Compute Roots of Polynomial via Clipping Method Reporter: Lei Zhang Date: 2007/3/21

Outline History Review Bézier Clipping Quadratic Clipping Cubic Clipping Summary

Stuff Nishita, T., T. W. Sederberg, and M. Kakimoto. Ray tracing trimmed rational surface patches. Siggraph, 1990, Nishita, T., and T. W. Sederberg. Curve intersection using Bézier clipping. CAD, 1990, 22.9, Michael Barton, and Bert Juttler. Computing roots of polynomials by quadratic clipping. CAGD, in press. Lei Zhang, Ligang Liu, Bert Juttler, and Guojin Wang. Computing roots of polynomials by cubic clipping. To be submitted.

History Review Quadratic Equation  祖冲之 (429~500) 、祖日桓  花拉子米 (780~850)

Cubic Equation (Cardan formula) Tartaglia (1499~1557) Cardano (1501~1576)

Quartic Equation (Ferrari formula) Ferrari (1522~1565)

Equation  Lagrange (1736~1813)  Abel (1802~1829)  Galois (1811~1832)

Bezier Clipping Nishita, T., and T. W. Sederberg. Curve intersection using Bézier clipping. CAD, 1990, 22.9, Convex hull of control points of Bézier curve

Find the root of polynomial on the interval

Polynomial in Bézier form

Convex hull construction

The new interval

Algorithm

Convergence Rate  Single root: 2  Double root, etc: 1

Quadratic Clipping Michael Barton, and Bert Juttler. Computing roots of polynomials by quadratic clipping. CAGD, in press. Degree reduction of Bézier curve

The best quadratic approximant  (n+1) dimensional linear space of polynomials of degree n on [0, 1]  Bernstein-Bezier basis :  inner product:  is given, find quadratic polynomial such that is minimal

Degree reduction  Dual basis to the BB basis  Subspace,, Bert Juttler. The dual basis functions of the Bernstein polynomials. Advanced in Comoputational Mathematics. 1998, 8,

Degree reduction matrix  n=5, k=2

Error bound  Raising best quadratic function to degree n  Bound estimation

Bound Strip

Algorithm Convergence Rate  Quadratic clipping 3 1  Bezier clipping Single rootDouble rootTriple root

Proof of Convergence Rate

Computation effort comparison

Time cost per iteration (μs)

Numerical examples  Single roots

Double roots

Near double root

Future work  System of polynomials  Quadratic polynomial Cubic polynomial  Cubic clipping 4 2  Quadratic clipping 3 1  Bezier clipping Single rootDouble rootTriple root

Cubic Clipping

Cardano Formula Given a cubic equation

Single Roots Clone from quadratic clipping Proof

Double roots Proof

Triple roots Proof

Summary Furture Work  Quartic clipping (conjecture): cubic ->quartic polynomial singledoubletriplequadruple quartic55/25/35/4 cubic424/31 quadratic33/211 bezier2111

Thanks for your attention!