Calculating approximate GCD of univariate polynomials with 'approximate' syzygies
-
Upload
akira-terui -
Category
Documents
-
view
369 -
download
0
Transcript of Calculating approximate GCD of univariate polynomials with 'approximate' syzygies
![Page 1: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/1.jpg)
Calculating approximate GCD of univariate polynomials
with ‘approximate’ syzygies
Akira TeruiFaculty of Pure and Applied Sciences
University of Tsukuba
Computer Algebra―The Algorithms, Implementations and the Next Generation
December 26, 2012
![Page 2: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/2.jpg)
Approximate GCDOne of the most well-known method
in approximate algebraic computations
F (x) = x2 + 0.9x� 6.55
G(x) = x2 � 5.7x + 11.25
F (x) = F (x)� 2.1x� 1.5 = (x� 3.5)(x + 2.3)
G(x) = G(x)� 2.3x + 4.5 = (x� 3.5)(x� 4.5)
Perturbations
F and G can be pairwise
relatively prime
ApproximateGCDof F and G
![Page 3: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/3.jpg)
Approximate GCDOne of the most well-known method
in approximate algebraic computations
F (x) = x2 + 0.9x� 6.55
G(x) = x2 � 5.7x + 11.25
Perturbations ApproximateGCDof F and G
F and G can be pairwise
relatively prime
F (x) = F (x)� 0.476722x� 0.19906 = (x� 2.39486)(x + 2.81814)
G(x) = G(x)� 1.18569x� 0.495096 = (x� 2.39486)(x� 4.49082)
![Page 4: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/4.jpg)
For the given F(x), G(x) and d,find H(x): an approximate GCD
of F and G of degree d
F (x) = F (x) + �F (x) = H(x) · F (x)
G(x) = G(x) + �G(x) = H(x) · G(x)
F (x), G(x) : pairwise relatively prime
: degree m, F (x) : degree nG(x) (m � n)
H(x) : degree d (d < n)
![Page 5: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/5.jpg)
An iterative method based onconstrained minimization
GPGCD: an approximate GCD algorithmGradient Projection
Calculates an approximate GCDwith similar amount of perturbations,
with much more efficiency,compared with optimization-based method
![Page 6: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/6.jpg)
Calculating approximate GCD for multiple polynomial inputs
• A method based on SVD of Sylvester matrix (Rupprecht, 1999)
• STLN-based method (Kaltofen, Yang, Zhi; 2006)
![Page 7: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/7.jpg)
The GPGCD method for multiple polynomials based on Rupprecht’s
1st method
My previous work (2010)
Nk(P1, . . . , Pn)
=
�
⇧⇧⇧⇤
Cd1�1�k(P2) Cd2�1�k(P1) 0 · · · 0Cd1�1�k(P3) 0 Cd3�1�k(P1) · · · 0
......
. . ....
Cd1�1�k(Pn) 0 · · · 0 Cdn�1�k(P1)
⇥
⌃⌃⌃⌅,
Generalized Sylvester matrix becomes large for many polynomials
![Page 8: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/8.jpg)
An attempt based on Rupprecht’s 2nd method
with calculating ‘approximate’ syzygies
Today’s talk
![Page 9: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/9.jpg)
Formulation of the problem
![Page 10: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/10.jpg)
Pairwise relatively prime in
general
P1(x), . . . , Pn(x) � R[x]
Inputs
Pi(x) = p(i)di
xdi + · · · p(i)1 x + p(i)
0
min{d1, . . . , dn} > 0
min{d1, . . . , dn} > d > 0
• Polynomials
• Degree d:
![Page 11: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/11.jpg)
• Perturbations
• Satisfying ...
• Minimizing ...
Find for...
