be a dense linear subspace, and let $q:Q\times Q\to \mathbb{C}$ be a Hermitian sesquilinear form: it is linear in its first argument, conjugate-linear in its second, and $q(u,v)=\overline{q(v,u)}$ for all $u,v\in Q$. In particular, $q(u,u)\in \mathbb{R}$. Write $q[u]=q(u,u)$. Assume $q[u]\ge -c\mathrm{\parallel }u{\mathrm{\parallel }}^2$ for some real $c\ge 0$, and assume $Q$ is complete in the form norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{q,c}^2=q[u]+(c+1)\mathrm{\parallel }u{\mathrm{\parallel }}^2$; this is the meaning of a closed semibounded form here.

For an integer $n\ge 1$, let ${\varphi }_1,\dots ,{\varphi }_n\in Q$ be linearly independent. The exact input matrices are the Gram matrix $G$ and form matrix $A$ with entries

$G_{ij}=⟨{\varphi }_j,{\varphi }_i⟩,A_{ij}=q({\varphi }_j,{\varphi }_i)(1\le i,j\le n)\mathrm{.}$

For every nonzero coefficient vector $z\in {\mathbb{C}}^n$, $z^{\ast }Gz=\mathrm{\parallel }{\sum }_j{z}_j{\varphi }_j{\mathrm{\parallel }}^2>0$, so $G$ is positive definite; the symmetry of $q$ makes $A$ Hermitian.

The procedure is:

1. Form $G$ and $A$ from the supplied vectors and form, using exact entries.
2. Diagonalize the positive definite Hermitian matrix $G$ and form its unique positive inverse square root $G^{-1\mathrm{/}2}$. Set $B=G^{-1\mathrm{/}2}A{G}^{-1\mathrm{/}2}$.
3. Diagonalize the Hermitian matrix $B$, listing its real eigenvalues ${\rho }_1\le \cdots \le {\rho }_n$ with multiplicity and selecting an orthonormal eigenbasis $y_1,\dots ,y_n$. For each $j$, set $c_j=G^{-1\mathrm{/}2}{y}_j$ and return $({\rho }_j,c_j)$.

Repeated eigenvalues may return any orthonormal basis of the corresponding eigenspace. The procedure uses trial vectors in $Q$ and does not require them to lie in the operator domain of an associated self-adjoint operator. It operates on exact matrix entries: if quadrature or an approximate eigensolver supplies numerical data, this procedure alone supplies no error enclosure or certification for those approximations.

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

Conformity means exactly that each trial vector lies in the stated form domain. A nonconforming grid vector is not automatically an element of that continuum form domain. The procedure returns the exact eigenvalues of its supplied matrices; any certified numerical Ritz interval must enclose integration error in both $G$ and $A$ and include the numerical eigensolver error separately.