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Computational Real Algebraic Geometry and Applications to Robotics JNCF Lecture, Part 2 Computational real algebraic geometry and applications to robotics JNCF lecture, part 2 Guillaume Moroz Inria Nancy - Grand Est March 4th, 2021 Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 1 / 52 Solving systems 1 With initial point Newton 2 Without initial point Symbolic approaches Numerical approaches Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 2 / 52 Newton f pxq “ 0 x0 “ initial point f px q x x n n`1 “ n ´ 1 f pxnq x2 x1 x0 Good for path tracking Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 3 / 52 Newton f pxq “ 0 x0 “ initial point ´1 xn`1 “ xn ´ Df pxnq f pxnq x2 x1 x0 Good for path tracking Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 4 / 52 Reliable Newton with Kantorovich ´1 2 Kpf , x0q “ sup }Df px0q D f pxq} x ´1 βpf , x0q “ }Df px0q f px0q} Theorem (Kantorovich) If βpf , x0qKpf , x0q ¤ 1{2, then f has a unique solution in Bpx0, 2βpf , x0qq. Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 5 / 52 Reliable Newton with Smale k ´1 D f px0q 1 γpf , x0q “ sup }Df px0q } k´1 k¥2 k! ´1 βpf , x0q “ }Df px0q f px0q} Theorem (Smale) ? If β f , x γ f , x 3 2 2, p 0q p 0q ¤ ´ ? 1´ 2{2 then f has a unique solution in Bpx0, q. γpf ,x0q Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 6 / 52 Reliable interval Newton B a box interval containing p 1 variable ´1 f pxq´f ppq Npxq “ p ´ x´p f ppq NpBqĂ B ñ´N has a¯ fix point in B ñ f has a solution in B 2 variables or more f ppq NpBq “ p ´ lJpBq´1 gppq ˆ ˙ NpBqĂ B ñ f “ g “ 0 has a solution in B Interval operations ra, bs ` rc, ds “ ra ` c, b ` ds ra, bs b rc, ds “ rminpac, ad, bc, bdq, maxpac, ad, bc, bdqs Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 7 / 52 Further reading >Ì j>̵ÕiÃÊEÊ««V>ÌÃÊx{ i>*iÀÀiÊ i`iÕ *ÌÃÊÝiÃ]ÊÊ <jÀÃÊiÌÊ>ÊjÌ `iÊÊ `iÊ iÜÌ Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 8 / 52 Further computing Computer Algebra Systems Maple Mathematica Mathemagix (mmx) Matlab SageMath Xcas ... Numerical Newton Interval and ball arithmetic Libraries Boost (C++) Arb (C, Python) [Johansson] MPSolve (C) [Bini, Fiorentino, Robol] GAOL (C++) [Goualard] GNU Scientific Library (C) Intlab (Octave/Matlab) [Rump] Roots, Optim, NLsolve (Julia) mpfi (C) [Revol] Scipy (Python) numerix (mmx, C++) [van der Hoeven] ... IntervalArithmetic (Julia) [Benet,Sanders] ... Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 9 / 52 Univariate Representation Reduce problem to univariate polynomial y 0 0 x ppxq “ 0 y “ qpxq Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 10 / 52 Resultant definition Given two polynomials in Crys : d P “ p0y ` ¨ ¨ ¨ ` pd with roots σ1, . , σd d Q “ q0y ` ¨ ¨ ¨ ` qd with roots τ1, . , τd Definition: Resultant d d RespP, Qq “ p0 q0 i,j pσi ´ τj q d ± “ p0 Q pσ1q ¨ ¨ ¨ Q pσd q d2 d “ p´1q q0 P pτ1q ¨ ¨ ¨ P pτd q Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 11 / 52 Resultant definition Bezout ϕ : Crysd´1 ˆ Crysd´1 Ñ Crys2d´1 U , V ÞÑ UP ` VQ ϕpU, V q “ 1 ô gcdpP, Qq “ 1 ô ϕ invertible Resultant r is the determinant of the Sylvester Matrix r P hP, Qi “ I Definition: Sylvester matrix 1 2 ¨ ¨ ¨ d ` 1 d ` 2 d ` 3 ¨ ¨ ¨ p0 q0 p p q q ¨ 1 0 1 0 ˛ p2 p1 p0 q2 q1 q0 ˚ p p p p q q q q ‹ ˚ 3 2 1 0 3 2 1 0 ‹ ˚ . .. .. ‹ ˚ . ‹ ˚ ‹ Guillaume Moroz˝ (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021‚ 12 / 52 Bivariate case Given two polynomials in Crx, ys of degree d in x and y: d P “ p0pxqy ` ¨ ¨ ¨ ` pd pxq RespP, Qq is a polynomial in x Opn3´1{ω`εq arithmetic operations d Q “ q0pxqy ` ¨ ¨ ¨ ` qd pxq [Villard 2018] Opn2`εq randomized in a finite field [van der Hoeven, Lecerf 2019] y 0 0 x Resd p Ppx, yq , Qpx, yq qpαq “ Resd p Ppα, yq , Qpα, yq q Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 13 / 52 First subresultant definition Given two polynomials in Crys : d P “ p0y ` ¨ ¨ ¨ ` pd with roots σ1, . , σd d Q “ q0y ` ¨ ¨ ¨ ` qd with roots τ1, . , τd Properties Sres1 “ s1y ` s0 has degree at most 1 D E Sres1 P P , Q “ I Definition: first subresultant d´1 Q