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1979•Mathematics of ComputationOpen access

Estimating the largest eigenvalue of a positive definite matrix

Dianne P. O’Leary, G. W. Stewart, James S. Vandergraft

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Abstract

The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the dominant eigenvalue after a very few iterations-even when the order of the matrix is large.

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What this paper is about

The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the dominant eigenvalue after a very few iterations-even when the order of the matrix is large.

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Available abstract

The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the dominant eigenvalue after a very few iterations-even when the order of the matrix is large.

Key concepts: Rayleigh quotient iteration, Mathematics, Rayleigh quotient, Eigenvalues and eigenvectors, Inverse iteration, Power iteration, Positive-definite matrix, Convergence (economics)

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