# Pentadiagonal Companion Matrices

Brydon Eastman; Kevin N. Vander Meulen

Special Matrices (2016)

- Volume: 4, Issue: 1, page 13-30
- ISSN: 2300-7451

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topBrydon Eastman, and Kevin N. Vander Meulen. "Pentadiagonal Companion Matrices." Special Matrices 4.1 (2016): 13-30. <http://eudml.org/doc/276929>.

@article{BrydonEastman2016,

abstract = {The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler factorization, up to transpose, given only its corner entries.},

author = {Brydon Eastman, Kevin N. Vander Meulen},

journal = {Special Matrices},

keywords = {companion matrices; pentadiagonal matrices; Fiedler companion matrices; Hessenberg matrices; algorithms; zeros of polynomials},

language = {eng},

number = {1},

pages = {13-30},

title = {Pentadiagonal Companion Matrices},

url = {http://eudml.org/doc/276929},

volume = {4},

year = {2016},

}

TY - JOUR

AU - Brydon Eastman

AU - Kevin N. Vander Meulen

TI - Pentadiagonal Companion Matrices

JO - Special Matrices

PY - 2016

VL - 4

IS - 1

SP - 13

EP - 30

AB - The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler factorization, up to transpose, given only its corner entries.

LA - eng

KW - companion matrices; pentadiagonal matrices; Fiedler companion matrices; Hessenberg matrices; algorithms; zeros of polynomials

UR - http://eudml.org/doc/276929

ER -

## References

top- [1] J.L. Aurentz, R. Vandebril, and D.S. Watkins, Fast computation of the zeros of a polynomial via factorization of the companion matrix, SIAM J. Sci. Comput.35 (2013) A255 – A269.[WoS] Zbl1264.65074
- [2] J.L. Aurentz, R. Vandebril, and D.S. Watkins, Fast computation of eigenvalues of companion, comrade, and related matrices, BIT Numer. Math.54 (2014) 7–30. Zbl1293.65052
- [3] T. Bella, V. Olshevsky, and P. Zhlobich, A quasiseparable approach to five-diagonal CMV and Fiedler matrices, Linear Algebra Appl.434(4) (2011) 957–976.[WoS] Zbl1215.15014
- [4] B. Bevilacqua, G.M. Del Corso, and L. Gemignani, A CMV–based eigensolver for companion matrices, SIAM J. Matrix Anal. Appl.36(3) (2015) 1046–1068.[WoS][Crossref] Zbl1321.65076
- [5] D.A. Bini, P. Boito, Y. Eidelman, L. Gemignani, and I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl.432(8) (2010) 2006–2031.[WoS] Zbl1188.65039
- [6] D.A. Bini, F. Daddi, and L. Gemignani, On the shifted QR iteration applied to companion matrices, Electron. Trans. Numer. Anal.18 (2004) 137–152. Zbl1066.65039
- [7] S. Chandrasekaran, M. Gu, J. Xia, and J. Zhu, A fast QR algorithm for companion matrices, Oper. Theory Adv. Appl.179 (2008) 111–143. Zbl1136.65041
- [8] F. De Terán, F. Dopico, and D.S. Mackey, Fiedler companion linearizations and the recovery of minimal indices, SIAM J. Matrix Anal. Appl., 31:4 (2010) 2181–2204.[WoS][Crossref] Zbl1205.15024
- [9] F. De Terán, F. Dopico, and J. Pérez, Condition numbers for inversion of Fiedler companion matrices, Linear Algebra Appl.439 (2013) 944–981.[WoS] Zbl1281.15004
- [10] B. Eastman, I.-J. Kim, B. Shader, and K.N. Vander Meulen, Companion matrix patterns, Linear Algebra Appl.463 (2014) 255–272.[WoS] Zbl1310.15015
- [11] M. Fiedler, A note on companion matrices, Linear Algebra Appl.372 (2003) 325–331. Zbl1031.15014
- [12] C. Garnett, B. Shader, C. Shader, and P. van den Driessche, Characterization of a family of generalized companion matrices Linear Algebra Appl. (2015), in press, .[Crossref] Zbl06570135
- [13] N.J. Higham, D.S. Mackey, N. Mackey, and F. Tisseur, Symmetric linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl.29 (2006) 143–159.[WoS] Zbl1137.15006
- [14] C. Ma and X. Zhan, Extremal sparsity of the companion matrix of a polynomial, Linear Algebra Appl.438 (2013) 621–625.[WoS] Zbl1269.15007
- [15] S. Vologiannidis and E.N. Antoniou, A permuted factors approach for the linearization of polynomial matrices, Math. Control Signals Systems22 (2011) 317–342.[WoS] Zbl1248.93045

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