Computing roots of polynomials by quadratic clipping

Transcription

1 Computing roots of polynomials by quadratic clipping Michael Bartoň, Bert Jüttler SFB F013, Project 15 Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria FSP Seminar, Strobl, 19 22/6/2006 1

2 Outlook Bézier clipping algorithm Quadratic clipping The best quadratic approximant Dual basis Error bound Convergence rate Single roots Double roots Comparison Examples Conclusion FSP Seminar, Strobl, 19 22/6/2006 2

3 Bézier clipping Polynomial function p(t) of degree n on interval [α 0, β 0 ] is given Algorithm: 1. Represent the polynomial as a parametric Bézier curve of degree n 2. Construct convex hull of the control polygon (CP) 3. Intersect the convex hull with t-axis 4. Obtain a new interval [α 1, β 1 ] if α 1 β 1 < 1 2 α 0 β 0 restrict p on [α 1, β 1 ], otherwise bisect [α 0, β 0 ]. FSP Seminar, Strobl, 19 22/6/2006 3

4 Bézier clipping FSP Seminar, Strobl, 19 22/6/2006 4

5 Bézier clipping p(t) 0 α 0 β 0 Find the root of polynomial p(t) on the interval [α 0, β 0 ]. FSP Seminar, Strobl, 19 22/6/2006 4

6 Bézier clipping p(t) 0 α 0 β 0 Polynomial p(t) in Bézier form. FSP Seminar, Strobl, 19 22/6/2006 4

7 Bézier clipping p(t) 0 α 0 β 0 Convex hull construction. FSP Seminar, Strobl, 19 22/6/2006 4

8 Bézier clipping p(t) 0 α 0 β 0 Convex hull construction. FSP Seminar, Strobl, 19 22/6/2006 4

9 Bézier clipping p(t) 0 α 1 β 1 The new interval [α 1, β 1 ]. FSP Seminar, Strobl, 19 22/6/2006 4

10 Quadratic clipping The same type of algorithm as Bézier clipping Convex hull Degree reduction 1. Find the best quadratic approximant q of p in L 2 norm 2. Compute error bound of p and q 3. Construct quadratic functions: upper bound M, lower bound m 4. Compute roots of M and m FSP Seminar, Strobl, 19 22/6/2006 5

11 Quadratic clipping FSP Seminar, Strobl, 19 22/6/2006 6

12 Quadratic clipping p(t) 0 α 0 β 0 Find the root of polynomial p(t) on the interval [α 0, β 0 ]. FSP Seminar, Strobl, 19 22/6/2006 6

13 Quadratic clipping p(t) 0 α 0 β 0 Polynomial p(t) in Bézier form. FSP Seminar, Strobl, 19 22/6/2006 6

14 Quadratic clipping p(t) q 0 0 α 0 β 0 The best quadratic approximant q. FSP Seminar, Strobl, 19 22/6/2006 6

15 Quadratic clipping M 0 p(t) q 0 m 0 0 α 0 β 0 Bounding quadratic functions M 0 and m 0 FSP Seminar, Strobl, 19 22/6/2006 6

16 Quadratic clipping M 0 p(t) q 0 m 0 0 α 0 α 1 β 1 β 0 Restriction on the new interval [α 1, β 1 ]. FSP Seminar, Strobl, 19 22/6/2006 6

17 Quadratic clipping p(t) q 1 0 α 1 β 1 2rd iteration FSP Seminar, Strobl, 19 22/6/2006 6

18 Quadratic clipping M 1 p(t) q 1 0 m 1 α 1 β 1 α 2 β 2 2rd iteration FSP Seminar, Strobl, 19 22/6/2006 6

19 Quadratic clipping p(t), q 2, M 2, m 2 0 α 2 β 2 3rd iteration FSP Seminar, Strobl, 19 22/6/2006 6

20 The best quadratic approximant Π n (n + 1)-dimensional linear space of polynomials of degree n on [0, 1] Bernstein-Bezier basis {B n i (t)}n i=0, p(t) = n i=0 b ib n i (t) L 2 inner product is defined by (f(t), g(t)) = 1 0 f(t)g(t)dt p is given, find quadratic polynomial q such that f q 2 is minimal p Π n q Π 2 Find orthogonal projection of p to subspace Π 2 FSP Seminar, Strobl, 19 22/6/2006 7

