computing an irreducible decomposition of A B looking for solutions where the Jacobian drops rank

Transcription

1 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates MCS 563 Lecture 37 Analytic Symbolic Computation Jan Verschelde, 16 April 2014 Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

2 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

3 intersecting solution sets Consider the intersection two solution sets A and B, given by their respective systems f A and f B. For example: A is the line x 2 = 0, defined as one of the solution components of f A (x 1, x 2 )=x 1 x 2 = 0; and B is the line x 1 x 2 = 0, defined as one of the solution components of f B = x 1 (x 1 x 2 )=0. If we just stack the defining equations like { fa = x f (x) = 1 x 2 = 0 f B = x 1 (x 1 x 2 )=0 then the problem is that A B =(0, 0) does not occur as an irreducible solution component of f 1 (0). Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

4 sphere cylinder The curve C defined by this intersection is C := { (x 1, x 2, x 3 ) x x = 0 (x ) 2 + x x = 0 } Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

5 equations The equations f A (x) =x1 2 + x = 0 and f B (x) =(x ) 2 + x2 2 + x = 0 define a witness set representation of the cylinder and the sphere. Both sets are two dimensional and of degree two. Each witness set contains two random hyperplanes and two points of intersection of these hyperplanes with the defining equations. As the input is given as two witness sets, the output consists also of witness sets. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

6 the idea Consider the product A B. The intersecting hyperplanes for A and B are respectively L A1, L A2 and L B1, L B2. Then we use u and v for the coordinates of A and B. A special instance of the diagonal homotopies is h(x, t) = (1 t) f A (u 1, u 2, u 3 )=0 f B (v 1, v 2, v 3 )=0 L A1 (u 1, u 2, u 3 )=0 L A2 (u 1, u 2, u 3 )=0 L B1 (v 1, v 2, v 3 )=0 L B2 (v 1, v 2, v 3 )=0 + t f A (u 1, u 2, u 3 )=0 f B (v 1, v 2, v 3 )=0 u 1 v 1 = 0 u 2 v 2 = 0 u 3 v 3 = 0 L AB (u 1, u 2, u 3 )=0. As t goes from 0 to 1, the deformation starts at pairs (α, β) A B, where α and β are witness point on A and B respectively satisfying f A (α) =0, f B (β) =0 and L A1 (α) =0 = L A2 (α), L B1 (β) =0 = L B2 (β). At t = 1 we find witness points on the curve of intersection. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

7 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

8 the singular locus The singular locus of the solution set of f (x) =0 is where the Jacobian matrix J f (x) of f is singular. Using n auxiliary multipliers λ, we consider the intersection of A = f 1 (0) with the solution set B of the system { Jf (x)λ = 0 h T λ = 1 where h is a random vector of complex coefficients to scale the multiplier coefficients in the combinations of the columns of the Jacobian matrix. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

9 a nontrivial example A nontrivial example from molecular chemistry: 1 2 (x x 2x 3 + x3 2)+a(x 2 2x 3 2 1) =0 f (x, a) = 1 2 (x x 3x 1 + x1 2)+a(x 3 2x 1 2 1) =0 1 2 (x x 1x 2 + x2 2)+a(x 1 2x 2 2 1) =0. For generic choices of the parameter a, the system f (x, a) =0 has 16 isolated solutions. Setting up the Jacobian system, we compute those values for a for which f (x, a) =0 has isolated singularities or positive dimensional solution sets. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

10 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

11 restricted systems We generalize coefficient-parameter homotopies, for f (x, y) =0 over X Y, where Y is the parameter space. We want the solutions to f (x, y )=0, forsomey Y. 1 Choose a general y Y (y y ). D =#{ x f (x, y )=0 } is maximal for all y Y. 2 Construct a curve B Y connecting y to y. 3 Construct a map c :[0, 1] Γ B, Γ={ γ C γ = 1 }, so that c(0, Γ) = y and c(1, Γ) = y. 4 Choose γ Γ at random and track D solution paths defined by the homotopy f (x, c(t,γ)) = 0, starting at t = 0 at the solutions of f (x, y )=0 and ending at t = 1 at the desired solutions of f (x, y )=0. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

