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| '''Rayleigh quotient iteration''' is an [[eigenvalue algorithm]] which extends the idea of the [[inverse iteration]] by using the [[Rayleigh quotient]] to obtain increasingly accurate [[eigenvalue]] estimates.
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| Rayleigh quotient iteration is an [[iterative method]], that is, it must be repeated until it [[Limit of a sequence|converges]] to an answer (this is true for all eigenvalue algorithms). Fortunately, very rapid convergence is guaranteed and no more than a few iterations are needed in practice. The Rayleigh quotient iteration algorithm [[rate of convergence|converges cubically]] for Hermitian or symmetric matrices, given an initial vector that is sufficiently close to an [[EigenVector|eigenvector]] of the [[Matrix (mathematics)|matrix]] that is being analyzed.
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| == Algorithm ==
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| The algorithm is very similar to inverse iteration, but replaces the estimated eigenvalue at the end of each iteration with the Rayleigh quotient. Begin by choosing some value <math>\mu_0</math> as an initial eigenvalue guess for the Hermitian matrix <math>A</math>. An initial vector <math>b_0</math> must also be supplied as initial eigenvector guess.
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| Calculate the next approximation of the eigenvector <math>b_{i+1}</math> by
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| <math>
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| b_{i+1} = \frac{(A-\mu_i I)^{-1}b_i}{||(A-\mu_i I)^{-1}b_i||},
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| </math><br>
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| where <math>I</math> is the identity matrix,
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| and set the next approximation of the eigenvalue to the Rayleigh quotient of the current iteration equal to<br>
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| <math>
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| \mu_i = \frac{b^*_i A b_i}{b^*_i b_i}.
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| </math>
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| To compute more than one eigenvalue, the algorithm can be combined with a deflation technique.
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| == Example == | |
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| Consider the matrix
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| :<math> | |
| A =
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| \left[\begin{matrix}
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| 1 & 2 & 3\\
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| 1 & 2 & 1\\
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| 3 & 2 & 1\\
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| \end{matrix}\right]
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| </math>
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| for which the exact eigenvalues are <math>\lambda_1 = 3+\sqrt5</math>, <math>\lambda_2 = 3-\sqrt5</math> and <math>\lambda_3 = -2</math>, with corresponding eigenvectors
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| :<math>v_1 = \left[
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| \begin{matrix}
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| 1 \\
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| \varphi-1 \\
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| 1 \\
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| \end{matrix}\right]</math>, <math>v_2 = \left[
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| \begin{matrix}
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| 1 \\
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| -\varphi \\
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| 1 \\
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| \end{matrix}\right]</math> and <math>v_3 = \left[
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| \begin{matrix}
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| 1 \\
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| 0 \\
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| 1 \\
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| \end{matrix}\right]</math>.
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| (where <math>\textstyle\varphi=\frac{1+\sqrt5}2</math> is the golden ratio).
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| The largest eigenvalue is <math>\lambda_1 \approx 5.2361</math> and corresponds to any eigenvector proportional to <math>v_1 \approx \left[
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| \begin{matrix}
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| 1 \\
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| 0.6180 \\
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| 1 \\
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| \end{matrix}\right].
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| </math>
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| We begin with an initial eigenvalue guess of
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| :<math>b_0 =
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| \left[\begin{matrix}
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| 1 \\
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| 1 \\
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| 1 \\
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| \end{matrix}\right], ~\mu_0 = 200</math>.
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| Then, the first iteration yields
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| :<math>b_1 \approx
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| \left[\begin{matrix}
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| -0.57927 \\
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| -0.57348 \\
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| -0.57927 \\
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| \end{matrix}\right], ~\mu_1 \approx 5.3355
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| </math> | |
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| the second iteration,
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| :<math>b_2 \approx | |
| \left[\begin{matrix}
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| 0.64676 \\
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| 0.40422 \\
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| 0.64676 \\
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| \end{matrix}\right], ~\mu_2 \approx 5.2418
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| </math>
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| and the third,
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| :<math>b_3 \approx | |
| \left[\begin{matrix}
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| -0.64793 \\
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| -0.40045 \\
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| -0.64793 \\
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| \end{matrix}\right], ~\mu_3 \approx 5.2361
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| </math>
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| from which the cubic convergence is evident.
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| == Octave Implementation ==
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| The following is a simple implementation of the algorithm in [[GNU Octave|Octave]].
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| <source lang="matlab">
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| function x = rayleigh(A,epsilon,mu,x)
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| x = x / norm(x);
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| y = (A-mu*eye(rows(A))) \ x;
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| lambda = y'*x;
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| mu = mu + 1 / lambda
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| err = norm(y-lambda*x) / norm(y)
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| while err > epsilon
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| x = y / norm(y);
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| y = (A-mu*eye(rows(A))) \ x;
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| lambda = y'*x;
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| mu = mu + 1 / lambda
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| err = norm(y-lambda*x) / norm(y)
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| end
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| end
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| </source>
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| == See also ==
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| * [[Power iteration]]
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| * [[Inverse iteration]]
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| ==References==
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| * Lloyd N. Trefethen and David Bau, III, ''Numerical Linear Algebra'', Society for Industrial and Applied Mathematics, 1997. ISBN 0-89871-361-7.
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| * Rainer Kress, "Numerical Analysis", Springer, 1991. ISBN 0-387-98408-9
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| {{Numerical linear algebra}}
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| [[Category:Numerical linear algebra]]
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