Jared L. Aurentz

Fast Computation of the Zeros of a Polynomial via Factorization of the Companion Matrix

A new fast algorithm for computing the zeros of a polynomial in

Fast and Backward Stable Computation of Roots of Polynomials

O(n^{2})

Chopping a Chebyshev Series

time using

Block Operators and Spectral Discretizations

O(n)

Fast and Backward Stable Computation of Roots of Polynomials, Part II: Backward Error Analysis; Companion Matrix and Companion Pencil

memory is developed. The eigenvalues of the Frobenius companion matrix are computed by applying a nonunitary analogue of Franciss implicitly shifted

A factorization of the inverse of the shifted companion matrix

QR

Fast computation of eigenvalues of companion, comrade, and related matrices

algorithm to a factored form of the matrix. The algorithm achieves high speed and low memory use by preserving the factored form. It also provides a residual and an error estimate for each root. Numerical tests confirm the high speed of the algorithm.

Fast and stable unitary QR algorithm

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Franciss implicitly shifted QR algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rank-one, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Franciss algorithm, and it is these properties that are exploited here. The matrix is represented as a product of 3n-1 Givens rotators plus the rank-one part, so only

A note on companion pencils

O(n)

Roots of Polynomials: on twisted QR methods for companion matrices and pencils

storage space is required. In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly. Franciss algorithm implemented on this representation requires only O(n) flops per iteration and thus