In minisymposium: Matrix Structures and Univariate Polynomial RootfindingThu 17:30–18:00, Room AV 91.12
The roots of a polynomial in Chebyshev basis can be computed by finding the eigenvalues of the colleague matrix. The colleague matrix is of symmetric-plus-rank-1 form; that is the sum of a symmetric matrix and a rank-1 matrix. We will show that the Cayley transformation can be used to transform this matrix to unitary-plus-rank-1 form. The unitary-plus-rank-1 eigenvalue problem is solved using the companion QR algorithm .
We will demonstrate that the transformation to unitary-plus-rank-1 form is fast. We will present some examples showing that the roots are computed accurately.
- J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, Fast and backward stable computation of roots of polynomials, SIAM Journal on Matrix Analysis and Applications, 36 (2015), pp. 942–973.