�P1(x), . . . ,�Pn(x) � R[x]
deg(�Pi) � deg(Pi)
Pi(x) = Pi(x) + �Pi(x) = H(x) · Pi(x)
⇥�P1(x)⇥22 + · · · + ⇥�Pn(x)⇥2
2
Approximate GCD of degree d
Pairwise relatively prime
![Page 12: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/12.jpg)
2Solve the
minimization problem
Solving the problemin two steps
1Transfer the original
problem into anconstrained
minimization problem
![Page 13: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/13.jpg)
2Solve the
minimization problem
Solving the problemin two steps
1Transfer the original
problem into anconstrained
minimization problem
Set up the objective function
Set up the constraints
![Page 14: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/14.jpg)
Derive the objective function
The given polynomials (known)
The polynomials to be calculated (unknown)
Pi(x) = p(i)di
xdi + · · · p(i)1 x + p(i)
0
Pi(x) = p(i)di
xdi + · · · + p(i)1 x + p(i)
0
![Page 15: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/15.jpg)
The objective function: f(x)
f(x) = ⇤�P1(x)⇤22 + · · · + ⇤�Pn(x)⇤2
2
=n
i=1
⇤⌥
⇧
di
j=0
�p(i)
j � p(i)j
⇥2⌅�
⌃
x = (p(1)d1
, . . . , p(1)0 , . . . , p(n)
dn, . . . , p(n)
0 )
Variable Constant
![Page 16: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/16.jpg)
Derive the constraintwith Rupprecht’s
1st method
![Page 17: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/17.jpg)
‘Convolution’ matrix P (x) = pnxn + · · · + p0x
0
Ck(P ) =
�
⇧⇧⇧⇧⇧⇧⇤
pn...
. . .p0 pn
. . ....
p0
⇥
⌃⌃⌃⌃⌃⌃⌅.
�⌥ ⌦k+1
![Page 18: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/18.jpg)
For 2 polynomials:Subresulant Matrix
(sub-matrix of the Sylvester matrix)Nk(P1, P2) =
�Cd2�1�k(P1) Cd1�1�k(P2)
⇥
=
⇤
⌥⌥⌥⌥⌥⌥⌥⇧
p(1)d1
p(2)d2
.... . .
.... . .
p(1)0 p(1)
d1p(2)0 p(2)
d2
. . ....
. . ....
p(1)0 p(2)
0
⌅
�������⌃
.
↵ ⌦ �d2�k
↵ ⌦ �d1�k
min{d1, d2} > k � 0
![Page 19: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/19.jpg)
For n polynomials: Generalized Sylvester Matrix
Nk(P1, . . . , Pn)
=
�
⇧⇧⇧⇤
Cd1�1�k(P2) Cd2�1�k(P1) 0 · · · 0Cd1�1�k(P3) 0 Cd3�1�k(P1) · · · 0
......
. . ....
Cd1�1�k(Pn) 0 · · · 0 Cdn�1�k(P1)
⇥
⌃⌃⌃⌅,
min{d1, . . . , dn} > k � 0
![Page 20: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/20.jpg)
A key proposition
deg(gcd(P1, . . . , Pn)) � k
has full rankNk(P1, . . . , Pn)
![Page 21: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/21.jpg)
U1Pi + UiP1 = 0
Derive the constraint
Ui(x)There exist (degree ≤ )satisfying
di � d
Nd�1(P1, . . . , Pn) is rank-deficient
![Page 22: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/22.jpg)
Derive the constraintU1Pi + UiP1 = 0
0
BBB@
Cd1�1�k(P2) Cd2�1�k(P1) 0 · · · 0Cd1�1�k(P3) 0 Cd3�1�k(P1) · · · 0
......
. . ....
Cd1�1�k(Pn) 0 · · · 0 Cdn�1�k(P1)
1
CCCA
0
B@
tu1...
tun
1
CA =
0
1
CA
g(x) = 0 (constraint)
![Page 23: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/23.jpg)
Dimension of the generalized Sylvester matrixNk(P1, . . . , Pn)
=
�
⇧⇧⇧⇤
Cd1�1�k(P2) Cd2�1�k(P1) 0 · · · 0Cd1�1�k(P3) 0 Cd3�1�k(P1) · · · 0
......
. . ....
