Rational Univariate Representation

Transcription

1 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple MCS 563 Lecture 12 Analytic Symbolic Computation Jan Verschelde, 10 February 2014 Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

2 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

3 RUR A Rational Univariate Representation (or RUR) is { R = p 0 (T) = 0, x i = p } i(t), i = 1, 2,...,n q(t) where p 0, p 1, p 2,..., p n, q C[T]. This set R represents the coordinates of the zeroes of a solution set V, #V = D <, of some system of polynomials in C[x]. D is counted with multiplicities. The number of distinct zeroes in V is denoted by d. RUR generalizes the Shape Lemma representation. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

4 separating linear form Let the linear form L(x) separate the zeroes: L(z i ) L(z j ), for z i, z j V, i j. Consider the multiplication map m L : C[x]/I(V) C[x]/I(V) : h ((h L) G> r) where G> represents the normal form algorithm implemented by the division algorithm using some Gröbner basis G >. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

5 choosing a separator Lemma If V has d distinct zeroes, at least one of the for u i (x 1, x 2, x 3,..., x n ) = x 1 + ix 2 + i 2 x i n 1 x n, ( d 0 i (n 1) 2 is separating, i.e.: z j, z k V, j k, u i (z j ) u i (z k ). ) Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

6 proof of the lemma Proof. For the pair (z j, z k ), j k, of two distinct zeroes in V with components z j = (z j1, z j2,..., z jn ) z k = (z k1, z k2,...,z kn ), consider the bad situation when u t (z j ) = u t (z k ), corresponding to p(t) = (z j1 z k1 ) + (z j2 z k2 )t + + (z jn z kn )t n 1. Because z j z k, p 0 and therefore p can have at most n 1 zeroes. So for each pair of zeroes of V we have at most n 1 bad choices for u i and the number of pairs of zeroes is d(d 1)/2, yielding a total of at most (n 1)d(d 1)/2 nonseparating u i s. But the set of u i s consists of (n 1)d(d 1)/2 + 1 elements, so there is at least one separating u i. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

7 Stickelberger s Theorem Theorem (Stickelberger s theorem) The multiplication map m L is a linear map with matrix M L. The eigenvalues of M L give values for L(z), for all z V, occurring with the same multiplicity µ z. As a consequence, the characteristic polynomial of M L is p 0 (T) = det(m L T I D ) = z V (T L(z)) µz, with µ z the multiplicity of the root. The trace of M L is trace(m L ) = z V µ z L(z). The determinant of M L is det(m L ) = z V L(z) µz. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

8 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

9 The Elbow Manipulator We consider a spatial robot arm with three links. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

10 inverse kinematics We consider a spatial robot arm with three links. The input to an inverse position problem is n x o x a x p x n y o y a y p y n z o z a z p z representing position and orientation of the robot hand: p = (p x, p y, p z ) is the position of the hand; n = (n x, n y, n z ), n 2 = 1, normal; o = (o x, o y, o z ), o 2 = 1, orientation; a = (a x, a y, a z ), a 2 = 1, approach vector; related by the cross product: n = o a. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

11 coordinate transformations Wanted: angles θ i, i = 1, 2,...,6, s i = sin(θ i ), c i = cos(θ i ), and s 2 i + c 2 i = 1. Lengths of links are L 2, L 3, and L 4. c 1 0 s 1 0 s 1 0 c c 3 s 3 0 c 3 L 3 s 3 c 3 0 s 3 L c 5 0 s 5 0 s 5 0 c c 2 s 2 0 c 2 L 2 s 2 c 2 0 s 2 L c 4 0 s 4 c 4 L 4 s 4 0 c 4 s 4 L c 6 s s 6 c = n x o x a x p x n y o y a y p y n z o z a z p z , Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

12 RUR and the Newton Identities To compute { R = we proceed along two steps: 1 apply Newton s formula for p 0 ; p 0 (T) = 0, x i = p } i(t), i = 1, 2,...,n q(t) 2 compute in the quotient algebra to find the rest. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

