The equivalence of the constrained Rayleigh quotient and Newton methods for matrix polynomials expressed in different polynomial bases along with the confluent case

References

• Amiraslani , A. 2006 . “ New algorithms for matrices, polynomials and matrix polynomials ” . Department of Applied Mathematics, University of Western Ontario . Ph.D. diss
   Google Scholar
• Amiraslani , A. , Aruliah , D. A. and Corless , R. M. 2007 . Block LU factors of generalized companion matrix pencils . Theoret. Comput. Sci. , 381 ( 1–3 ) : 134 – 147 .
   Google Scholar
• Amiraslani , A. , Corless , R. M. and Lancaster , P. 2009 . Linearization of matrix polynomials expressed in polynomial bases . IMA J. Numer. Anal. , 29 : 141 – 157 .
   Web of Science ®Google Scholar
• Aruliah , D. A. , Corless , R. M. , Gonzalez-Vega , L. and Shakoori , A. 2007 . “ Geometric applications of the Bézout matrix in the bivariate tensor-product Lagrange basis ” . Research Report, Ontario Research Centre for Computer Algebra (ORCCA), TR-07-02, London, ON, Canada
   Google Scholar
• Barnett , S. 1983 . Polynomials and Linear Control Systems , New York : Marcel Dekker .
   Google Scholar
• Berrut , J. and Trefethen , L. 2004 . Barycentric Lagrange Interpolation . SIAM Rev. , 46 ( 4 ) : 501 – 517 .
   Google Scholar
• Corless , R. M. and Litt , G. 2001 . “ Generalized companion matrices for polynomials not expressed in monomial bases ” . London, ON, Canada : Department of Applied Mathematics, University of Western Ontario . Tech. Report
   Google Scholar
• Crandall , S. H. 1951 . Iterative procedures related to relaxation methods for eigenvalue problems . R. Soc. Lond. Proc. Ser. A , 207 : 416 – 423 .
   Google Scholar
• Demmel , J. W. 1997 . Applied Numerical Linear Algebra , Philadelphia, PA : SIAM .
   Google Scholar
• Farin , G. 1997 . Curves and Surfaces for Computer-Aided Geometric Design , San Diego : Academic Press .
   Google Scholar
• Farouki , R. T. , Goodman , T. N.T. and Sauer , T. 2003 . Construction of orthogonal bases for polynomials in Bernstein form on triangular simplex domains . Comput. Aided Geom. Design. , 20 : 209 – 230 .
   Web of Science ®Google Scholar
• Frazer , R. A. , Duncan , W. J. and Collar , W. J. 1952 . Elementary Matrices and Some Applications to Dynamics and Differential Equations , UK : Cambridge University Press .
   Google Scholar
• Gantmacher , F. R. 1990 . The Theory of Matrices , Vol. 1 , New York : American Mathematical Society Chelsea Publication .
   Google Scholar
• Gautschi , W. 2004 . Orthogonal Polynomials: Computation and Approximation , Oxford : Clarendon .
   Google Scholar
• Gohberg , I. , Lancaster , P. and Rodman , L. 1982 . Matrix Polynomials , New York : Academic Press .
   Google Scholar
• Golub , G. H. and Van Loan , C. F. 1996 . Matrix Computations , Baltimore, MD : Johns Hopkins University Press .
   Google Scholar
• Higham , N. 2002 . Accuracy and Stability of Numerical Algorithms , Philadelphia, PA : SIAM .
   Google Scholar
• Higham , N. 2004 . The numerical stability of barycentric Lagrange interpolation . IMA J. Numer. Anal. , 24 : 547 – 556 .
   Web of Science ®Google Scholar
• Jenkins , M. A. and Traub , J. F. 1970 . A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration . Numer. Math. , 14 : 252 – 263 .
   Google Scholar
• Jónsson , G. F. and Vavasis , S. 2005 . Solving polynomials with small leading coefficients . SIAM J. Matrix Anal. Appl. , 26 : 400 – 414 .
   Google Scholar
• Kahan , W. and Fateman , R. J. 1999 . Symbolic computation of divided differences . ACM SIGSAM Bull. , 33 ( 2 ) : 7 – 28 .
   Google Scholar
• Lakshman , Y. and Saunders , B. 1995 . Sparse polynomial interpolation in non-standard bases . SIAM J. Comput. , 24 ( 2 ) : 387 – 397 .
   Google Scholar
• Lancaster , P. 1961 . A generalized Rayleigh quotient iteration for lambda-matrices . Arch. Rat. Mech. Anal. , 8 : 309 – 322 .
   Google Scholar
• Lancaster , P. 1966 . Lambda-matrices and Vibrating Systems , Oxford : Pergamon . (Dover, Mineola, NY, 2002)
   Google Scholar
• Lorentz , G. , Jetter , K. and Riemenschneider , S. D. 1983 . Birkhoff Interpolation , Reading, MA : Addison-Wesley .
   Google Scholar
• Ostrowski , A. M. 1958 . On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, II . Arch. Rat. Mech. Anal. , 1 : 233 – 241 . 2 (1959), pp. 423–428
   Google Scholar
• O'Leary , D. P. and Stewart , G. W. 1998 . On the convergence of a new Rayleigh quotient method with applications to large eigenproblems . Electron. Trans. Numer. Anal. , 7 : 182 – 189 . (electronic)
   Google Scholar
• Parlett , B. N. 1974 . The Rayleigh quotient iteration and some generalizations for nonnormal matrices . Math. Comput. , 28 : 679 – 693 .
   Web of Science ®Google Scholar
• Parlett , B. N. 1980 . The Symmetric Eigenvalue Problem , 16 – 35 . Englewood Cliffs, NJ : Prentice-Hall .
   Google Scholar
• Pearson , C. E. 1990 . Handbook of Applied Mathematics , New York : Van Nostrand Reinhold .
   Google Scholar
• Rivlin , T. 1990 . Chebyshev Polynomials , New York : Wiley .
   Google Scholar
• Schwetlick , H. and Lösche , R. 2000 . A generalized Rayleigh quotient iteration for computing simple eigenvalues of nonnormal matrices . Z. Angew. Math. Mech. , 80 ( 1 ) : 9 – 25 .
   Google Scholar
• Shakoori , A. 2007 . “ Bivariate polynomial solver by values ” . University of Western Ontario . Ph.D. Diss., Applied Mathematics
   Google Scholar
• Tapia , R. A. and Whitley , D. L. 1998 . The project of Newton method had order 1+√2 for the symmetric eigenvalue problem . SIAM J. Numer. Anal. , 25 ( 6 ) : 1376 – 1382 .
   Google Scholar
• Toh , K. C. and Trefethen , L. N. 1994 . Pseudozeros of polynomials and pseudospectra of companion matrices . Numer. Math. , 68 : 403 – 425 .
   Web of Science ®Google Scholar
• Trefethen , L. N. 2001 . Spectral Methods in Matlab , Philadelphia, PA : SIAM .
   Google Scholar