Real root finding for rank defects in linear Hankel matrices Didier - PowerPoint PPT Presentation

Real root finding for rank defects in linear Hankel matrices Didier Henrion 1 , 2 Simone Naldi 1 Mohab Safey El Din 3 1 LAAS - CNRS (Toulouse), Universit´ e de Toulouse 2 Czech Technical University (Prague) 3 Lip6 CNRS (Paris), INRIA Paris Rocquencourt, UPMC, Sorbonne Universit´ ees I S S A C 2 0 1 5 Bath — 7th July 2015 0/10

Linear matrices and semidefinite programming A linear matrix is a polynomial A linear Hankel matrix matrix of degree 1:   x 1 + x 2 x 4 + x 2 x 3 − 1 A ( x ) = A 0 + x 1 A 1 + . . . + x n A n , x 4 + x 2 x 3 − 1 x 4   x 3 − 1 x 4 x 5 + 4 with x = ( x 1 , . . . , x n ) . Suppose: ◮ A i ∈ Q m × m ◮ A i are Hankel 1/10

Linear matrices and semidefinite programming A linear matrix is a polynomial A linear Hankel matrix matrix of degree 1:   x 1 + x 2 x 4 + x 2 x 3 − 1 A ( x ) = A 0 + x 1 A 1 + . . . + x n A n , x 4 + x 2 x 3 − 1 x 4   x 3 − 1 x 4 x 5 + 4 with x = ( x 1 , . . . , x n ) . Suppose: ◮ A i ∈ Q m × m ◮ A i are Hankel For A i = A T i , a linear matrix Lyapunov Stability is an LMI inequality is the formula Solve in P A 0 + x 1 A 1 + . . . + x n A n � 0 d f P ≻ 0 dt = M · f → M T P + PM ≺ 0 where � 0 is positive semidefinite . Semidefinite programming The set S = { x ∈ R n : A ( x ) � 0 } is min c 1 x 1 + . . . + c n x n called a spectrahedron . x ∈ R n | A ( x ) � 0 � � x ∈ S = s . t . 1/10

A Hankel spectrahedron The Carath´ eodory orbitope Consider: ◮ Linear action of SO (2 , R ) = { M ∈ R 2 × 2 : MM T = I 2 , det M = 1 } ( X , Y ) T �→ M ( X , Y ) T ◮ over the set of homogeneous binary quartics f ( X , Y ) = x 0 X 4 + 4 x 1 X 3 Y + 6 x 2 X 2 Y 2 + 4 x 3 XY 3 + x 4 Y 4 The convex hull of the orbit of f ( X , Y ) = X 4 is the Hankel spectrahedron   x 0 x 1 x 2  � 0 x 1 x 2 x 3  x 2 x 3 x 4 s.t. x 0 + 2 x 2 + x 4 = 1 . 2/10

Problem statement Given X 2 ◮ m , n , r ∈ N , 0 ≤ r ≤ m − 1 ◮ A 0 , A 1 . . . A n ∈ Q m × m ◮ A i Hankel matrices let ◮ A ( x ) = A 0 + x 1 A 1 + . . . + x n A n X 1 � x ∈ C n � � ◮ D r = � rank A ( x ) ≤ r . � Pb. : Compute one point in each connected component of D r ∩ R n . Fact Henrion-N.-Safey Let A ( x ) be symmetric.

Problem statement Given X 2 ◮ m , n , r ∈ N , 0 ≤ r ≤ m − 1 ◮ A 0 , A 1 . . . A n ∈ Q m × m ◮ A i Hankel matrices let ◮ A ( x ) = A 0 + x 1 A 1 + . . . + x n A n X 1 � x ∈ C n � � ◮ D r = � rank A ( x ) ≤ r . � Pb. : Compute one point in each connected component of D r ∩ R n . Fact Henrion-N.-Safey � � Let A ( x ) be symmetric. Let R ( A ) = min rank A ( x ) | x ∈ S

Problem statement Given X 2 ◮ m , n , r ∈ N , 0 ≤ r ≤ m − 1 ◮ A 0 , A 1 . . . A n ∈ Q m × m ◮ A i Hankel matrices let ◮ A ( x ) = A 0 + x 1 A 1 + . . . + x n A n X 1 � x ∈ C n � � ◮ D r = � rank A ( x ) ≤ r . � Pb. : Compute one point in each connected component of D r ∩ R n . Fact Henrion-N.-Safey � � Let A ( x ) be symmetric. Let R ( A ) = min rank A ( x ) | x ∈ S and C a connected component of D R ( A ) ∩ R n such that C ∩ S � = ∅ .