21 Degree reduction Standard technique used for obtaining the best polynomial approximant Lutterkort D., Peters J., Reif U.: Polynomial degree reduction in the L 2 -norm equals best Euclidean approximation of Bézier coefficients Eck M.: Degree reduction of Bézier curves 46 related papers (MathSciNet) Our approach: via dual basis FSP Seminar, Strobl, 19 22/6/2006 8

22 Dual basis Dual basis {D n i (t)}n i=0 to the BB basis {Bn i (t)}n i=0 Subspace Π 2, { B 2 i (t)} 2 i=0, { D 2 i (t)} 2 i=0 (B n i (t), D n j (t)) = δ ij, i, j = 0... n. q(t) = q(t) = 1 0 ( 2 j=0 2 j=0 (p(t), Dj 2 (t)) }{{} B2 j (t), (1) n b i Bi n (t))dj 2 (t)dt, i=0 n i=0 b i (B n i (t), D 2 j (t) }{{} )B2 j (t). (2) β n,2 i,j FSP Seminar, Strobl, 19 22/6/2006 9

23 Degree reduction matrix n=5, k=2 (β 5,2 i,j ) i=0,..,5;j=0,...,2 = (3) The coefficients of q are obtained by multiplying the row vector (b 0,..., b 5 ) of the coefficients of p by this matrix. FSP Seminar, Strobl, 19 22/6/

24 Error bound Best quadratic function q is found, its degree is raised to n n q(t) = c i Bi n (t) p(t) Let us denote i=0 ɛ := max i=0...n b i c i ǫ q(t) p(t) q(t) = n b i c i Bi n (t) i=0 n ɛbi n (t) = ɛ α β i=0 M(t) = q(t) + ɛ, m(t) = q(t) ɛ FSP Seminar, Strobl, 19 22/6/

25 Convergence rate Single root G(t) q the best quadratic approximant of p in L 2 norm Construct G quadratic Taylor expansion of p in the root t 0 p G = O(h 3 ) O(h 3 ) p(t) q(t) t 0 t 0 + h FSP Seminar, Strobl, 19 22/6/

26 Convergence rate Single root q(t) the best approximant of p in L 2 norm = O(h 3 ) G(t) p(t) p q 2 p G 2 p q = O(h 3 ) 3rd equivalent norm maximum norm on BB coefficient s vector p q BB = O(h 3 ) = ɛ t 0 q(t) t 0 + h FSP Seminar, Strobl, 19 22/6/

27 Convergence rate Single root Vertical width of bounding strip is O(h 3 ) Single root horizontal width O(h 3 ) Lenght of interval h O(h 3 ) Convergence rate is 3 O(h 3 ) G(t) p(t) q(t) t 0 t 0 + h FSP Seminar, Strobl, 19 22/6/

28 Convergence rate Double root The same idea of the proof like in the single root case Construct G quadratic Taylor expansion of p in the root t 0,... Convergence rate is 1.5 Special case of double root: if p (3) (t 0 ) = 0 then convergence rate is 2 FSP Seminar, Strobl, 19 22/6/

29 Comparison Convergence rate Bézier clipping: single root 2 double root 1 Quadratic clipping: single root 3 double root 1.5 Asymptotical advantage of Quadratic clipping Necessary to compare time complexity (Regula falsi vs. Newton) Secant method (1 + 5)/ Newton 2 FSP Seminar, Strobl, 19 22/6/

30 Number of operations per one iteration n Quadratic clipping Bézier clipping ± All ± All Table 1: Convex control polygon Quadratic clipping fixed numbers Bézier clipping numbers may vary Both algorithms O(n 2 ) operations FSP Seminar, Strobl, 19 22/6/