Cd1�1�k(Pn) 0 · · · 0 Cdn�1�k(P1)
⇥
⌃⌃⌃⌅,
# of Rows: d1 + d2 + · · · + dn � n k + (n� 1)d1
# of columns: d1 + d2 + · · · + dn � n k
inreases in propotion to the sum of degrees of the polynomials
![Page 24: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/24.jpg)
Consider Rupprecht’s 2st method
using ‘approximate’ syzygies
![Page 25: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/25.jpg)
Outline of Rupprecht’s 2nd method (exact case)
(1) Calculate n-1 syzygies of satisfyingP1, . . . , Pn
U (j)1 P1 + · · · + U (j)
n Pn = 0
U (j)1 , . . . , U (j)
n
![Page 26: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/26.jpg)
Outline of Rupprecht’s 2nd method(exact case)
(2) Calculate cofactors of
from calculated syzygies satisfying
P1, . . . , Pn
U (j)1 , . . . , U (j)
n
Pj = H · Pj , gcd(P1, . . . , Pj) = 1
GCD of P1, . . . , Pn
![Page 27: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/27.jpg)
Outline of Rupprecht’s 2nd method
(approximate case)(1) Calculate n-1 ‘approximate’ syzygies
and perturbations
by the Singular Value Decomposition (SVD)
U (j)1 , . . . , U (j)
n
�P1, . . . ,�Pn
U (j)1 (P1 + �P1) + · · · + U (j)
n (Pn + �Pn) = 0
![Page 28: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/28.jpg)
Outline of Rupprecht’s 2nd method
(approximate case)
(2) Calculate cofactors of
from calculated syzygies
P1, . . . , Pn
U (j)1 , . . . , U (j)
n
Pj + �Pj = H · Pj , gcd(P1, . . . , Pj) = 1
Approximate GCD of P1, . . . , Pn
![Page 29: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/29.jpg)
GCD calculationin exact case
(review)
![Page 30: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/30.jpg)
columnsn(r + 1)� (d1 + · · · + dn)
Another generalized Sylvester matrix of degree r
Nr(P1, . . . , Pn)
=�Cr�d1(P1) Cr�d2(P2) · · · Cr�dn(Pn)
�
=
0
BBBBBBBBBBB@
p(1)d1
p(2)d2
· · · p(n)d1
.... . .
.... . .
.... . .
... p(1)d1
... p(2)d2
· · ·... p(n)
d2
p(1)0
... p(2)0
... · · · p(n)0
.... . .
.... . .
.... . .
...p(1)0 p(2)
0 · · · p(n)0
1
CCCCCCCCCCCA
r + 1 rows
![Page 31: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/31.jpg)
Syzygy of of degree r
then there exists a tuple of polynomials
satisfying
Syzygy of degree r
If there exists a vector satisfying
Nr(P1, . . . , Pn) · v = 0v 6= 0
R1, . . . , Rn
R1P1 + · · · + RnPn = 0P1, . . . , Pn
![Page 32: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/32.jpg)
Calculate cofactors from n - 1 ‘independent’ syzygies
U =
0
BB@
U (1)1 · · · U (1)
n
......
U (n�1)1 · · · U (n�1)
n
1
CCA
![Page 33: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/33.jpg)
satisfying
Lemma 5.1(Rupprecht, 1999)
For matrix U, there exists a tuple of polynomials
V1, . . . , Vn
���������
U (1)1 · · · U (1)
n
......
U (n�1)1 · · · U (n�1)
n
V1 · · · Vn
���������
= 1
![Page 34: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/34.jpg)
Lemma 5.2(Rupprecht, 1999)
Then, we have
�i =
��������
U (1)1 · · · U (i�1)
1 U (i+1)1 · · · U (1)
n
......
......
U (n�1)1 · · · U (i�1)
n U (i+1)1 · · · U (n�1)
n
��������
Let
Pi = H · �i
GCD cofactor
A minor of U obtained by eliminating the i-th column
![Page 35: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/35.jpg)
Approximate calculation of syzygies: an example
GCDH(x) = x + 1
P1(x) = H(x) · (x� 1) = x2 � 1
P2(x) = H(x) · (x� 1)(x + 2) = x3 + 2x2 � x� 2P3(x) = H(x) · (x + 2)(x� 4)(x + 5)
= x4 + 4x3 � 15x2 � 58x� 40
P4(x) = H(x) · (x� 4)(x3 + x2 + x + 1)
= x5 � 2x4 � 6x3 � 6x2 � 7x� 4
![Page 36: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/36.jpg)
Generalized Sylvester matrix of degree 5
N5(P1, P2, P3, P4)
=
0
BBBBBB@
1 0 0 0 1 0 0 1 0 10 1 0 0 2 1 0 4 1 �2�1 0 1 0 �1 2 1 �15 4 �60 �1 0 1 �2 �1 2 �58 �15 �60 0 �1 0 0 �2 �1 �40 �58 �70 0 0 �1 0 0 �2 0 �40 �4
1
CCCCCCA
![Page 37: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/37.jpg)