Problem statement Given X 2 ◮ m , n , r ∈ N , 0 ≤ r ≤ m − 1 ◮ A 0 , A 1 . . . A n ∈ Q m × m ◮ A i Hankel matrices let ◮ A ( x ) = A 0 + x 1 A 1 + . . . + x n A n X 1 � x ∈ C n � � ◮ D r = � rank A ( x ) ≤ r . � Pb. : Compute one point in each connected component of D r ∩ R n . Fact Henrion-N.-Safey � � Let A ( x ) be symmetric. Let R ( A ) = min rank A ( x ) | x ∈ S and C a connected component of D R ( A ) ∩ R n such that C ∩ S � = ∅ . Then C ⊂ S . 3/10

Problem statement Given ◮ m , n , r ∈ N , 0 ≤ r ≤ m − 1 ◮ A 0 , A 1 . . . A n ∈ Q m × m ◮ A i Hankel matrices let ◮ A ( x ) = A 0 + x 1 A 1 + . . . + x n A n � x ∈ C n � � ◮ D r = � rank A ( x ) ≤ r . � Pb. : Compute one point in each connected component of D r ∩ R n . Fact Henrion-N.-Safey � � Let A ( x ) be symmetric. Let R ( A ) = min rank A ( x ) | x ∈ S and C a connected component of D R ( A ) ∩ R n such that C ∩ S � = ∅ . Then C ⊂ S . 3/10

State of the art and main contributions Compute at least one point in each conn. comp. of D r ∩ R n . D r : f 1 ( x 1 , . . . , x n ) = 0 , . . . , f s ( x 1 , . . . , x n ) = 0 ◮ Cylindrical Algebraic Decomposition: Collins (1975) doubly exp ◮ Critical point method: Grigoriev, Vorobjov (1988) singly exp ; → D O ( n ) o (1993) . . . − Renegar (1992), Heintz, Roy, Solern´ ◮ Polar varieties: Bank, Giusti, Heintz, Mbakop, Pardo, Safey, Schost ◮ Bounds on the exponents: Safey, Schost (2001-present) ◮ smooth equidimensional algebraic sets: O ( D 3n ) ; ◮ singular algebraic sets O ( D 4n ) . Main contributions: ◮ A dedicated algorithm for the Real Root Finding for D r ∩ R n � n +2 m − r − 1 ◮ Complexity ≈ polynomial in � (poly in n when m fixed) n ◮ Implementation as maple library: s.p.e.c.t.r.a. (freely available soon) 4/10

Strategy How to avoid singularities? Input : P 1 = . . . = P a = 0 A system Q 1 = . . . = Q b = 0 − → V ( P 1 , . . . , P a ) possibly singular V ( Q 1 , . . . , Q b ) good properties How to reduce the dimension? A system R 1 = . . . = R c = 0 dim V ( Q 1 , . . . , Q b ) > 0 − → V ( R 1 , . . . , R c ) is finite Main idea : variant of critical points method for determinantal varieties with Hankel structure 5/10

Exploiting structures Determinantal structure Room-Kempf desingularization:  y 1 , 1 . . . y 1 , m − r  . . Q = A( x ) · Y = A( x ) · . .  = 0 .   . .  y m , 1 . . . y m , m − r An incidence relation : ker A ( x ) = � Y � Q = Bilinear equations in ( x , y ) 6/10