12 embedding polynomial systems Let X be a reduced pure n-dimensional algebraic set and consider f in a cascade of embeddings E n (f )=E(f ) and E 0 (f )=f, where E i (f ) is restricted to Y i. E(f, x, z, Y )= [ f (x)+a T 2 z z A 0 A 1 x ] Y =(A0, A 1, A 2 ), A 0 C n 1, A 1 C n n, A 2 C n n. The stratification of the parameter space is Y 0 Y 1 Y n,last N i rows of Y i are zero. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

13 a generalized cascade For random γ i C, γ i = 1, the homotopy H i (x, z, t, Y,γ i )= ( ) Ei 1 (f )(x, z, Y γ i (1 t) E i (f )(x, z, Y i )+t i 1 ) = 0, defines paths starting at t = 0 at the solutions of E i (f ), ending at t = 1 at the solutions of E i 1 (f ). Theorem For the homotopy H i (x, z, t, Y,γ i )=0 the following holds: 1 Solutions of H i (x, z, t = 0, Y,γ i )=0 with z =(z 1,z 2,...,z i ) 0 are regular, and stay regular for all t < 1. 2 Solutions of H i (x, z, t = 1, Y,γ i )=0 contain all witness sets on the (i 1)-dimensional components of f 1 (0). z i Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

14 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

15 problem statement Input: two irreducible components A and B, givenby polynomial systems f A and f B (possibly identical), random hyperplanes L A and L B, and the solutions to { fa (x) =0 L A (x) =0 #L A = dim(a) =a { α 1,α 2,...,α deg A } deg A generic points }{{} a witness set for A and Output: witness sets for all components of A B. { fb (x) =0 L B (x) =0 #L B = dim(b) =b { β 1,β 2,...,β deg B } deg B generic points }{{} a witness set for B Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

16 a numerical embedding Let X be an N-dimensional solution component of g(x) =0, a system of n equations g =(g 1, g 2,...,g n ) in x C m. Randomize g to have as many equations as co-dimension of X : G(x) :=R(g(x), m N) =Λg(x), Λ C (m N) n, where Λ is a random matrix. In the cascade of homotopies, replace E i (f ) by [ G(x) E i (f )(x, z) ]. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

17 decomposing the diagonal Given two irreducible components A and B in C k, consider their product X := A B C k+k. Then A B = X where is the diagonal of C k+k defined by u 1 v 1 = 0 u 2 v 2 = 0 δ(u, v) := on X.. u k v k = 0 Notice: δ plays role of f in the abstract embedding. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

18 input data for diagonal homotopies Let A C k be an irreducible component of f 1 A (0), dima = a; B C k be an irreducible component of f 1 B (0), dimb = b. Assuming a b and B A, then dim(a B) b 1. Randomize: F A (u) :=R(f A, k a) and F B (v) :=R(f B, k b). A B is a solution component of F(u, v) := [ FA (u) F B (v) ] = 0. Let {α 1,α 2,...,α deg A } satisfy F A (u) =0 and L A (u) =0; and {β 1,β 2,...,β deg B } satisfy F B (v) =0 and L B (v) =0, where L A (u) =0 is a system of a general hyperplanes; and L B (v) =0 is a system of b general hyperplanes. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

19 diagonal homotopies, when a + b < k Randomize the diagonal D(u, v) :=R(δ(u, v), a + b). At the start of the cascade (denote z 1:b =(z 1, z 2,...,z b ) T ): The homotopy E b (u, v, z 1:b )= t γ F(u, v) L A (u) L B (v) z 1:b F(u, v) R(D(u, v), z 1,...,z b ; a + b) z 1:b R(1, u, v; b) +(1 t) E b(u, v, z 1:b ) = 0. = 0 starts the cascade at t = 1, at the deg A deg B solutions, at the product {(α 1,β 1 ), (α 1,β 2 ),...,(α deg A,β deg B )} C 2k. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

20 diagonal homotopies, when a + b k As A B dim(a B) a + b k, the cascade starts at E b (u, v, z (a+b k+1):b )= F(u, v) R(δ(u, v), z a+b k+1,...,z b ; k) R(1, u, v; a + b k) z (a+b k+1):b R(1, u, v; k a) = 0, where z (a+b k+1):b =(z a+b k+1,...,z b ) T.Use F(u, v) t γ L A (u) L B (v) +(1 t)e b(u, v, z (a+b k+1):b ) = 0 z (a+b k+1):b as before to start the cascade at t = 1. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

