Numerically reliable computation of characteristic polynomials

Misra, P. K.; Quintana, Enrique S.; Dooren, Paul M. Van

doi:10.1109/acc.1995.532688

Cited by 5 publications

(4 citation statements)

References 10 publications

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“…Misra, Quintana and Van Dooren [24] proved the backward stability of the Hyman algorithm for the general eigenvalue problem, which is consistent with the other backward stability results.…”

Section: Introductionsupporting

confidence: 86%

Hyman’s method revisited

Galántai¹,

Hegeds²

2009

Journal of Computational and Applied Mathematics

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“…Misra, Quintana and Van Dooren [24] proved the backward stability of the Hyman algorithm for the general eigenvalue problem, which is consistent with the other backward stability results.…”

Section: Introductionsupporting

confidence: 86%

Hyman’s method revisited

Galántai¹,

Hegeds²

2009

Journal of Computational and Applied Mathematics

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“…However b 1 , B 2 , and b 2 are functions of λ. To recover the coefficients of λ i requires the solution of n upper triangular systems [24].…”

Section: Krylov's Methods and Variantsmentioning

confidence: 99%

“…A structured backward error bound under certain conditions has been derived in [24], and iterative refinement is suggested for improving backward accuracy. However, it is not clear that this will help in general.…”

Section: Krylov's Methods and Variantsmentioning

confidence: 99%

La Budde's Method for Computing Characteristic Polynomials

Rehman,

Ipsen

2011

Preprint

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La Budde's method computes the characteristic polynomial of a real matrix A in two stages: first it applies orthogonal similarity transformations to reduce A to upper Hessenberg form H, and second it computes the characteristic polynomial of H from characteristic polynomials of leading principal submatrices of H. If A is symmetric, then H is symmetric tridiagonal, and La Budde's method simplifies to the Sturm sequence method. If A is diagonal then La Budde's method reduces to the Summation Algorithm, a Horner-like scheme used by the MATLAB function poly to compute characteristic polynomials from eigenvalues.We present recursions to compute the individual coefficients of the characteristic polynomial in the second stage of La Budde's method, and derive running error bounds for symmetric and nonsymmetric matrices. We also show that La Budde's method can be more accurate than poly, especially for indefinite and nonsymmetric matrices A. Unlike poly, La Budde's method is not affected by illconditioning of eigenvalues, requires only real arithmetic, and allows the computation of individual coefficients.

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“…. , n, can be obtained using proper numerically reliable algorithms (e.g., [10,11]). Alternatively, the following well-known Faddeev-Leverrier algorithm can also be applied:…”

Section: Equivalent Formsmentioning

confidence: 99%

An explicit solution to the matrix equation AX−XF=BY

Zhou

Duan

2005

Linear Algebra and its Applications

100

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A complete, general and explicit solution to the generalized Sylvester matrix equation AX − XF = BY , with the matrix F in a companion form, is proposed. The solution is in an extremely neat form represented by a symmetric operator matrix, a Hankel matrix and the controllability matrix of the matrix pair (A, B). Furthermore, several equivalent forms of this solution are also presented. Based on these presented results, explicit solutions to the normal Sylvester equation and the well-known Lyapunov matrix equation are also established. The results provide great convenience to the analysis of the solution to the equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, a simple and effective approach for parametric pole assignment is proposed.

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Numerically reliable computation of characteristic polynomials

Cited by 5 publications

References 10 publications

Hyman’s method revisited

Hyman’s method revisited

La Budde's Method for Computing Characteristic Polynomials

An explicit solution to the matrix equation AX−XF=BY

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