13 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

14 the elementary symmetric polynomials A polynomial p in one variable x defined by its roots x 1, x 2, x 3, and x 4, written as a monic polynomial: p(x) = (x x 1 )(x x 2 )(x x 3 )(x x 4 ) = x 4 (x 1 + x 2 + x 3 + x 4 )x 3 + (x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 )x 2 (x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 3 x 4 + x 2 x 3 x 4 )x + x 1 x 2 x 3 x 4 = x 4 e 1 (x 1, x 2, x 3, x 4 )x 3 + e 2 (x 1, x 2, x 3, x 4 )x 2 e 3 (x 1, x 2, x 3, x 4 )x + e 4 (x 1, x 2, x 3, x 4 ). The polynomials e 1, e 2, e 3, and e 4 are the elementary symmetric polynomials. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

15 the power sums Substituting the roots x 1, x 2, x 3, and x 4 into p(x) = x 4 e 1 x 3 + e 2 x 2 e 3 x + e 4 gives 0 = p(x 1 ) = x 4 1 e 1x e 2x 2 1 e 3x 1 + e 4, 0 = p(x 2 ) = x 4 2 e 1x e 2x 2 2 e 3x 2 + e 4, 0 = p(x 3 ) = x 4 3 e 1x e 2x 2 3 e 3x 3 + e 4, 0 = p(x 4 ) = x 4 4 e 1x e 2x 2 4 e 3x 4 + e 4. Adding up: 0 = s 4 e 1 s 3 + e 2 s 2 e 3 s 1 + 4e 4, where s 1, s 2, s 3, and s 4 are the power sums: s 1 = x 1 + x 2 + x 3 + x 4, s 2 = x1 2 + x x x 4 2, s 3 = x1 3 + x x x 4 3, s 4 = x1 4 + x x x 4 4. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

16 the Newton Identities Expressing the power sums in terms of the elementary symmetric polynomials: s 1 = e 1, s 2 = e 1 s 1 2e 2, s 3 = e 1 s 2 e 2 s 1 + 3e 3, s 4 = e 1 s 3 e 2 s 2 + e 3 s 1 4e 4. The relations above allow the derivations of the power sums from the coefficients of a monic polynomial. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

17 computing coefficients from power sums Writing the Newton identities in another way: e 1 = s 1, 2e 2 = e 1 s 1 s 2, 3e 3 = e 1 s 2 e 2 s 1 + s 3, 4e 4 = e 1 s 3 e 2 s 2 + e 3 s 1 s 4, we see that, given the power sums of the roots, we can derive the coefficients of the monic polynomial that vanishes at those roots. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

18 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

19 the characteristic polynomial To derive the characteristic polynomial p 0, the trace of L i is s i = z V µ z L i (z). If p 0 (T) = D b i T D i, D = #V, b i C[x], i=0 then p 0 (T) p 0 (T) = z V D 1 We have p 0 (T) = µ z T L(z) = j 0 l=0 D l 1 j=0 Apply Newton s formula: (D i)b i = This gives a linear system in the b i s. trace(l j ) T j+1. trace(l j )b l T D l j 1. i trace(l j )b i j. j=0 Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

20 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

21 separating form L For any v A, A is the quotient algebra, we define g L (v, T) = µ z v(z) (T y). z V y V(p 0 ) y L(z) Observe g L (v, L(z)) g L (1, L(z)) = µ z v(z) z V (L(z) y) y V(p 0 ) y L(z) µ z (L(z) y) z V y V(p 0 ) y L(z) = v(z). Then we let v become a coordinate x i... Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

22 representing coordinates As g L(v, L(z)) g L (1, L(z)) = v(z), we let v = x i and we have Note x i = g L(x i, T) g L (1, T), g L (1, T) = i = 1, 2,...,n. p 0 (T) GCD(p 0 (T), p 0(T)). If all roots occur with multiplicity one, the denominator g L (1, T) is just the derivative of p 0 (T). Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

23 computing g L (v, T) To compute g L (v, T), we define and p 0 (T) = p 0 (T) GCD(p 0 (T), p 0(T)), d 1 g L (v, T) = j=0 d j 1 k=0 trace(vl j )a i T d j k 1, p 0 = d a i T d i. i=0 Then we set q(t) = g L (1, T) and p i (T) = g L (x i, T) and obtain a representation for RUR. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

