Jean-Claude Yakoubsohn - unican.es · Jean-Claude Yakoubsohn Institut de Math´ematiques de...
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Tracking multiplicities
Jean-Claude Yakoubsohn
Institut de Mathematiques de Toulouse
Santalo’s Summer SchoolRecent Advances in Real Complexity and Computation
Work in collaboration with Marc Giusti.
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The problem
1– x = (x1, . . . , xn) ∈ Cn.
2– a polynomial system f = (f1, . . . , fm) ∈ Cm[x ].
3– w an isolated multiple root of f , i.e. Df (w) has not a full rank.
The quadratic convergence of the Newton method is lost in aneighborhood of w . To recover it
1– determine a regular system at w from the initial.
fi =∑9
k=1 xk + x2i − 2xi − 8,
i = 1 : 9
(1, ..., 1)multiplicity 256(Dayton − Zeng)
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Contents
1– Dimension and number of roots
2– Multiplicity : algebraic point of view.
3– Rouche theorem and dimension
4– Multiplicity and duality via linear algebra.
5– On methods recovering quadratic convergence.
6– Another way : deflating and kernelling.
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References
1– B.H. Dayton, T.Y. Li, Z. Zeng, Multiple zeros of nonlinearsystems, Math of Comp. 80, 2143-2168, 2011.
2– N. Li, L. Zhi, Computing isolated singular solutions ofpolynomials systems : case of breadth one, SIAM J. Numer.Math., 50,1 ,354-372, 2012.
3– A. Leykin, J. Vershelde, A. Zhao, Newton’s method withdeflation for isolated singularities of polynomial systems,Theoretical Comput. Sci.,359, 1112-122,2006.
4– Angelos Mantzalaris and Bernard Mourrain, Deflation andCertified Isolation of Singular Zeros of Polynomial Systems,arXiv1101.340v1, 17-01-2011.
5– Two papers of M. Giusti,G.Lecerf,B. Salvy, J.-C. Yak., in FOCM,2005, 2007.
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References
7– J.-P. Dedieu, M. Shub, On Simple Double Zeros and BadlyConditioned Zeros of Analytic Functions of n Variables.Mathematics of Computation, 70 (2001) 319-327.
Book
6– D.A. Cox, J. Little, D.O’Shea, Using Algebraic Geometry,Springer, 2005.
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Number of roots of a polynomial system
Theorem Let I =< f1, . . . , fm > and V (I ) ⊂ Cn the associatedvariety.
1– The dimension of C [x ]/I is finite iff the dimension of V (I ) iszero.
2– In the finite dimension case one has
dimC [x ]/I = #V (I )
where #V (I ) is the number of points of V (I ) counted withmultiplicities.
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Example
f1(x , y ) = x2 + x3, f2(x , y ) = x3 + y 2 and V (I ) = {(0, 0), (−1, 1)}.A Groebner basis of I is :g1(x , y ) = x0y 4 − y 2, g2(x , y ) = x1y 2 + y 2, g3(x , y ) = x2y 0 − y 2,We deduce
dimC [x ]/I = 6
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Multiplicities and local rings
Let w an isolated root of f = (f1, . . . , fm) and I =< f1, . . . fm >.Let C{x − w} the local ring of convergent power series in n variableswhich the maximal ideal is generated by x1 − w1, . . . xn − wn.Let IC{x − w} the ideal generated by I in C{x − w}We note Aw = C{x − w}/IC{x − w}.Theorem.
Let V (I ) = {w1,w2, . . .wN}. Then
1– C[x ]/I ∼ Aw1× . . .× AwN
.
2– dimC[x ]/I =N∑
i=1
dim Awi.
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Example
f1(x , y ) = x2 + x3, f2(x , y ) = x3 + y 2 andV (I ) = {(0, 0), (−1, 1), (−1, 1)}.1– A standard basis of IC{x , y} is : g1 = x2, g2 = y 2.Hence dimA(0,0) = 4.2– A standard basis of IC{x + 1, y − 1} (resp. IC{x + 1, y + 1}) is :g1 = 3x , g2 = 2y .Hence dimA(−1,1) = dimA(−1,−1) = 1.
= + +
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Multiple roots and clusters of roots
Let w a multiple root of f with multiplicity µ. Let f a pertubatedsystem of f . In an open neighborhood Uw where the Rouche theoremholds, i.e. if for all x ∈ ∂Uw we have
||f (x)− f (x)|| < ||f (x)||
then f has µ roots in Uw .
dim Afw =
∑
w∈f −1(0)∩Uw
dimAfw .
