25/6/2006 Solving Multivariate Nonlinear Polynomial Systems Massarwi Fady Computer Aided Geometric Design (236716) Spring 2006

25/6/2006 References  "Computation of the solutions of nonlinear polynomial systems", by E. C. Sherbrooke and N. M. patrikalakis. Computer Aided Geometric Design, Vol 10, No 5, pp , 1993  "Geometric Constraint Solver using Multivariate Rational Spline Functions", by G. Elber and M. S. Kim. The Sixth ACM/IEEE Symposium on Solid Modeling and Applications, Ann Arbor, Michigan, pp 1-10, June  "Subdivision methods for solving polynomial equations", by B. Mourrain and J. P. Pavone, Technical Report 5658, Inria, Sophia-Antipolis, 2005.

25/6/2006 Problem definition  Given a set of n rational polynomial functions of m variables:  And m dimensional box  Find solution such that

25/6/2006 Definitions  Multi Index I is an ordered m-tuple of non-negative integers  Example : if M=(1,1) then  Bernstein i’th polynomial of degree m :  Bernstein basis function determined by multi index I and bounded by multi index M is defined by:

25/6/2006 Problem definition  Since each function f is polynomial it can be represented by a basis change in Bernstein basis  Define the graph F of the function f as :

25/6/2006 Problem definition  Since  We can write  And write the graph F of each function as :  The set of V are the control points of F, this representation is much powerful and allows to use different properties of Bernstein basis ( like convex hull property )  u is a solution to the polynomial system if the point (u,0) contained in all the graphs Fi

25/6/2006 Subdivision-based techniques  Numerical methods to find areas (box) where a single root ( solution ) could be found  In each step, If the current box dimensions are larger than some tolerance split it into two boxes and search in each of them recursively.  Advantages: Speed Stability Simple  Disadvantages: Designed for zero dimensional solution set -- No guarantee that all the roots are found Doesn’t supply information on the root multiplicity

25/6/2006 Subdivision-based techniques  We will review The following: Subdivision while restricting the search domain  PP ( projection polyhedron ), LP (linear programming ) - ( Sherbrooke & Patrikalakis) Subdivision with numerical improvements ( Elber & Kim )  Usage examples Further improvements ( Mourrain & Pavoni)

25/6/2006 Subdivision with domain restriction  Representing the graph of each function in the Bernstein basis allows us to use the convex hull property – The graph is contained in the convex hull of its coefficients ( v i )  If there is a solution for all the equations then the point should be contained in all the graphs, thus in the intersection of the convex hulls of the graphs coefficients

25/6/2006 Algorithm  Assume domain box is B = [0,1] n  Form the convex hulls of all F k and intersect them with one another and with the hperplane u n+1 =0, call the intersection set A, if A is smaller than a tolerance stop.  A=A’X{0}, find a box B’=[a 1,b 1 ] X[a 2,b 2 ] X….. X[a n,b n ] that encloses A’  If B’ is not singificantly smaller than [0,1] n, split B into equally sizes sub domains and work with each sub domain separately  For each k, define new function f’ k by  Can be computed by De Casteljau multivariate subdivision  Update B with B’ and continue from step 2

25/6/2006 Algorithm ( cont’d)  Step 2 contains two heavy operations Computing the convex hull in dimension n Intersecting all the convex hulls  Actually what we need is the bounding box of the intersection of the convex hulls, this can be achieved in different simple way

25/6/2006 Algorithm ( updated )  (1) -Start with initial search box  (2) - Perform transformation that brings the search box to [0,1] n  (3) - Find a sub box of [0,1] n which contains the roots  (4) - Use the reverse of the transformation used in step 2 to determine if the sub box is too small in R n, if yes there is a root there and report for the mid point of the box.  (5) - if the dimensions of the box is close to 1, split the box into two parts in each dimension.  (6) - Go back to step 2, once for each box.

25/6/2006 Finding the Bounding Box of the intersection of the CHs  Projected Polyhedron ( PP ) Find the width of the bounding box in each dimension by projecting it in that dimension For dimension j:  For each graph F k project each of it’s control points u to (u j,u n+1 ) ( call it x-y plane )  Form the convex hull in 2d of the projected points  Return as [a j,b j ] the intersection of the convex hull with the X axis in the projection plane ( between 0 and 1 for all the graphs ). Finally the bounding box will be  B = [a 1,b 1 ]X [a 2,b 2 ]X…… X[a n,b n ]

