Rayleigh quotient
/ RAY-lee /
How can you estimate the lowest natural frequency of a drum without solving the drum's equation exactly? The Rayleigh quotient is the answer: a single number, built from any trial shape you guess, that overestimates the lowest eigenvalue and gets closer the better your guess. It converts an eigenvalue problem into a minimization, which is often far easier to handle approximately.
For a Sturm-Liouville operator the Rayleigh quotient of a trial function u (satisfying the boundary conditions) is R[u] = (integral of [p (u')^2 - q u^2] dx) divided by (integral of u^2 w dx). The numerator is, physically, a stored energy (a bending or stretching energy plus a potential term); the denominator is a weighted norm. Two facts make it powerful. First, if you plug in the exact eigenfunction y_n you get back its eigenvalue lambda_n exactly. Second, the quotient is stationary at each eigenfunction, and minimized over all admissible u by the lowest eigenvalue: lambda_1 equals the minimum of R[u]. So for any trial u that is not the true ground state, R[u] is greater than lambda_1 — an upper bound you can compute.
This is the engine of the Rayleigh-Ritz method and a staple of physics and engineering. Guess a simple trial function with a couple of free parameters, compute R[u] as a function of those parameters, and minimize; the result is a rigorous upper bound on the fundamental eigenvalue (the lowest vibration frequency, the ground-state energy in quantum mechanics, the buckling load of a column). Even a crude guess often lands within a few percent, because the quotient is stationary near the true mode — small errors in the shape produce only second-order errors in the eigenvalue.
For y'' + lambda y = 0 on [0, 1] with y(0) = y(1) = 0, the true lambda_1 is pi^2 = 9.8696. Try the parabola u = x(1 - x): R[u] = (integral of (u')^2) / (integral of u^2) = (1/3)/(1/30) = 10. That is just 1.3 percent above the exact value, from a one-line guess.
A crude trial shape overestimates the fundamental eigenvalue by barely a percent — the hallmark of the variational method.
The minimum of the Rayleigh quotient gives only the lowest eigenvalue, and always from above; to reach lambda_2 you must restrict trial functions to be orthogonal to the ground state, and the estimate is still an over-, never under-, estimate.