Certifying solutions to square systems of polynomial-exponential equations

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1 Certifying solutions to square systems of polynomial-exponential equations Jonathan D Hauenstein Viktor Levandovskyy September 8, 0 Abstract Smale s α-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations For a given system of polynomial-exponential equations, we consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system This bound yields a complete algorithm for certifying solutions to polynomial-exponential systems, which is implemented in alphacertified Examples are presented to demonstrate this certification algorithm Key words and phrases certified solutions, alpha theory, polynomial system, polynomialexponential systems, numerical algebraic geometry, alphacertified 00 Mathematics Subject Classification Primary 65G0; Secondary 65H05, 4Q99, 68W30 Introduction A map f : C n C n is called a square system of polynomial-exponential functions if f is polynomial in both the variables x,, x n and finitely many exponentials of the form e βxi where β C That is, there exists a polynomial system P : C n+m C n, analytic functions g,, g m : C C, and integers σ,, σ m {,, n} such that fx,, x n ) = P x,, x n, g x σ ),, g m x σm )) where each g i satisfies some linear homogeneous partial differential equation PDE) with complex coefficients In particular, for each i =,, m, there exists a positive integer r i and a linear function l i : C ri+ C such that l i g i, g i,, gri) i ) = 0 Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX jhauenst@mathtamuedu, wwwmathtamuedu/~jhauenst) This author was supported by Texas A&M University, the Mittag-Leffler Institute, and NSF grants DMS-095 and DMS-4336 Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 506 Aachen Germany ViktorLevandovskyy@mathrwth-aachende, wwwmathrwth-aachende/~viktorlevandovskyy)

2 Consider the square polynomial-exponential system F : C n+m C n+m where P x,, x n, y,, y m ) y g x σ ) Fx,, x n, y,, y m ) = ) y m g m x σm ) Since the projection map x, y) x defines a bijection between the solutions of Fx, y) = 0 and fx) = 0, we will only consider certifying solutions to square systems of polynomial-exponential equations of the form Fx, y) = 0 For a square system g : C n C n of analytic functions, a point x C n is an approximate solution of g = 0 if Newton iterations applied to x with respect to g quadratically converge immediately to a solution of g = 0 The certificate returned by our approach that a point is an approximation solution of F = 0 is an α-theoretic certificate In short, α-theory, which started for systems of analytic equations in [9], provides a rigorous mathematical foundation for the fact that if the Newton residual at the point is small and the higher order derivatives at the point are controlled, then the point is an approximate solution For polynomial systems, by exploiting the fact that there are only finitely many nonzero derivatives, Shub and Smale [8] provide a bound on all all of the higher order derivatives For polynomial-exponential systems, our approach uses the structure of F together with the linear functions l i to bound the higher order derivatives Systems of polynomial-exponential functions naturally arise in many applications including engineering, mathematical physics, and control theory to name a few On the other hand, such functions are typical solutions to systems of linear partial differential equations with constant coefficients Systems, including ubiquitous functions like sinx), cosx), sinhx), and coshx), can be equivalently reformulated as systems of polynomial-exponential functions, since these functions can be expressed as polynomials involving e βx for suitable β C Since computing all solutions to such systems is often nontrivial, methods for approximating and certifying some solutions for general systems is very important, especially in the aforementioned applications In the rest of this section, we introduce the needed concepts from α-theory Section formulates the bounds for the higher order derivatives of polynomial-exponential systems and presents a certification algorithm for polynomial-exponential systems In Section 3, we discuss methods for generating numerical approximations to solutions of polynomial-exponential systems Section 4 describes the implementation of the certification algorithm in alphacertified with Section 5 demonstrating the algorithms on a collection of examples Smale s α-theory We provide a summary of the elements of α-theory used in the remainder of the article as well as in alphacertified Hence, this section closely follows [4, ] expect polynomial is replaced by analytic We focus on square systems, which are systems with the same number of variables and functions, with more details provided in [] Let f : C n C n be a system of analytic functions with zeros Vf) = {ξ C n fξ) = 0} and Dfx) be the Jacobian matrix of f at x For a point x C n, the point N f x) is called the Newton iteration of f at x where the map N f : C n C n is defined by { x Dfx) N f x) = fx) if Dfx) is invertible, x otherwise

