On the Convergence of the QR Algorithm with Origin Shifts for Normal Matrices

Huang, C. P.

doi:10.1093/imanum/1.1.127

Cited by 2 publications

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“…The proof of the convergence of the basic QR algorithm can be found in numerous papers [20,21,22], many of them with focus on the various shift strategies or the different types of matrices like unitary Hessenberg or real symmetric tridiagonal [5,23,24]. Whereas Parlett [21] and…”

Section: Proof Of Convergencementioning

confidence: 99%

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An extended QR‐solver for large profiled matrices

Ruess

Pahl

2009

Numerical Meth Engineering

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A new method for the solution of the standard eigenvalue problem with large symmetric profile matrices is presented. The method is based on the well-known QR-method for dense matrices. A new, flexible and reliable extension of the method is developed that is highly suited for the independent computation of any set of eigenvalues. In order to analyze the weak convergence of the method in the presence of clustered eigenvalues, the QR-method is studied. Two effective, stable and numerically cheap extensions are introduced to overcome the troublesome stagnation of the convergence. A repeated preconditioning process in combination with Jacobi rotations in the parts of the matrix with the strongest convergence is developed to significantly improve both local and global convergence. The extensions preserve the profile structure of the matrix. The efficiency of the new method is demonstrated with several examples.

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Section: Proof Of Convergencementioning

confidence: 99%

“…A has a q-fold eigenvalue −λ : Λ b only contains the submatrices Λ 1 , Λ 2 and A 22 . In analogy to case 2, A 11 converges to diagonal form.…”

Section: Eigenvalues Then the Iterated Matrix A S Converges To A Blomentioning

confidence: 99%

An extended QR‐solver for large profiled matrices

Ruess

Pahl

2009

Numerical Meth Engineering

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Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices

Wang

Gragg

2001

Math. Comp.

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Abstract. This paper shows that for unitary Hessenberg matrices the QR algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

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On the Convergence of the QR Algorithm with Origin Shifts for Normal Matrices

Cited by 2 publications

References 0 publications

An extended QR‐solver for large profiled matrices

An extended QR‐solver for large profiled matrices

Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices

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