21 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

22 witness set for a cylinder The file cylinder contains 1 3 x^2 + y^2 + z - z - 1; With phc -l we intersect with a line: $ phc -l Welcome to PHC (Polynomial Homotopy Continuation) v Witness Set for Hypersurface cutting with Random Line. Reading the name of the file with the polynomial. Give a string of characters : cylinder Reading the name of the output file. Give a string of characters : out1 $ Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

23 witness set for a sphere The file sphere contains 1 3 (x+0.5)^2 + y^2 + z^2-1; With phc -l we intersect with a line: $ phc -l Welcome to PHC (Polynomial Homotopy Continuation) v Witness Set for Hypersurface cutting with Random Line. Reading the name of the file with the polynomial. Give a string of characters : sphere Reading the name of the output file. Give a string of characters : out2 $ Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

24 Witness Set Intersection $ phc -w Welcome to PHC (Polynomial Homotopy Continuation) v Witness Set Intersection using Diagonal Homotopies. Reading a file name for witness set 1. Give a string of characters : cylinder_w2 Reading a file name for witness set 2. Give a string of characters : sphere_w2 Reading the name of the output file. Give a string of characters : out3 $ Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

25 the output file The file out3_w1 contains a witness set: 4 x^2+y^2+( E E-01*i)*zz1-1; x^2+y^2+z^2+x+( e E-01* E-01; zz1; ( E E-02*i)*x +( E E+00*i)*y +z+zz1+( E E+00*i); TITLE : witness set of dimension 1, see out3 for diagnostics The rest of the file contains the witness points, as solutions of the polynomial system. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

26 an illustrative example f = (y x 2 )(x 2 + y 2 + z 2 1)(x 0.5) (z x 3 )(x 2 + y 2 + z 2 1)(y 0.5) (y x 2 )(z x 3 )(x 2 + y 2 + z 2 1)(z 0.5) This example has been constructed in a factored form so that it is easy to identify the decomposition of Z = f 1 (0) into its irreducible solution components, as Z = Z 2 Z 1 Z 0 = {Z 21 } {Z 11 Z 12 Z 13 Z 14 } {Z 01 } where Z 21 is the sphere, Z 11, Z 12, Z 13 are lines, Z 14 is the twisted cubic, and Z 01 is a point.. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

27 flow in the solver deg(f 1 )=5 deg(f 2 )=6 2 common satisfy f 3 (5 2, 6 2) to continue (5 2) (6 2) =12 solution paths 5 at infinity 6satisfyf 3 to classify 1 to continue deg(f 3 )=8 8 4satisfyf 1 3satisfyf 2 = 1left (8 7) 1 = 1 solution paths 1 to classify Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

28 Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an Algebraic Set generalized setup of coefficient-parameter homotopies 4 diagonal homotopies homotopies to intersect solution sets a session with phc intrinsic coordinates Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

29 intrinsic coordinates To avoid the doubling of the number of variables, we use parametric representations of the linear spaces. Let f be one polynomial defining a hypersurface: f (x 1, x 2,...,x n )=0 c 1,0 + c 1,1 x 1 + c 1,2 x c 1,n x n = 0 c 2,0 + c 2,1 x 1 + c 2,2 x c 2,n x n = 0. c n 1,0 + c n 1,1 x 1 + c n 1,2 x c n 1,n x n = 0 then we cut f 1 (0) with a line, represented as x = b + λv, b, v C n and we solve a univariate equation in λ: f (x = b + λv) =0. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

30 solving equation-by-equation Diagonal homotopies lead to a numerical algorithm to solve polynomials systems one equation after the other. Symbolic approaches to solve equation-by-equation: lifting fibers for a geometric resolution, the algorithm F5 for a Gröbner basis. These methods compensate for a disappointing complexity and in practice, the polynomials in many systems occur in a regular sequence. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31

31 Summary + Exercises Diagonal homotopies provide numerical algorithms to intersect algebraic sets. Exercises: 1 Consider the intersection of {x = 0, y = 0} with {z = 0, w = 0}. Describe how the diagonal homotopy works for this example. 2 Set up the system for the nontrivial example to compute its singular locus. Use the diagonal homotopies available in phc -c to compute the singular values for a. Compute a witness set for those a which lead to positive dimensional solution sets. Apply deflation to treat isolated singular solutions. 3 Give algorithms to define the linear algebra operations to convert between extrinsic and intrinsic coordinate representations of witness sets for hypersurfaces. Analytic Symbolic Computation (MCS 563) Diagonal Homotopies L April / 31