3 For k N, let N k f x) be the kth Newton iteration of f at x, that is, Nf k x) = N f N f x) }{{} k times The following defines an approximate solution of f to be a point which converges quadratically in the standard Euclidean norm on C n to a point in Vf) Definition Let f : C n C n be an analytic system A point x C n is an approximate solution of f = 0 with associated solution ξ Vf) if, for every k N, N k f x) ξ ) k x ξ Clearly, every solution of f = 0 is an approximate solution of f = 0 Additionally, when Dfx) is not invertible, then a point x is an approximate solution of f = 0 if and only if x Vf) When Dfx) is invertible, the results of α-theory provide a certificate that x is an approximate solution of f = 0 This certificate is based on the constants αf, x), βf, x), and γf, x) which are defined as αf, x) = βf, x) γf, x), βf, x) = x N f x) = Dfx) fx), and γf, x) = sup Dfx) D k fx) k k! k ) where D k fx) is the k th derivative of f see [6, Chap 5]) When Dfx) is not invertible, we define βf, x) as zero and γf, x) as infinity The constant αf, x) is then the indeterminate form 0 which is defined based on the value of fx) If fx) = 0, then αf, x) is defined as zero, otherwise αf, x) is defined as infinity The following lemma, which is a conclusion of Theorem of [, Chap 8], shows that, when x is an approximate solution of f = 0, the distance between x and its associated solution can be bounded in terms of βf, x) Moreover, this bound can be used to produce a certificate that two approximate solutions have distinct associated solutions Lemma Let f : C n C n be an analytic system If x C n is an approximate solution of f = 0 with associated solution ξ, then x ξ βf, x) Moreover, if x, x C n are approximate solutions of f = 0 with associated solutions ξ, ξ, respectively, then ξ ξ provided that x x > βf, x ) + βf, x )) Proof Both results immediately follow from the triangle inequality In particular, x ξ x N f x) + N f x) ξ βf, x) + x ξ yields x ξ βf, x) Additionally, x x x ξ + ξ ξ + ξ x βf, x ) + βf, x )) + ξ ξ 3

4 yields that ξ ξ when x x > βf, x ) + βf, x )) The following theorem, called an α-theorem, is a version of Theorem of [, Chap 8] which shows that the value of αf, x) can be used to produce a certificate that x is an approximate solution of f = 0 Theorem 3 If f : C n C n is an analytic system and x C n with αf, x) < then x is an approximate solution of f = , The following theorem, called a robust α-theorem and is a version of Theorem 4 and Remark 6 of [, Chap 8], shows that the value of αf, x) and γf, x) can be used to produce a certificate that x and another point y have the same associated solution Theorem 4 Let f : C n C n be an analytic system and x C n with αf, x) < 003 If y C n such that x y < 0γf, x), then x and y are both approximate solutions of f = 0 with the same associated solution Let π R : C n R n be the real projection map defined by π R x) = x + x where x is the conjugate of x If f is an analytic system such that N f x) = N f x) for all x such that Dfx) is invertible, then N f defines a real map, ie, N f R n ) R n In particular, if x is an approximate solution of f = 0 with associated solution ξ, then x is also an approximate solution of f = 0 with associated solution ξ and βf, x) = βf, x) The following proposition, which is a summary of the approach in [4, ], can be used to determine if the associated solution of an approximation solution is real Proposition 5 Let f : C n C n be a polynomial system such that N f x) = N f x) for all x C n such that Dfx) is invertible Let x C n is an approximate solution of f = 0 with associated solution ξ If x π R x) > βf, x), then ξ / R n If αf, x) < 003 and x π R x) < 0γf, x), then ξ Rn Proof Since x x = x π R x) and βf, x) = βf, x), Item follows by concluding ξ ξ using Lemma Item follows from Theorem 4 together with the fact that π R x) R n and N f R n ) R n Bounding higher order derivatives The constant γf, x) defined in ) yields information regarding the higher order derivatives of f evaluated at x Even though, for polynomial systems, γf, x) is actually a maximum of finitely many values, it is often computationally difficult to compute exactly However, in the polynomial case, it can be bounded above based in part on the degrees of the polynomials [8] 4

