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Rayleigh-Ritz Variational Technique

A technique for computing eigenfunctions and eigenvalues. It proceeds by requiring

J=int_a^b[p(x)y_x^2-q(x)y^2]dx
(1)

to have a stationary value subject to the normalization condition

int_a^by^2w(x)dx=1
(2)

and the boundary conditions

py_xy|_a^b=0.
(3)

This leads to the Sturm-Liouville equation

d/(dx)(p(dy)/(dx))+qy+lambdawy=0,
(4)

which gives the stationary values of

F[y(x)]=(int_a^b(py_x^2-qy^2)dx)/(int_a^by^2wdx)
(5)

as

F[y_n(x)]=lambda_n,
(6)

where lambda_n are the eigenvalues corresponding to the eigenfunction y_n.

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References

Arfken, G. "Rayleigh-Ritz Variational Technique." §17.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 957-961, 1985.Rayleigh, J. W. "In Finding the Correction for the Open End of an Organ-Pipe." Phil. Trans. 161, 77, 1870.Ritz, W. "Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik." J. reine angew. Math. 135, 1-61, 1908.Whittaker, E. T. and Robinson, G. "The Rayleigh-Ritz Method for Minimum Problems." §184 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 381-382, 1967.

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Rayleigh-Ritz Variational Technique

Cite this as:

Weisstein, Eric W. "Rayleigh-Ritz Variational Technique." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rayleigh-RitzVariationalTechnique.html

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