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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!