25/6/2006 Finding the Bounding Box of the intersection of the CHs  Linear Programming ( LP ) minimization function - If we look at the i’th coordinate of the points lying in the feasible area, what is the smallest ( a i ), and what is the biggest ( b i ) Constrains – for each point x in the feasible area, it must satisfy  x n+1 =0  For each k between 1 and n, x is contained in the convex hull of V k (the control points of F k )

25/6/2006 Linear Programming  Formally  Minimization functions:  Constrains Each point u is in the convex hull

25/6/2006 Linear Programming  The unknowns are the coefficients  The number of the unknowns is

25/6/2006 Analysis  The algorithm terminates The subdivision factor (p)  Linear convergence in the PP method  Quadratic convergence in the LP method

25/6/2006 Linear convergence in the PP method  Given a box B=[a 1,b 1 ]x[a 2,b 2 ]x…x[a n,b n ] which contains one and only one simple root x 0 of a system of polynomials f 0,k :[0,1] n  R where k ranges from 1 to n. let f k represent the subdivision of f 0,k over B. Let the new box after executing one step of the PP method, denoted by B ’ =[a’ 1,b’ 1 ]x[a’ 2,b’ 2 ]x…x[a’ n,b’ n ]. Let j be an integer between 1 and n, and let ( the lengths of the j’th side ) then

25/6/2006 Linear convergence in the PP method - Proof  Let and g(u) is the linear approximation of f,  Then the following holds :

25/6/2006 Linear convergence in the PP method - Proof  The projected control points of the graphs of the functions f, g ( (u,f(u)), (u,g(u)) ), are separated by vertical distances no more than  The projection of G on the (u j,u n+1 ) plane is planar region ( because G is linear approximation ). For a fixed u j, the difference between the maximum y-coordinate and minimum y coordinate of this region is given by:  This is obtained from g(x 1 )-g(x 0 ), where x 0 ={0,0,…,x j,0,0,…,0}, x 1 ={1,1,…,x j,1,1,…,1}

25/6/2006 Linear convergence in the PP method - Proof  Then we can say that the whole projected control points of F are enclosed by two parallel lines that has a distance of from each other and slope of  Then if we look at the intersection of these two lines with the x axis ( between 0 and 1), the distance between the intersections ( which is translated to the box dimension later ) is :

25/6/2006 Linear convergence in the PP method - Proof  Since   But we do scaling to transform the box to R n, by scaling by, and this is the new box we work with so the new box dimension is

25/6/2006 Quadratic convergence in the LP method  Given a box B=[a 1,b 1 ]x[a 2,b 2 ]x…x[a n,b n ] which contains one and only one root u 0 of a system of polynomials f 0,k :[0,1] n  R where k ranges from 1 to n. let f k represent the subdivision of f 0,k over B. After executing one step of the LP method, and let the new box denoted by B ’ =[a’ 1,b’ 1 ]x[a’ 2,b’ 2 ]x…x[a’ n,b’ n ]. Define. There exists a neighborhood U of u 0 and a depending only on U such that if and Let Then

25/6/2006 Quadratic convergence in the LP method-proof  The linearization of the n f k is  Lets denote by p={p 1,p 2,…,p n,0} the intersection ( in R n+1 ) of all g k (k=1..n) with x n+1 =0 ( i.e g 1 =g 2 =…=g n =x n+1 =0 ) ( such a point exists since the Jacobian matrix is not zero )  Pick a point q=(q 0,q 1,…,q n,0) that lies in the intersection of F 1, F 2,…, F n with hyperplane x n+1 =0  We know that  Substitute (q 0, q 1,…, q n ) for x, since all f k are zero there, then

25/6/2006 Quadratic convergence in the LP method-proof  letting r k =g k (q 1,q 2,…,q n ), then there exists n points r k =(q 1,q 2,…,q n,r k ) such that r k lies on linear approximation of F k,  For point p we know that  Subtracting * and **, and define we get

25/6/2006 Quadratic convergence in the LP method-proof  By Cramer’s rule we get ( assuming the Jacobian matrix is non singular ) that where J is the Jacobian matrix, and J k is the Jacobian matrix after exchanging k’th column with the column vector (r 1,r 2,…,r n ) T  Since then and since we get  Then

25/6/2006 Quadratic convergence in the LP method-proof  q is chosen arbitrary and its distance from other fixed point ( p ) is, then the distance between two points in the bounding box we want is also  Then the length of each side of the bounding box is also  The real size then ( in R n ) is multiplied by factor of the bounding box,then the dimensions of the new bounding box is  This hold for all K, then

