Finding one root of a polynomial system Smales 17th problem Pierre - PowerPoint PPT Presentation

Finding one root of a polynomial system Smale’s 17th problem Pierre Lairez Inria Saclay FoCM 2017 Foundations of computational mathematics 15 july 2017, Barcelona

Solving polynomial systems subdivision finding one root finding all roots generic/typical case deflation, augmentation singular cases dense systems sparse/structured A well studied problem complex roots optimization real roots numerical methods Gröbner bases exact methods solving polynomial systems 1

Finding one root: a purely numerical question Bézout bound vs. input size Maybe, but only with numerical methods. Polynomial complexity? One may approximate one root disregarding the others. Numerical computation Having one root is having them all (generically). Exact computation To compute a single root, do we have to pay for #roots? 2 degree #roots input size n polynomial equations #roots ≫ input size n variables, degree D ( D + n ) D n D n n ∼ 1 2 n 3 2 n 2 1 1 2 4 n n n n ∼ � π n 1 ( n − 1)! D n D n D ≫ n ∼

Smale 17th problem approximately, on the average, in polynomial time with a uniform algorithm?” — S. Smale, 1998 approximate root A point from which Newton’s iteration converges quadratically. polynomial time with respect to the input size. on the average with respect to some input distribution. uniform algorithm A Blum–Shub–Smale machine (a.k.a. real random access machine): • registers store exact real numbers, • unit cost arithmetic operations, • branching on positivity testing. Infinite precision?! Yes, but we still have to deal with stability issues. The model is very relevant for this problem. 3 “Can a zero of n complex polynomial equations in n unknowns be found

Another brick in the wall Problem solved! Shub, Smale (1990s) Quantitative theory of Newton’s iteration Complexity of numerical continuation Beltrán, Pardo (2009) Randomization Smoothed analysis Lairez (2017) Derandomization 4 Bürgisser, Cucker (2011) Deterministic polynomial average time when D ≪ n or D ≫ n

Numerical continuation

Newton's iteration output • Convergerges quadratically fast close to a regular root. • May diverge on a open set of initial point. 5 F : C n → C n a polynomial map, z k + 1 = z k − d z k F − 1 · F ( z k ). z − d z F − 1 F ( z )

The geometry of the basins of attraction is complex... Convergence of Newton’s iteration for In red, the points from which Newton’s iteration do not converge. (Picture by Henning Makholm.) 6 the polynomial z 3 − 2 z + 2 .

... but we can give sufgicient conditions Smale (1986) then 7 F : C n → C n , a polynomial map. 1 � � k − 1 . k ! d x F − 1 · d k γ ( F , x ) ≜ sup � � 1 � x F � k > 1 γ -Theorem � If F ( ζ ) = 0 and ∥ z − ζ ∥ γ ( F , ζ ) � 3 − 7 2 � � � � 2 1 − 2 k . � Newton ( k ) ( z ) − ζ � �

Numerical continuation output • How many steps do we need to go • How to choose a path? • Solves any generic system 8 F t : C n → C n a polynomial system depending continuously on t ∈ [0,1] ; z 0 a root of F 0 . z z k + 1 = z k − d z k F † t k · F t k ( z k ) − d z F † t k + 1 = t k + δ t k t F t ( z ) δ t • How to set the step size δ t ? • How to choose the start system F 0 ? t from F 0 to F 1 ?

Condition number of a root unitary change of variables. a corner stone 9 H the space of homogeneous polynomial systems of n equations of degree D in n + 1 variables, embedded with some Hermitian norm that is invariant under F a polynomial system in H z a root of F in P n µ ( F , z ) = sup d P ( z , z ′ ) with F ′ ∼ F and F ′ ( z ′ ) = 0 ∥ F ′ − F ∥ � � (d z F ) † � 1 � � = � = least singular value of d z F 1 where z is a singular root of F ′ ≃ sup ∥ F − F ′ ∥ � 2 D − 3 2 γ proj ( F , z ).

Complexity of numerical computation Newton’s iteration Theorem (Shub 2009) Choosing the step size . 10 F t + δ t = 0 We have d ( ζ , ζ ′ ) ≲ µ ( F t , ζ ) ∥ ˙ z ′ F t ∥ δ t . ζ ′ F t = 0 1 We need d ( z , ζ ′ ) ≲ 3 2 µ ( F t + δ t , ζ ′ ) D 1 ζ z It sufgices that δ t ≲ 3 2 µ ( F t , ζ ) 2 . D One can compute an approximate root of F 1 given an approximate root of F 0 with ∫ 1 3 µ ( F t , ζ t ) 2 ∥ ˙ # steps � 136 D F t ∥ d t . 2 0

How to choose the path? a better path? We can imagine the notion of adaptative path, but it is difgicult to make it works. singular system better path 11 linear interpolation F t = tF 1 + (1 − t ) F 0 F 0 F 1

How to choose the start system? Deterministic start systems B 12 on what happen along the continuation path. A difgiculty It is not enough to control the conditioning of the start system, we need a grasp { x i = 0, 1 � i � n (homogeneized in degree D ) with its root (0,...,0) . Works well when D ≫ n (Armentano, Beltrán, Bürgisser, Cucker, Shub 2016). { x D i = 1, 1 � i � n with its D n roots. Works well when D ≪ n (Bürgisser, Cucker 2011). C F ( x 1 ,..., x n ) − F (0,...,0) = 0 with its root (0,...,0) .