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Duality and Multiplicities
We define
Dkw (f ) = {L =
∑
|α|≤k
Lα∂α[w ] : L(f ) = 0
and φi(L) ∈ Dk−1w , ∀i = 1 : n}
where the φ′is are the linear anti-differentiation transformations :
φi(∂α[w ]) = ∂β[w ] with βj =
{
βj if j 6= i
βi − 1 if j = i
Theorem
1– The root w is isolated iff there exists l s.t. Dl−1w = Dl
w .
2– In this case the dimension of Dlw is equal to the multiplicity of w .
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Macaulay matrices and multiplicity
We consider the Macaulay matrices
Sk = (∂α[w ]((x − w)βfi(x))) for |β| ≤ k − 1 and i = 1 : m.
Theorem Dkw is isomorphic to the kernel of Sk .
Consequently D l−1w = D l
w when nullity (Sl−1) = nullity (Sl)
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Example
f1 = x2 + y 2 − 2, f2 = xy − 1. w = (1, 1).
∂00 ∂10 ∂01 ∂20 ∂11 ∂02
S0 f1 0 | 2 2 | 2 0 2f2 0 | 1 1 | 0 1 0
−−− −| |S1 |
− − − − − −|S2 (x − 1)f1 0 0 0 4 2 0
(x − 1)f2 0 0 0 2 1 0(y − 1)f1 0 0 0 0 2 4(y − 1)f2 0 0 0 0 1 2
rank(S0) = 0, corank(S0) = 1rank(S1) = 1, corank(S1) = 2rank(S2) = 4, corank(S2) = 2. Hence µ = 2.Jean-Claude Yakoubsohn (IMT) Tracking multiplicities July , 2012 13 / 30
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Recovering the quadratic convergence
The idea is to determine a sequence of systems which the last isregular at the multiple root of the original system.
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The method of LVZ
Let r the rank of the jacobian matrix J of f at w .The LVZ method consist to add at each step of deflation r + 1equations and unknowns at the initial system.
f (x) = 0
J(x)Bλ = 0 B ∈ Cn×(r+1) randommatrix
λTh − 1 = 0 h ∈ Cr+1 random vector
The unknowns of this new system are (x , λ) ∈ Cn+r+1.The corank of the system (J(x)Bλ = 0, hTλ− 1 = 0) is generalicallyequal to 1 and the multiplicty of the new system is less than theinitial system.We have added m + r + 1 equations and n + r + 1 unknowns.The number of step of deflations to restore quadratic convergence forthe Gauss-Newton method is bounded by the multiplicity of the root.
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The first method of DLZ
f (x) = 0
J(x)λ = 0
λTh − 1 = 0
At the end of this of this process we can to add 2µ−1 ×m equationsand 2µ−1 × n unknowns.
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The second method of DLZ : case of breadth one.
Breadth one : if at each deflation step corank(system) = 1.In this case the second method of DLZ constructs a regular systemwith at most µ×m equations and µ× n unknowns.
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Mantzalaris and Mourrain improvement.
Step 1- Compute a basis for the dual space of the local quotient ringAw : Λ = (Λ1, . . . ,Λµ)Step 2- Compute the system f Λ := (Λ(f1), . . . ,Λ(fm)).
Theorem The system f Λ is regular at w .
Example.
f1(x , y ) = x2 + y 2 − 2, f2 = xy − 1. w = (1, 1).Λ = (1, ∂1 − ∂2).f Λ = (f1, f2, x − y ).
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Recovering quadratic convergence : another way.
Newton method does not converge quadratically closed to an isolatedroot. To recover this quadratic convergence the idea is to determinea polynomial system which admits w as regular root, i.e, its jacobianis full rank at w .To do that we use both numerical and symbolic computation. Twoingredients
1– Deflation based on numerical computation of derivatives.
2– Determination of new polynomials based on computation ofnumerical ranges.
This way does not require to compute the whole structure of thelocal qotient algebra which is too huge.
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Recovering quadratic convergence : deflating and kernelling.
We proceed by two actions :
1– Deflating
2– Kerneling
We do that without to add new variables. The number of equationsonly increases.
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Deflating
We can replace an equation g(x) = 0 by the n equations ∂g(x)∂xi
= 0,i = 1 : n if
g(w) = 0,∂g(w)
∂xi= 0, i = 1 : n.
We decide of this thanks to a point closed to w .
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Deflating. Example
f1 = x3/3 + y 2x + x2 + 2yx + y 2, f2 = x2y + x2 + 2yx + y 2
The multiplicity of (0, 0) is 6.We have
∂1 ∂2 ∂1 ∂2x2 + y 2 + 2x + 2y 2yx + 2x + 2y 2xy + 2x + 2y x2 + 2x + 2y
∂11 ∂12 ∂21 ∂22 ∂11 ∂12 ∂11 ∂122x + 2 2y + 2 2y + 2 2x + 2 2y + 2 2x + 2 2x + 2 2
The new polynomial system consists of tree polynomials
x2 + y 2 + 2x + 2y , xy + x + y , x2 + 2x + 2y .
