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, 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.

History Review

Quadratic Equation

2 0ax bx c

祖冲之 (429~500)、祖日桓花拉子米 (780~850)

Cubic Equation (Cardan formula)3 2 0x ax bx c

Tartaglia (1499~1557)

Cardano (1501~1576)

Quartic Equation (Ferrari formula)

4 3 2 0x ax bx cx d

Ferrari (1522~1565)

Equation 5n

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, 538-549.

Convex hull of control points of Bézier curve

Find the root of polynomial on the interval 0 0[ , ] ( )p t

Polynomial in Bézier form( )p t

Convex hull construction

Convex hull construction

The new interval 1 1[ , ]

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

n

0

( )nn

i iB t

0( ) ( )

n ni ii

p t b B t

2L

1

0( ), ( ) ( ) ( )f t g t f t g t dt

p q

2p q

Degree reduction Dual basis to the BB basis

Subspace , ,

0

( )nn

i iD t

0( )

nni iB t

2 22

0( )i i

B t

22

0( )i i

D t

22 2

0

( ) ( ), ( ) ( )j jj

q t p t D t B t

1 2

00

( ) ( )n

ni i j

i

b B t D t dt

22 2

0 0

( ) ( ), ( ) ( )n

ni i j j

j i

q t b B t D t B t

,2,ni j

( ), ( )n ni j ijB t D t

Bert Juttler. The dual basis functions of the Bernstein polynomials. Advanced in Comoputational Mathematics. 1998, 8, 345-352.

Degree reduction matrix n=5, k=2

Error bound Raising best quadratic function to degree n

Bound estimation

q

0

( ) ( )n

ni i

i

q t c B t

0...max i ii n

b c

0

0

( ) ( ) ( )

( )

nn

i i ii

nni

i

p t q t b c B t

B t

Bound Strip

( ) ( )M t q t

( ) ( )m t q t

Algorithm Convergence Rate

Quadratic clipping 3 1

Bezier clipping 2 1 1

Single root Double root Triple root

3

2

Proof of Convergence Rate

3

1 For any given polynomial , there exists a constant

depending solely on , such that for all interval [ , ] [0 1] the

bound generated in line 3 of quadclip satisfies , where

.

p

p

p C

p

C h

h

Lemma

，

[ , ] [3

2 For any given polynomial there exist constant , and

depending solely on , such that for all intervals [ , ] [0,1] the quadratic

polynomial obtained by applying degree reduction to satisfies

, ' '

p p p

p

p V D A

p

q p

p q V h p q

Lemma

, ] [ , ]2 , and '' '' .p pD h p q A h

*1 If the polynomial has a root in [ , ] and provided that this

root has multiplicity1, then the sequence of the lengths of the intervals gene-

-rated by quadclip which contain that root has the convergence rate 3.

p t

d

Theorem

*2 If the polynomial has a root in [ , ] and provided that this

root has multiplicity 2, then the sequence of the lengths of the intervals gene-

3-rated by quadclip which contain that root has the convergence rate .

2

p t

d

Theorem

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 2 1 1

Single root Double root Triple root

3

2

4

3

Cubic Clipping

3 9 31 11( ) ( )*(2 ) *( 5) *( )

3 10p t t t t t

Cardano Formula

Given a cubic equation 3 2 0x ax bx c

3 31

23 32

2 3 32

2 2 3

2 2 3

2 2 3

q q ax

q q ax

q q ax

2 31 2 1,

3 27 3p a b q a ab c

22 33,

4 27

iq pe

4

1 For any given polynomial , there exists a constant

depending solely on , such that for all interval [ , ] [0 1] the

bound generated in line 3 of cubicclip satisfies , where

.

p

p

p C

p

hC

h

Lemma

，

[ , ] 4

2 For any given polynomial there exist constant , and

depending solely on , such that for all intervals [ , ] [0,1] the quadratic

polynomial obtained by applying degree reduction to satisfies

, ' '

p p p p

p

p V D A B

p

q p

p q V h p q

Lemma

[ , ] 3

[ , ] [ , ]2

,

'' '' , ''' ''' .

p

p p

D h

p q A h p q B h

Single Roots

*1 If the polynomial has a root in [ , ] and provided that this

root has multiplicity1, then the sequence of the lengths of the intervals gene-

-rated by cubicclip which contain that root has the convergence rate 4.

p t

d

Theorem

Clone from quadratic clipping

Proof

Double roots

*2 If the polynomial has a root in [ , ] and provided that this

root has multiplicity 2, then the sequence of the lengths of the intervals gene-

-rated by cubicclip which contain that root has the convergence rate 2.

p t

d

Theorem

Proof

Triple roots

*3 If the polynomial has a root in [ , ] and provided that this

root has multiplicity 3, then the sequence of the lengths of the intervals gene-

4-rated by cubicclip which contain that root has the convergence rate .

3

p t

d

Theorem

Proof

Summary

Furture Work Quartic clipping (conjecture): cubic ->quartic polynomial

single double triple quadruple

quartic 5 5/2 5/3 5/4

cubic 4 2 4/3 1

quadratic 3 3/2 1 1

bezier 2 1 1 1

Thanks for your attention!