24 Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities the Newton identities computing the characteristic polynomial computing the coordinates 4 Algorithm and Software pseudo code to compute a RUR, using Maple Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

25 pseudo code Input:. G >, a Gröbner basis for an ideal I with term order >, #V(I) = D <. Output: a rational univariate representation for V( I). compute N > the basis vector for the quotient ring; let D = #N > = #V(I), counted with multiplicities; compute TrM and deduce d = #distinct zeroes; choose a separating element u as one of the u i s; compute for m from 1 to D: trace(u m ) and use u to form p 0 (T); compute p 0 for I, if deg( p 0 ) < d then choose another u; for j from 1 to D for i from 0 to d compute trace(x j u i ) and deduce g u (x j, T); set q(t) = g u (1, T) and p i = g u (x i, T), i = 1, 2,...,n. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

26 using Maple In Maple, we compute a RUR as follows: [> f := [2*x[1]^2 + 2*x[2]^2 + 2*x[3]^2 + x[4]^2 - x[4], 2*x[1]*x[2] + 2*x[2]*x[3] + 2*x[3]*x[4] - x[3], 2*x[1]*x[3] + x[3]^2 + 2*x[2]*x[4] - x[2], 2*x[1] + 2*x[2] + 2*x[3] + x[4] - 1]; [> v := x[4],x[3],x[2],x[1]; [> Groebner[Basis](f,plex(v)); [> Groebner[RationalUnivariateRepresentation] (f,v,output=factored); Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

27 Summary + Exercises RUR generalizes the Shape Lemma representation. Exercises: 1 Construct an example of a solution set V in two variables and #V > 1 where all except for one choice of the u i s fail to be separating. 2 Use a lexicographic term order to compute a Gröbner basis for the system Katsura, for n = 3 (see lecture 3). How many decimal places does the largest coefficient in this basis have? Compare with the size of the coefficients in the lecture note. 3 Use Maple s Groebner[RationalUnivariateRepresentation] on the example of the previous exercise. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

28 more exercises 4 Create a Maple worksheet to define the polynomial system for the elbow manipulator, for general choices of the position. Solve the system using the choices in the lecture note, either by Maple or by Sage.Singular. 5 Consider the system: 24x 1 x 2 x1 2 x 2 2 x 1 2x = 0 f(x) = 24x 2 x 3 x2 2 x 3 2 x 2 2x = 0 24x 3 x 1 x3 2 x 1 2 x 3 2x = 0. 1 Solve the system to verify that no variable is separating. 2 Find a separating element of the form as u i an run through the steps of Algorithm to compute a RUR. Use a worksheet or a notebook to guide the computations. 3 Use the builtin commands in Maple or Sage to compute a rational univariate representation. Compare the output with the outcome of the step-by-step execution of Algorithm to compute a RUR. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29

29 one last exercise 6 Consider the following modification of the cyclic 5-roots problem: x 1 + x 2 + x 3 + x 4 + x 5 = 0 x 1 x 2 + x 2 x 3 + x 3 x 4 + x 4 x 5 + x 5 x 1 = 0 f(x) = x 1 x 2 x 3 + x 2 x 3 x 4 + x 3 x 4 x 5 + x 4 x 5 x 1 + x 5 x 1 x 2 = 0 x 2 x 3 x 4 + x 2 x 3 x 4 x 5 + x 3 x 4 x 5 x 1 + x 4 x 5 x 1 x 2 + x 5 x 1 x 2 x 3 = 0 x 1 x 2 x 3 x 4 x 5 1 = 0. where the monomial x 1 x 2 x 3 x 4 in the original cyclic 5-roots system is replaced by x 2 x 3 x 4. Compute a Gröbner basis with the graded lexicographical order to determine the number of roots of this modified cyclic 5-roots system. Compute a rational univariate representation for this system. Compare the size of the coefficients between the lexicographical Gröbner basis (shape lemma) and the RUR. Analytic Symbolic Computation (MCS 563) Rational Univariate Representation L February / 29