Exploiting structures Determinantal structure Room-Kempf desingularization:  y 1 , 1 . . . y 1 , m − r  . . Q = A( x ) · Y = A( x ) · . .  = 0 .   . .  y m , 1 . . . y m , m − r An incidence relation : ker A ( x ) = � Y � Q = Bilinear equations in ( x , y ) Hankel structure The kernel of a Hankel matrix is structured! Heinig, Rost (1984)  x 1 x 2 x 3 x 4 x 5   y 1 0 0   f 1 f 2 f 3  x 2 x 3 x 4 x 5 x 6 y 2 y 1 0 f 2 f 3 f 4       x 3 x 4 x 5 x 6 x 7 y 3 y 2 y 1  = f 3 f 4 f 5             x 4 x 5 x 6 x 7 x 8 0 y 3 y 2 f 4 f 5 f 6      x 5 x 6 x 7 x 8 x 9 0 0 y 3 f 5 f 6 f 7 Theorem: Q = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 ) is radical and V ( Q ) is smooth and equidimensional 6/10

Exploiting structures Determinantal structure Room-Kempf desingularization:  y 1 , 1 . . . y 1 , m − r  . . Q = A( x ) · Y = A( x ) · . .  = 0 .   . .  y m , 1 . . . y m , m − r An incidence relation : ker A ( x ) = � Y � Q = Bilinear equations in ( x , y ) Hankel structure The kernel of a Hankel matrix is structured! Heinig, Rost (1984)  x 1 x 2 x 3 x 4 x 5   y 1 0 0   f 1 f 2 f 3  x 2 x 3 x 4 x 5 x 6 y 2 y 1 0 f 4 ✗ ✗       x 3 x 4 x 5 x 6 x 7 y 3 y 2 y 1  = ✗ ✗ f 5             x 4 x 5 x 6 x 7 x 8 0 y 3 y 2 ✗ ✗ f 6      x 5 x 6 x 7 x 8 x 9 0 0 y 3 ✗ ✗ f 7 Theorem: Q = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 ) is radical and V ( Q ) is smooth and equidimensional 6/10

Bilinearity of critical points systems Let a = [ a 1 , . . . , a n ] T ∈ R n and consider Π a ( x , y ) = a 1 x 1 + . . . + a n x n . Critical points → solutions of the bi-linear Lagrange system R : � jac X Q ( x , y ) jac Y Q ( x , y ) � z ′ Q ( x , y ) = 0 = 0 . a 1 , . . . , a n 0 · · · 0 Theorem: The set of solutions is finite 7/10

Bilinearity of critical points systems Let a = [ a 1 , . . . , a n ] T ∈ R n and consider Π a ( x , y ) = a 1 x 1 + . . . + a n x n . Critical points → solutions of the bi-linear Lagrange system R : � jac X Q ( x , y ) jac Y Q ( x , y ) � z ′ Q ( x , y ) = 0 = 0 . a 1 , . . . , a n 0 · · · 0 Theorem: The set of solutions is finite Hope : Bilinear structure on x , y , z − → Degree bounds 7/10

Bilinearity of critical points systems Let a = [ a 1 , . . . , a n ] T ∈ R n and consider Π a ( x , y ) = a 1 x 1 + . . . + a n x n . Critical points → solutions of the bi-linear Lagrange system R : � jac X Q ( x , y ) jac Y Q ( x , y ) � z ′ Q ( x , y ) = 0 = 0 . a 1 , . . . , a n 0 · · · 0 Theorem: The set of solutions is finite Hope : Bilinear structure on x , y , z − → Degree bounds By multilinear B´ ezout bounds : � 3 � n + 2m − r − 1 degree of V ( R ) = number of complex solutions ≤ n Remark! when m is fixed, degree is polynomial in n Multilinear bounds ≪ Classical B´ ezout bound 7/10

Complexity and output degree estimates Output representation: rational parametrization x 1 = q 1 ( t ) q 0 ( t ) , . . . , x n = q n ( t ) q 0 ( t ) , q ( t ) = 0 . Algorithms for computing rational parametrizations: ◮ [Giusti, Lecerf, Salvy] − → geometric resolution ◮ [Jeronimo, Matera, Solern´ o] − → symbolic homotopy Output degree � 3 � n + 2 m − r − 1 δ ( m , n , r ) ≤ n 8/10