pσj q Sres1pP, Qq “ p0 py ´ σ1q j‰1 σ ´σ ˜ j 1 ` ¨ ¨ ¨ ` ± Q pσj q py ´ σd q j‰d σ ´σ j d ¸ Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and± Robotics March 4th, 2021 14 / 52 First subresultant definition ϕ : Crysd´2 ˆ Crysd´2 Ñ Crys2d´2 U , V ÞÑ UP ` VQ The s0 and s1 are the determinants of minors of the Sylvester Matrix Sylvester matrix 1 2 ¨ ¨ ¨ d ` 1 d ` 2 d ` 3 ¨ ¨ ¨ p0 q0 p p q q ¨ 1 0 1 0 ˛ p2 p1 p0 q2 q1 q0 ˚ p p p p q q q q ‹ ˚ 3 2 1 0 3 2 1 0 ‹ ˚ . .. .. ‹ ˚ . ‹ ˚ ‹ ˝ ‚ Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 15 / 52 Parametrization of y y 0 0 x s1pxqy ` s0pxq “ 0 rpxq “ 0 Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 16 / 52 Three variables and more f1px1, ¨ ¨ ¨ , xnq “ ¨ ¨ ¨ “ fnpx1, ¨ ¨ ¨ , xnq “ 0 Problem Find ppx1q “ q1f1 ` ¨ ¨ ¨ ` qnfn D Matrix of 1, x1,..., x1 modulo hf1,..., fni D 1 ¨ ¨ ¨ x1 m1 M “ . ¨ ˛ mk ˚ ‹ ˚ ‹ ˝ ‚ D ñ v P KerpMq iff v0 ` ¨ ¨ ¨ ` vDx P hf1,..., fni Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 17 / 52 Three variables and more f1px1, ¨ ¨ ¨ , xnq “ ¨ ¨ ¨ “ fnpx1, ¨ ¨ ¨ , xnq “ 0 Problem Find ppx1q “ q1f1 ` ¨ ¨ ¨ ` qnfn Matrix of multiplication by x1 modulo hf1,..., fni x1m1 ¨ ¨ ¨ x1mk m1 M “ . ¨ ˛ mk ˚ ‹ ˚ ‹ ˝ ‚ ñ χM px1q P hf1,..., fni Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 18 / 52 Three variables and more f1px1, ¨ ¨ ¨ , xnq “ ¨ ¨ ¨ “ fnpx1, ¨ ¨ ¨ , xnq “ 0 Problem Find ppx1q “ q1f1 ` ¨ ¨ ¨ ` qnfn Matrix of multiplication by x1 modulo f1px1q 0 0 ... 0 ´c0 1 0 ... 0 ´c1 ¨ ˛ M “ 0 1 ... 0 ´c2 ˚. .. ‹ ˚. ‹ ˚ ‹ ˚0 0 ... 1 ´c ‹ ˚ d´1‹ ˝ ‚ ñ χM px1q “ f1px1q P hf1i Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 19 / 52 Normal form and univariate representation Normal form x1mi mod hf1,..., fni Euclidean division by Gröbner basis Groebner Bases computation n ¡ 2 in Oppndqpω`1qnq operations [Bardet, Faugère, Salvy 2015] n “ 2 in Opd2q with terse representation [van der Hoeven, Larrieu 2018] r Univariate Representation r Multivariate subresultant Rational Univariate Representation [Kronecker 1882, Rouillier 1999] u-resultant and its derivatives at point pt, a1,..., anq u-resultant “ C pu0 ` u1ζn ` ¨ ¨ ¨ ` unζnq ζ F ζ 0 | ¹p q“ Lexicographical Gröbner bases Polynomial parametrization generically, bigger size [Dahan, Schost 2004] Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 20 / 52 Univariate Representation with Geometric Resolution f px, yq “ 0 Theorem (Hensel lifting) y0 “ root of f p0, yq “ 0 f px,ynpxqq 2n`1 yn 1pxq “ ynpxq ´ mod x $ ` Bf px,y pxqq & By n 2n Then ynpxq is a% root of f px, yq “ 0 mod x . Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 21 / 52 Univariate Representation with Geometric Resolution f px, yq “ 0 Theorem (Hensel lifting) y0 “ root of f p0, yq “ 0 f px,ynpxqq 2n`1 yn 1pxq “ ynpxq ´ mod x $ ` Bf px,y pxqq & By n 2n Then ynpxq is a% root of f px, yq “ 0 mod x . Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 21 / 52 Univariate Representation with Geometric Resolution f px, yq “ 0 Theorem (Hensel lifting) y0 “ root of f p0, yq “ 0 f px,ynpxqq 2n`1 yn 1pxq “ ynpxq ´ mod x $ ` Bf px,y pxqq & By n 2n Then ynpxq is a% root of f px, yq “ 0 mod x . Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 21 / 52 Univariate Representation with Geometric Resolution f px, yq “ 0 Theorem (Hensel lifting) y0 “ root of f p0, yq “ 0 f px,ynpxqq 2n`1 yn 1pxq “ ynpxq ´ mod x $ ` Bf px,y pxqq & By n 2n Then ynpxq is a% root of f px, yq “ 0 mod x . Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 21 / 52 Univariate Representation with Geometric Resolution f px, yq “ 0 Theorem (Hensel lifting) y0 “ root of f p0, yq “ 0 f px,ynpxqq 2n`1 yn 1pxq “ ynpxq ´ mod x $ ` Bf px,y pxqq & By n 2n Then ynpxq is a% root of f px, yq “ 0 mod x . Guillaume Moroz (Inria Nancy - Grand Est) Real Algebraic Geometry and Robotics March 4th, 2021 21 / 52 Univariate Representation with Geometric Resolution f px, yq “ 0 Theorem (Hensel lifting) y0 “ root of f p0, yq “ 0 f px,ynpxqq 2n`1 yn 1pxq “ ynpxq ´ mod x $ ` Bf px,y pxqq & By n 2n Then ynpxq is a% root of f px, yq “ 0 mod x .
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