5 Due to the nature of polynomial-exponential systems, this bound will be used in our algorithm presented in Section for certifying solutions to polynomial-exponential systems Let g : C n C be a polynomial of degree d where gx) = ρ d a ρx ρ and g = d! ρ! d ρ )! a ρ ρ d is the standard unitarily invariant norm on the homogenization of g For a polynomial system f : C n C n with fx) = [f x),, f n x)] T, we have f = n f i For a point x C n, define x = + x = + n x i The following is an affine version of Propositions and 3 from [8] Proposition 6 If g : C n C is a polynomial of degree d, then, for all x C n and k, gx) g x d and D k gx) d d ) d k + ) g x d k Let k Lemma 3 of [8] yields d d ) d k + ) d / k! ) k d / d ) d3/ Additionally, since x, we know x d x d k These facts together with Proposition 6 yield D k gx) k! k d / D k gx) d k ) d / k! ) k d d ) d k + ) g x d k d / k! d / x d k g ) d3/ x d / x d g which we summarize in the following proposition ) k k d d ) d k + ) d / k! ) k ) k Proposition 7 If g : C n C is a polynomial of degree d, then, for all x C n and k, D k gx) k! k ) d3/ d / x d k g x Let f : C n C n be a polynomial system with deg f i = d i Define D = max d i and µf, x) = max{, f Dfx) d) x) } 3) 5

6 assuming Dfx) is invertible where d) x) = d / x d d / n x dn 4) Since µf, x), µf, x) k µf, x) for any k The following version of Proposition 3 of [8, I-3] yields an upper bound for γf, x) Proposition 8 Let f : C n C n be a polynomial system with deg f i = d i and D = max d i For any x C n such that Dfx) is invertible, Proof For k, we have Dfx) D k fx) k! k γf, x) µf, x) D3/ x f Dfx) d) x) ) k n µf, x) µf, x)d3/ x f i f d 3/ i x d) x) D k fx) f k! ) k ) k ) k Certifying solutions Since the bound provided in Proposition 8 does not apply to a polynomial-exponential system F, we develop a new bound based on the solutions of linear homogeneous partial differential equations With this bound, algorithms for certifying approximate solutions, distinct associated solutions, and real associated solutions of [4] apply to F Consider gx) = e βx for some β C Clearly, for any k 0, g k) x) = β k gx) By letting Bx) = gx) and C = max{, β }, we have g k) x) C k Bx) 5) The following lemma shows that a similar bound holds in general Lemma Let c 0,, c r C, lx 0,, x r ) = x r r i=0 c ix i, and g : C C be an analytic function such that lg, g,, g r) ) = 0 and r is minimal with such a property If Bx) = max{ gx), g x),, g r ) x) } and C = max{, c 0,, c r }, then, for any x C and k 0, we have { Bx) if k < r g k) x) C) k r r Bx) C if k r In particular, g k) x) C) k r Bx) C = k r C k Bx) 6

7 Proof We know g r) = r i=0 c ig i) x) For any k > r, by differentiation, we know r g k) = c i g i+k r) x) We will now proceed by induction starting at k = r In particular, i=0 r r g r) x) c i g i) x) Bx) C = r Bx) C i=0 For k > r with p = k r, we have g k) x) r c i g i+p) x) C i=0 C r Bx) + r Bx) C max{r p,0} r i=r p ) p r Bx) C + C p i p r Bx) C p+ = C) k r r Bx) C i=0 i=0 i=0 g i+p) x) + C) i+p r r i=max{0,r p} g i+p) x) The remaining statement follows from the fact that C and r The following lemma will also be used to deduce our bound Lemma If δ 0 0 and α, δ,, α m, δ m, then sup k δ k ) 0 + k ) α k i δ i ) ) k ) δ 0 + αi δ i Proof Fix k Since k ) and 4k ) k, we know α 4k ) i δ k ) i δi for i =,, m The lemma now follows since δ 0 + ) k ) αi δ i δ k ) m ) k ) 0 + k ) αi δ i δ k ) 0 + k ) δ k ) 0 + k ) α 4k ) i δ k ) i α k i δ i α k i and Throughout the remainder of this section, we assume that F : C n+m C n+m is a polynomialexponential system such that there exists a polynomial system P : C n+m C n, analytic func- 7