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Kerneling
Let us consider a polynomial system f (x) = 0 such that Df (w) isrank deficient and Dfi(w) 6= 0, i = 1 : n. Hence the rank of Df (w) isr > 0. If
Df (w) =
(
A(w) B(w)C (w) D(w)
)
∈ Cm×n
with A(w) invertible matrix of size r × r .The Jacobian matrix Df (w) has rank r iff the Schur complement iszero
D(w)− C (w)A(w)−1B(w) = 0.
i.e., w is a root of
D(x)− C (x)A(x)−1B(x).
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Kerneling using Schur complement
Then we add to the initial system at most the (m − r)× (n − r)equations given by:
numer(D(x)− C (x)A−1(x)B(x)) = 0.
Alternative : we can also deal with
1– the rational functions D(x)− C (x)A−1(x)B(x) = 0.
2– the power series D(x)− C (x)A1(x)B(x) where A1(x) is thepower series of A−1(x) at w .
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Kerneling using Schur Complement. Example
f1 = x2 + y 2 + 2x + 2y , f2 = xy + x + y , f3 = x2 + 2x + 2y .The root (0, 0) has multiplicity 2.
J =
2x + 2 2y + 2y + 1 x + 12x + 2 2y + 2
and Df (0, 0, 0) =
2 21 12 2
with rank 1.
Then there are two equations which are added to f since :
numer(J(2..3,2)−J(2..3,1)J(1,1)−1 J(1,2))=
2x2 + 4x − 2y2y
Moreover the rank of the new system is 2.
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Recovering quadratic convergence
Let GNf (x) = x − (Df (x)TDf (x))−1Df (x)T f (x) and
xk+1 = GNf (xk), k ≥ 0.
Considerf1 = x2 + y 2 + 2x + 2y , f2 = xy + x + y , f3 = x2 + 2x + 2yf4 = x2 + 2x − y , f5 = y .
e1 = 0.22e2 = 0.0088e3 = 3× 10−5
e4 = 6× 10−10
e5 = 1.7× 10−19
e6 = 1.4× 10−38
e7 = 1× 10−76
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Example from Kobayashi-Suzuki-Sakai
fi =9
∑
k=1
xk + x2i − 2xi − n + 1, i = 1 : n. w = (1, . . . , 1).
J(x) =
2x1 − 1 1 . . . 11 2x2 − 1 . . . 1
1 1 . . . 2xn − 1
Jw = (1)i ,j . Hence rank(J) = 1.
The Shur complement is
2x2 − 1 1 . . . 11 2x3 − 1 . . . 1
.
.
.
.
.
. . . .
.
.
.1 1 . . . 2xn − 1
−
11
.
.
.1
12x1−1
(1, . . . , 1).
The deflated system is
(f1, . . . , fn, x1 − 1, (2x1 − 1)(2xi − 1)− 1, i = 2 : n)
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Lecerf example
f =
2 x + 2 x2 + 2 y + 2 y 2 + z2 − 1
(x + y − z − 1)3 − x3
(2 x3 + 2 y 2 + 10 z + 5 z2 + 5)3− 1000 x5
The multiplicity of the root (0, 0,−1) is 18.The deflated system is :
x
y
1+z
y−z−1
914
x5+ 528 (2 x
3+2 y2+10 z+5 z2+5)x2− 625126
x
x+y−z−1
x+x2+y+y2+1/2 z2−1/2
RemarkJean-Claude Yakoubsohn (IMT) Tracking multiplicities July , 2012 28 / 30
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Theoretical results
f = (f1, . . . fm) and w a multiple root of f .
Theorem Let f (w) = 0 and ∇f1(w) = 0 such that ∇∂i f1(w) 6= 0,i = 1 : n. Then
multiplicityw (f ) > multiplicityw (∇f1, f )
Theorem Let f (w) = 0, rank(Df (w)) < n and ∇fi(w) 6= 0,i = 1 : n. Let S = { numerators of a Schur complement of Df (x)}.Then
multiplicityw (f ) > multiplicityw (f , S)
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Complexity
1- At each step of deflating-kernelling the multiplicity decreases atless by one.
2- The number of steps of deflating-kernelling is bounded by themultiplicity of the root.
Jean-Claude Yakoubsohn (IMT) Tracking multiplicities July , 2012 30 / 30