25/6/2006 Subdivision with numerical improvements  Subdivide the multivariate functions bounded within a certain resolution  Using the convex hull property if for a constrain F i all the control points have the same sign then it won’t have a roots.  For each cell with dimensions lower than the wanted resolution report for a root ( center of the cell )  Subdivision is terminated when it is known that there is an isolated root in the domain, in this case Newton- Raphson iterations is applied ( with quadratic convergence )  If the zero set has a dimension larger than 0, then fit a multivariate surface to the set of the discrete points found by means of least squares ( minimizing the L2 norm )

25/6/2006 Root Finding Algorithm

25/6/2006 Numerical improvement of the solution  for a given approximated solution u 0 found, such that for each constrain F i (u 0 )~0, using first order approximation, each graph F i constrains the solution to be on the hyperplane which is defined in the point u 0 as following : Consider the normal to the surface F i (u) at the point u 0 : Then the hyper plane at the point u 0 is defined as

25/6/2006 Numerical improvement of the solution  There are m such planes, and with the additional constraint that the component u m+1 =0, we end with n+1 linear equations( with m+1 variables)  If n=m then the system is full constrained and there is one solution  Otherwise if n < m a least square solution is found in an L2-norm ( finds the closest solution to u 0 from the possible solutions set)

Uniqueness of a solution  If it is known that there is a single solution then it would be better to stop the subdivision and move to more efficient numerical procedure for root finding.  Definitions: Cone: normal cone of hypersurface F i : set of all possible normal vectors on F i. Complementary (Tangent) cone:

25/6/2006 Uniqueness of a solution  Theorm: Given m implicit hypersurfaces F i (u)=0. i=1,…m in R m, there is at most one common solution if  Proof: Consider u 0 (in R m ) is a common solution of the m equations F i (u)=0. Consider, from the relation we have

25/6/2006 Examples  Ray-Traps Definition:  Given a set of n planar curves, it’s ray-trap of length n is a set of n points {P i =C i (u i )} such that a ray bouncing from P i towards P (i+1)mod n will be reflected toward P (i+2)mod n The incoming ray from P i-1 and the outgoing ray towards P i+1 should form an identical angle with the normal of C i at P i This can be written as:

25/6/2006 Examples- ray-traps

25/6/2006 Examples  Surface-Surface Bisectors Let S 1 (u,v) and S 2 (u,v) be two regular rational surfaces in R 3 The points B=(x,y,z) on the bisector surface of S 1 and S 2 must satisfy the following constraints

25/6/2006 Examples-Bisectors  Let  Then we can choose and the problem is reduced to

25/6/2006 Examples- Bisectors  4 variables with 2 equations – under constrained problem  Can fit a bi-variate surface to discrete points sampled on the solution space.

25/6/2006 Examples- Bisectors

25/6/2006 Further improvements  Preconditioning Transformation of the system f=0 into equivalent system Mf=0 where M is n X n non singular matrix global transformation  Useful when two or more functions have closed graphs on the domain, aims to increase the distance between them  The distance between two graphs is not trivial, so it is replaced by the distance between the vectors of the Bernstein basis coefficients Local straightening:  Interesting situation of reduction is when the zero-set of the function f i are orthogonal to the x i direction

25/6/2006 Preconditioning- global transformation  If then define the inner product between functions  Let and let E be a matrix of unitary eigenvectors of Q. Then f’=Ef is a system of polynomials which are orthogonal for the inner product above.  Proof:

25/6/2006 Preconditioning- global transformation

25/6/2006 Preconditioning- Local straightening Local straightening A better domain reduction can be done when the zero level of the functions f i are orthogonal to the x i axis. This can be done by transforming the system f=0 in order to be closed to case (b). Transform locally the system into a system J f -1 (u 0 )f, where J f is the Jacobian matrix of f at the point u 0

25/6/2006 Questions ??? Thanks

25/6/2006 Backup

25/6/2006 Improvements- Examples

25/6/2006 Further improvements - Reduction  For a function f and j=1,…,n let  Then  For any root u=(u 1, u 2,…, u n ) of the equation f(u)=0, we have where t1/t2 is either a root of m j (f;u j )=0 or M j (f;u j )=0 in [a j,b j ] or a j /b j if m j (f;u j )=0/ M j (f;u j )=0 has no roots in [a j,b j ] m j (f;u j )<=0<=M j (f;u j )

25/6/2006 Further improvements  Reduction Similar to the PP ( projected Polyhedron ), reduce the domain of search PP is based on the convex hull property The improvement consists in computing the first/last root of the polynomial m j (f k,u j )/M j (f k,u j ) in the interval [a j,b j ] and keep the intervals [t 1,t 2 ]  Analysis shows that with these improvements we get a quadratic convergence.