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!