Approximate syzygies caluclated by the SVD
U
(1)1 (x) = �0.529832950578389639x
3
� 0.392106355437565890x
2
+ 0.555395351048165176x
� 0.235241591480521151,
U
(1)2 (x) = 0.419758249032335162x
2
� 0.0861045050752664576x
� 0.00560217899865405114,
U
(1)3 (x) = �0.0223458873839458561x
� 0.00560217899865405114,
U
(1)4 (x) = 0.132420588930000
![Page 38: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/38.jpg)
Approximate syzygies caluclated by the SVD
We obtain
as well
(U (2)1 (x), U (2)
2 (x), U (2)3 (x), U (2)
4 (x)),
(U (3)1 (x), U (3)
2 (x), U (3)3 (x), U (3)
4 (x)),
(U (4)1 (x), U (4)
2 (x), U (4)3 (x), U (4)
4 (x)),
(U (5)1 (x), U (5)
2 (x), U (5)3 (x), U (5)
4 (x))
![Page 39: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/39.jpg)
Calculated cofactors do not satisfy degree condition(s)
�1 =
�������
U
(1)2 (x) U
(1)3 (x) U
(1)4 (x))
U
(2)2 (x) U
(2)3 (x) U
(2)4 (x))
U
(3)2 (x) U
(3)3 (x) U
(3)4 (x))
�������
Degree 2
Degree 1
Degree 0
P1(x) = (x + 1)(x� 1)
GCD cofactor
We have to calculate a
polynomial of degree 1 as the
cofactor
Degree 3 in total
![Page 40: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/40.jpg)
Basis of syzygy module of ideal generated by P1, P2, P3, P4
(Calculated with Singular)
U =
0
@0 x
2 + x� 20 �x + 1 090 47x� 203 �9x + 7 9
x + 2 �1 0 0
1
A
We have to obtain ‘reduced’ form of syzygies in some sense by somehow
![Page 41: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/41.jpg)
Generalized Sylvester matrix of smaller degree will also cause a problem
P1(x) = H(x) · (x� 1)P2(x) = H(x) · (x� 1)(x + 2)
We don’t know apriorideg(gcd(P1, P2)) > deg(gcd(P1, P2, P3, P4))
![Page 42: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/42.jpg)
Applicable case:Proposition
dj = deg(Pj), d = deg(H)
r: degree of generalized Sylvester matrix involving all input polynomials
P1, . . . , Pn
Let
![Page 43: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/43.jpg)
Applicable case:Proposition
and the degrees of syzygies calculatedby the SVD of the generalized Sylvestermatrix do not decrease, then we have
If we have
(n� 1)r + d = d1 + · · · + dn
deg(�i) = deg(Pi)
![Page 44: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/44.jpg)
Applicable case:example
H = x2 + x + 1,
P1 = H · (x2 + 3x + 1)
= x4 + 4x3 + 3x2 + 2x� 1,
P2 = H · (x3 + 2x2 � x + 2)
= x5 + 3x4 + 2x3 + 3x2 + x + 2,
P3 = H · (x5 + 2x4 + x3 � 4x2 + x� 5)
= x7 + 3x6 + 4x5 � x4 � 2x3 � 8x2 � 4x� 5
d = 2
d1 = 4
d2 = 5
d3 = 7
![Page 45: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/45.jpg)
This example satisfies degree condition
(n� 1)r + d = d1 + · · · + dn
4 + 5 + 7 = 162 · 7 + 2 = 16
r = 7
![Page 46: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/46.jpg)
Approximate syzygiescalculated with the SVD on generalized
Sylvester matrix of degree 7
U
(1)1 = �0.0665708260100711796 x
3 + 0.640458735891240138 x
2 � 0.251464889020164006 x
+ 0.270849661666912989,
U
(1)2 = 0.212695343726665154 x
2 � 0.573887909881168556 x� 0.229886463458029544,
U
(1)3 = �0.146124517716594
U
(2)1 = �0.381649984182229784 x
3 + 0.0328566615969874409 x
2 + 0.0543411474203011752 x
� 0.347529283829036639
U
(2)2 = +0.532035125397169573 x
2 + 0.348793322585241872 x� 0.549727494951867457
U
(2)3 = �0.150385141214940
![Page 47: Calculating approximate GCD of univariate polynomials with 'approximate' syzygies](https://reader036.fdocuments.net/reader036/viewer/2022062418/55402ef94a79599a5f8b45a3/html5/thumbnails/47.jpg)
Calculated cofactors (with normalized coefficients)
�1 �! 1.0 x
2 + 2.99999999999998312 x� 0.99999999999999600,
�2 �! 1.0 x
3 + 1.99999999999999334 x
2 � 0.99999999999999656 x
+ 1.99999999999999489,
�3 �! +1.0 x
5 + 2.00000000000000844 x
4 + 0.99999999999999789 x
3
� 4.00000000000001066 x
2 + 1.00000000000000577 x
� 5.00000000000000888