31 Time per one iteration Convex control polygon t QR 2.4 t BC time vs. degree time (in s) obtained from the implementation in C n QR BC n 10 5 iterations (in order to obtain measurable quantity) QR BC 1 for n FSP Seminar, Strobl, 19 22/6/

32 Single root Number of iterations vs. Accuracy ɛ = 10 2 ɛ = 10 4 ɛ = 10 8 ɛ = ɛ = ɛ = ɛ = n QR BC QR BC QR BC QR BC QR BC QR BC QR BC Table 2: Number of iterations in single root case f 2 (t) = (t 1 )(3 t), 2 f 4 (t) = (t 1 3 )(2 t)(t + 5)2, f 8 (t) = (t 1 3 )(2 t)3 (t + 5) 4, f 16 (t) = (t 1 3 )(2 t)5 (t + 5) 10, on the interval [0, 1] FSP Seminar, Strobl, 19 22/6/

33 Single root Time vs. Accuracy 1.4 t BC QR 2.5 t BC QR n = 4 log 2 log 1 ǫ n = 8 log 2 log 1 ǫ Exact time values up to ɛ = (C code), extrapolated for higher accuracy (time per one iteration number of iterations) FSP Seminar, Strobl, 19 22/6/

34 Double root Number of iterations vs. Accuracy ɛ = 10 2 ɛ = 10 4 ɛ = 10 8 ɛ = ɛ = ɛ = ɛ = n QR BC QR BC QR BC QR BC QR BC QR BC QR BC Table 3: Number of iterations in double root case f 2 (t) = (t 1 2 )2, f 4 (t) = (t 1 2 )2 (t + 2)(3 t), f 8 (t) = (t 1 2 )2 (4 t) 3 (t + 5) 2 (t + 7), f 16 (t) = (t 1 2 )2 (4 t) 7 (t + 5) 6 (t + 7) on the interval [0, 1] FSP Seminar, Strobl, 19 22/6/

35 Double root Time vs. Accuracy t 100 t BC 60 BC QR 20 QR log 2 log 1 ǫ log 2 log 1 ǫ n = 4 n = 8 FSP Seminar, Strobl, 19 22/6/

36 Near double root Number of iterations vs. Accuracy ɛ = 10 2 ɛ = 10 4 ɛ = 10 8 ɛ = ɛ = ɛ = ɛ = n QR BC QR BC QR BC QR BC QR BC QR BC QR BC Table 4: Number of iterations in near double root case f 2 (t) = (t 0.56)(t 0.57), f 4 (t) = (t 0.4)(t )(t + 1)(2 t), f 8 (t) = (t )(t )(t + 5) 3 (t + 7) 3, f 16 (t) = (t )(t )(6 t) 7 (t + 5) 6 (t + 7) on the interval [0, 1] FSP Seminar, Strobl, 19 22/6/

37 Near double root Time vs. Accuracy 8 t BC 8 t BC QR 4 QR log 2 log 1 ǫ log 2 log 1 ǫ n = 4 n = 8 FSP Seminar, Strobl, 19 22/6/

38 Conclusion Asymptotic advantage of Quadratic clipping (Convergence rate) Comparable results in single roots with BC Significantly faster in double root cases Better in near double roots FSP Seminar, Strobl, 19 22/6/

39 References Jüttler B. (1998), The dual basis functions of the Bernstein polynomials. Adv. Comput. Math. 8, Nishita, T., T. Sederberg, and M. Kakimoto (1990), Ray tracing trimmed rational surface patches. Siggraph Proceedings, ACM, Ko, K., T. Sakkalis, and N. Patrikalakis (2005). Resolution of multiple roots of nonlinear polynomial systems, Int. J. Shape Model., 11.1, Mourrain, B., and J.-P. Pavone (2005), Subdivision methods for solving polynomial equations, Technical Report no. 5658, INRIA Sophia Antipolis, FSP Seminar, Strobl, 19 22/6/

40 Future Work Degree reduction to cubics Multivariate algorithm FSP Seminar, Strobl, 19 22/6/