8 tions g,, g m : C C, and integers σ,, σ m {,, n} such that P x,, x n, y,, y m ) y g x σ ) Fx,, x n, y,, y m ) = 6) y m g m x σm ) Also, for i =,, n, we define d i = deg P i and D = max d i We assume that each g i satisfies some nonzero linear homogeneous PDE with complex coefficients For each i =,, m, let r i be the smallest positive integer such that there exists a nonzero linear function l i : C ri+ C with l i g i, g i,, gri) i ) = 0 By construction, the coefficient of z ri in l i z 0, z,, z ri ) must be nonzero Upon rescaling l i, we will assume that this coefficient is one, that is, we have l i z 0, z,, z ri ) = z ri c i,ri z ri c i,0 z 0 7) which yields g ri) i = r i j=0 c i,jg j) i We note that the minimal integer r i with such a property is called the order of g i For example, for nonzero λ, µ C, if g x) = e λx, g x) = cosµx), and g 3 x) = x sinx), then the order of g i is,, and 4, respectively The corresponding differential equations are with linear functions g x λg = 0, g x + µ g = 0, and 4 g 3 x 4 + g 3 x + g 3 = 0 l z 0, z ) = z λz 0, l z 0, z, z ) = z + µ z 0, and l 3 z 0, z, z, z 3, z 4 ) = z 4 + z + z 0 The bound obtained in Proposition 8 depends upon µf, x) defined in 3) for polynomial systems We extend this to polynomial-exponential systems by defining { [ ] } µf, x, y)) = max, DFx, d) x, y) P y) 8) I m assuming that DFx, y) is invertible The matrix d) x, y) is the n n diagonal matrix defined in 4) and I m is the m m identity matrix We note that 8) reduces to 3) when m = 0 The following theorem yields a bound for γf, x, y)) Theorem 3 For i =,, m and z C, define B i z) = max{ g i z),, g ri ) i z) } and C i = max{, c i,0,, c i,ri } Then, for any x, y) C n+m such that DFx, y) is invertible, ) D 3/ γf, x, y)) µf, x, y)) + Ci max{, r i B i x σi )} 9) x, y) [ ] d) x, y) P Proof Let M = and k We have I m DFx, y) D k Fx, y) k! DFx, y) M M D k Fx, y) k! µf, x, y)) M D k Fx, y) k! 8

9 By Proposition 7 and Lemma, M D k Fx, y) n D k P i x, y) k! = d / + D k g i x σi ) i x, y) di P k! k! ) n P i d 3/ k ) i P + k r i Ci k B i x σi ) ) x, y) ) D 3/ k ) m + k ) ri Ci k B i x σi ) ) x, y) This yields γf, x, y)) = sup DFx, y) D k Fx, y) k k k! ) D 3/ k ) m µf, x, y)) sup + k ) k x, y) ) D 3/ k ) µf, x, y)) sup + k x, y) The result now follows from Lemma k ) m ri Ci k B i x σi ) ) ) k ) C k i max{, r i B i x σi )} ) ) k ) Remark 4 When m = 0, the bounds provided in Theorem 3 and Proposition 8 agree The following is an algorithm to certify approximate solutions of F = 0 Procedure B = CertifySolnF, z) Input A polynomial-exponential system F : C n+m C n+m and a point z C n+m Output A boolean B which is True if z can be certified as an approximate solution of F = 0, otherwise, False Begin If Fz) = 0, return True, otherwise, if DFz) is not invertible, return False Set β := DFz) Fz) and γ to be the upper bound for γf, z) provided in Theorem 3 3 If β γ < 3 3 7, return True, otherwise return False 4 The algorithms CertifyDistinctSoln and CertifyRealSoln from [4] apply to polynomialexponential systems using the bound provided in Theorem 3 The algorithm CertifyDistinct- Soln determines if two approximate solutions have distinct associated solutions The algorithm CertifyRealSoln applies to polynomial-exponential systems F such that N F R n+m ) R n+m and determines if the associated solution to a given approximate solution is real We conclude this section with a refinement of Theorem 3 applied to polynomial-exponential systems depending on exp, sin, cos, sinh, and cosh This refinement uses the following lemma 9

