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Compute Roots of Polynomial via Clipping Method

Compute Roots of Polynomial via Clipping Method. Reporter: Lei Zhang Date: 2007/3/21. Outline. History Review B é zier Clipping Quadratic Clipping Cubic Clipping Summary. Stuff.

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Compute Roots of Polynomial via Clipping Method

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  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

  12. 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

  17. 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

  18. Bert Juttler. The dual basis functions of the Bernstein polynomials. Advanced in Comoputational Mathematics. 1998, 8, 345-352. • Degree reduction • Dual basis to the BB basis • Subspace , ,

  19. Degree reduction matrix • n=5, k=2

  20. Error bound • Raising best quadratic function to degree n • Bound estimation

  21. Bound Strip

  22. Algorithm • Convergence Rate Single root Double root Triple root • Quadratic clipping 3 1 • Bezier clipping 2 1 1

  23. Proof of Convergence Rate

  24. Computation effort comparison

  25. Time cost per iteration (μs)

  26. Numerical examples • Single roots

  27. Double roots

  28. Near double root

  29. Future work • System of polynomials • Quadratic polynomial Cubic polynomial Single root Double root Triple root • Cubic clipping 42 • Quadratic clipping 3 1 • Bezier clipping 2 1 1

  30. Cubic Clipping

  31. Cardano Formula • Given a cubic equation

  32. Single Roots Proof Clone from quadratic clipping

  33. Double roots Proof

  34. Triple roots Proof

  35. Summary • Furture Work • Quartic clipping (conjecture): cubic ->quartic polynomial

  36. Thanks for your attention!

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