Symbolic Computation on Complex Polynomial Solution of Differential Equations

Zhang, Jun

doi:10.1006/jsco.1996.0055

Cited by 3 publications

(1 citation statement)

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“…One flaw is that overlapping disks do not tessellate the complex plane efficiently. Disks can be replaced by ellipses by interpolating at points on the ellipse [36] (not a line segment inside the ellipse) and by more general regions by using Faber polynomials [54,28].…”

Section: The Radical Failure Of Radicalsmentioning

confidence: 99%

Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix

Boyd¹

2013

SIAM Rev.

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Abstract. When a function f (x) is holomorphic on an interval x ∈ [a, b], its roots on the interval can be computed by the following three-step procedure. First, approximate f (x) on [a, b] by a polynomial f N (x) using adaptive Chebyshev interpolation. Second, form the ChebyshevFrobenius companion matrix whose elements are trivial functions of the Chebyshev coefficients of the interpolant f N (x). Third, compute all the eigenvalues of the companion matrix. The eigenvalues λ which lie on the real interval λ ∈ [a, b] are very accurate approximations to the zeros of f (x) on the target interval. (To minimize cost, the adaptive phase can automatically subdivide the interval, applying the Chebyshev rootfinder separately on each subinterval, to keep N bounded or to solve rare "dynamic range" complications.)We also discuss generalizations to compute roots on an infinite interval, zeros of functions singular on the interval [a, b], and slightly complex roots. The underlying ideas are undergraduate-friendly, but link the disparate fields of algebraic geometry, linear algebra, and approximation theory.

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Section: The Radical Failure Of Radicalsmentioning

confidence: 99%