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

2. Outline History Review Bézier Clipping Quadratic Clipping Cubic Clipping Summary

3. Stuff Nishita, T., T. W. Sederberg, and M. Kakimoto. Ray tracing trimmed rational surface patches. Siggraph, 1990, 337-345. Nishita, T., and T. W. Sederberg. Curve intersection using Bézier clipping. CAD, 1990, 22.9, 538-549. 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.

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

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

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

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

8. Bezier Clipping Nishita, T., and T. W. Sederberg. Curve intersection using Bézier clipping. CAD, 1990, 22.9, 538-549. Convex hull of control points of Bézier curve

9. Find the root of polynomial on the interval

10. Polynomial in Bézier form

11. Convex hull construction

13. The new interval

14. Algorithm

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

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

25. 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

26. 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, 345-352.

27. Degree reduction matrix  n=5, k=2

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

29. Bound Strip

30. Algorithm Convergence Rate  Quadratic clipping 3 1  Bezier clipping 2 1 1 Single rootDouble rootTriple root

31. Proof of Convergence Rate

34. Computation effort comparison

35. Time cost per iteration (μs)

36. Numerical examples  Single roots

37. Double roots

38. Near double root

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

40. Cubic Clipping

41. Cardano Formula Given a cubic equation

43. Single Roots Clone from quadratic clipping Proof

44. Double roots Proof

45. Triple roots Proof

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

47. Thanks for your attention!