10 Lemma 5 If λ 0,, λ m 0 and µ,, µ m, then sup k λ k ) 0 + µ i λ k i k! ) k ) λ 0 + µ i λ i Proof follows from Fix k Since k ) and µ i, we know λ 0 + ) k ) µ i λ i λ k ) 0 + λ k ) 0 + λ k ) 0 + λ k ) 0 + µi m µ i λ i µi ) k ) µi ) The lemma ) k ) ) k ) λ k ) i µ i λk ) i µ i λ k ) i k! Let a, b, c, e, h Z 0, δ i, ɛ j, ζ k, η p, κ q C, and σ i, τ j, φ k, χ p, ψ q {,, n} The following considers the following polynomial-exponential system ) Gx,, x n, u,, u a, v,, v b, w,, w c, y,, y d, z,, z e ) = P x,, x n, u,, u a, v,, v b, w,, w c, y,, y d, z,, z e ) u i expδ i x σi ), i =,, a v j sinɛ j x τj ), j =,, b w k cosζ k x φk ), k =,, c y p sinhη p x χp ), p =,, e z q coshκ q x ψq ), q =,, h 0) Corollary 6 Let G be defined as in 0) where P : C N C n is a polynomial system with N = n + a + b + c + e + h, d i = deg P i and D = max d i For any λ, θ C, define Aλ, θ) = max{ λ, λ expλθ)/ }, Bλ, θ) = max{ λ, λ sinλθ)/, λ cosλθ)/ }, and Cλ, θ) = max{ λ, λ sinhλθ)/, λ coshλθ)/ } Then, for any X = x, u, v, w, y, z) C N such that DGX) is invertible, D 3/ a γg, X) µg, X) + Aδ i, x σi ) + X b Bɛ j, x τj ) + j= + c Bζ k, x φk ) k= e Cη p, x χp ) + p= ) h Cκ q, x ψq ) ) q= 0

11 Proof Let k The following table lists the bounds on the higher derivatives together with associated quantities λ and µ used when applying Lemma 5 gx) bound for g k) x) λ µ expθx) θ k expθx) θ max{, θ expθx) } sinθx) cosθx) θ k max{ sinθx), cosθx) } θ max{, θ sinθx), θ cosθx)} sinhθx) coshθx) θ k max{ sinhθx), coshθx) } θ max{, θ sinhθx), θ coshθx)} The result now follows immediately by modifying the proof of Theorem 3 incorporating the bounds presented in this table together with Lemma 5 Based on Lemma 5, the functions A, B, and C are one-half of the product of the entries in the λ and µ columns 3 Approximating solutions In order to certify that a point is an approximate solution of F = 0, where F is a polynomialexponential system, one needs to first have a candidate point In some applications, candidate points arise naturally from the formulation of the problem One systematic approach to yield candidate points is to replace each analytic function g i by a polynomial g p i and solve the resulting polynomial system, namely F p x,, x n, y,, y m ) = P x,, x n, y,, y m ) y g p x σ ) y m g p mx σm ) ) When the degree of the polynomial approximations are sufficiently large, the numerical solutions of F p = 0 are candidates for being approximate solutions of F = 0 In Section 3, we discuss using regeneration [3] to solve F p = 0 If a numerical solution of F p = 0 is not an approximate solution of F = 0, one can try to apply Newton s method for F directly to these points to possibly yield an approximate solution of F = 0 Another approach is to construct a homotopy between F p and F and numerically approximate the endpoint of the path starting with a solution of F p = 0 We note that neither method is guaranteed to yield an approximate solution of F = 0 3 Regeneration and polynomial-exponential systems Regeneration [3] solves a polynomial system by using solutions to related, but easier to solve, polynomial systems In particular, we will utilize the linear product [3] structure of F p in ) Suppose that g is a univariate polynomial of degree d The polynomial y gx) has a linear product structure of x, y, x, x, }{{} d times This means y gx) can be written as a finite sum of polynomials of the form L x, y) L d x, y) where L x, y) = ay + b x + c and, for i =,, d, L i x, y) = b i x + c i