Randomization of the start system

A randomized start system Conditioning of a random system Theorem (Beltrán, Pardo 2011; Bürgisser, Cucker 2011) the input size 13 • F ∈ H , random polynomial system, uniformly distributed in S ( H ) . • ζ a random root of F = 0 , uniformly chosen among the D n roots. E ( µ ( F , ζ ) 2 ) � n · dim H � �� � • How to sample ( F , ζ ) ? Chicken-and-egg problem?

Complexity of numerical continuation with random endpoints (Shub 2009) (Beltrán, Pardo 2011; Bürgisser, Cucker 2011) (Tonelli’s theorem) 14 F 0 , F 1 random polynomial systems of norm 1 , uniformly distributed. ζ 0 a random root of F 0 , uniformly distributed. F t linear interpolation (normalized to have norm 1 ). ζ t continuation of ζ 0 . lemma ∀ t , F t is uniformly distributed and ζ t is uniformly distributed among its roots. ∫ 1 3 µ ( F t , ζ t ) 2 d t 2 d S ( F 0 , F 1 ) #steps � 136 D 0 [∫ 1 ] 3 µ ( F t , ζ t ) 2 d t 2 E E [ #steps ] � 136 π D 0 ∫ 1 3 [ µ ( F t , ζ t ) 2 ] � 136 π D d t E 2 0 ( ) 3 2 ( input size ) = O nD

How to sample uniformly a random system and a root? Beltrán, Pardo (2009) . Solves Smale’s problem with randomization . . 15 first try Sample ζ ∈ P n uniformly, sample F uniformly in { F s.t. F ( ζ ) = 0 and ∥ F ∥ = 1} .  F is not uniformly distributed. BP method Sample a linear system L uniformly, compute its unique root ζ ∈ P n , { } sample F uniformly in F s.t. F ( ζ ) = 0, d ζ F = L and ∥ F ∥ = 1  F and ζ are uniformly distributed. ( 2 ( input size ) 2 ) 3 Total average complexity O nD

Smoothed analysis only approximate. worst-case average-case w.r.t. the noise Bürgisser, Cucker (2011) 16 worst-case analysis is irrelevant here (unbounded close to a system with a singular root). average analysis gives little information on the complexity of solving one given system. smoothed analysis bridges the gap and gives information on a single system F pertubed by a Gaussian noise ε of variance σ 2 . This models an input data that is [ ] 3 = O ( σ − 1 nD 2 N 2 ). sup E cost of computing one root of F + ε system F

Derandomization

Truncation and noise extraction duplication of random variables truncation noise extraction 17 x , a random uniformly distributed variable in [0,1] . 0.505290197465315910133226678885000016210273 x = 0.6044025624180895161178081249104686505290197465315910133226678885000016210273 0.6044025624180895161178081249104686 • The truncation is a random variable that is close to x . • The noise is an independent from x and uniformly distributed in [0,1] .

Truncation and noise extraction on an odd-dimensional sphere (truncation) (noise) • The noise is nearly uniformly distributed and nearly independent from the truncation. 18 mesh on S 2 n − 1 S ( H ) ≃ S 2 n − 1 S [0,1] 2 n − 1 S 2 n − 1 • S is a measure preserving map due to Sibuya (1962).

Derandomization approx. target sys. Randomness is in Smale’s question from its very formulation asking for an average analysis. Solves Smale’s problem with a deterministic algorithm . increase truncation order No? noise ext. truncation target system noise BP randomization start system Beltràn and Pardo’s randomization numerical continuation approximate root? Lairez’s derandomization randomness BP randomization target system start system numerical continuation approximate root 19

Beyond Smale's problem

Complexity of numerical algorithms theory gap applications and observations structured system Can we have interesting complexity bounds, supported by probabilistic analysis, for structured systems, especially sparse systems and low evaluation complexity systems? singular roots Can we design algorithms that find singular roots within nice complexity bounds? better complexity In the setting of Smale’s question, can we reach a quasi-optimal 20 ( input size ) 1 + o (1) average complexity?

Complexity exponent in Smale's problem cost of Newton’s iteration Armentano, Beltrán, Bürgisser, Cucker, Shub (2016) 21 ( ) total cost = O ( input size ) · # steps . � �� � Beltrán, Pardo (2009) E (# steps ) = ( input size ) 1 + o (1) 1 2 + o (1) E (# steps ) = ( input size ) work in progress E (# steps ) = poly( n , D ) = ( input size ) o (1)