Rayleigh–Ritz method

Remarks

The Ritz values are the stationary values of the Rayleigh quotient $Q(u, u) / \ip{u}{u}$ on the span $V$ of the trial vectors. If the spectrum of $A$ below the essential spectrum consists of eigenvalues $\lambda_1 \le \lambda_2 \le \cdots$, the min-max theorem gives $\lambda_k \le \mu_k$ for $k = 1, \dots, m$. For a Hermitian $n \times n$ matrix $A$ and a matrix $W$ with $m$ orthonormal columns, the Ritz values are the eigenvalues of $W^* A W$.

Example: $H = L^2(0, 1)$ and $A = -\frac{d^2}{dx^2}$ with Dirichlet conditions, so $D(Q) = H^1_0(0, 1)$, $Q(u, u) = \int_0^1 \abs{u'}^2 \, dx$ and $\lambda_1 = \pi^2$. With $m = 1$ and $\varphi_1(x) = x (1 - x)$, which vanishes at $0$ and $1$, one gets $K = \int_0^1 (1 - 2x)^2 \, dx = \frac{1}{3}$ and $M = \int_0^1 x^2 (1 - x)^2 \, dx = \frac{1}{30}$. So $\mu_1 = 10$, an upper bound for $\pi^2 \approx 9.8696$.

Non-example: the trial function $\varphi_1 = 1$ is not in $H^1_0(0, 1)$, because it does not vanish at the ends. Its Rayleigh quotient is $0 < \pi^2$, so the upper bound fails when the trial vectors leave the form domain.