12 for some a, b i, c i C For i =,, m, let r i = deg g p i and a i, b i,,, b i,ri C Similar to the algorithms proposed in [3], we note that the following arguments and proposed algorithm depend on the genericity of a i and b i,j Define L i, x, y) = a i y + b i, x + and, for j =,, r i L i,j x, y) = b i,j x + Let ν = ν,, ν m ) such that ν i r i Consider the polynomial systems Q ν : C n+m C n+m defined by P x,, x n, y,, y m ) L,ν x σ, y ) Q ν x,, x n, y,, y m ) = 3) L m,νm x σm, y m ) For =,, ), we first compute the solutions of Q = 0 We note that in practice, Q is solved by working intrinsically on the linear space defined by L,ν x σ, y ) = = L m,νm x σm, y m ) = 0 Numerical approximations of these solutions can be obtained using standard numerical solving methods for square polynomial systems see [0, 4]) including, for example, polyhedral homotopies [] or basic regeneration [3] In order to compute the nonsingular isolated solutions of F p = 0, we need to compute the nonsingular isolated solutions of Q ν = 0 for all possible ν By the theory of coefficientparameter homotopies [7], the nonsingular isolated solutions of Q ν = 0 can be obtained by using a homotopy from Q to Q ν starting with the nonsingular isolated solutions of Q = 0 We note that if i j such that σ i = σ j and ν i, ν j >, then Q ν = 0 has no solutions After solving Q ν = 0 for all possible ν, we thus have all nonsingular isolated solutions of Px,, x n, y,, y m ) = P x,, x n, y,, y m ) r j= L,jx σ, y ) rm j= L m,jx σm, y m ) = 0 4) The final step is to use a homotopy deforming P to F p starting with the nonsingular isolated solutions of P = 0 The finite endpoints of this homotopy form a superset of the isolated nonsingular solutions of F p = 0 4 Implementation details The certification of polynomial-exponential systems is implemented in alphacertified [5] The systems must be of the form G in 0) where the coefficients of P as well as the constant in the argument of exp, sin, cos, sinh, and cosh must be rational complex numbers with the bound for γ presented in ) Due to the nature of exponential functions, the computations are performed using arbitrary precision floating point arithmetic Since floating point errors arising from the internal computations are not fully controlled, the results of alphacertified for polynomialexponential systems are said to be soft certified See [4, 5] for more details regarding input syntax, internal computations, and output

13 a θ a θ E O Figure : RR dyad 5 Examples The following examples used Bertini [] and alphacertified [5] with a 4 GHz Opteron 50 processor running 64-bit Linux with 8 GB of memory All files for running these examples can be found at the website of the first author 5 A rigid mechanism Consider the algebraic kinematics problem [4] of the inverse kinematics of the RR dyad The RR dyad, which is displayed in Figure, consists of two legs of fixed length, say a and a, which are connected by a pin joint The mechanism is anchored with a pin joint at the point O, which we take as the origin Given a point E = e, e ), the problem is compute the angles θ and θ so that the end of the second leg is at E That is, we want to solve fθ, θ ) = 0 where [ ] a cosθ fθ, θ ) = ) + a cosθ ) e a sinθ ) + a sinθ ) e The polynomial-exponential system G : C 6 C 6 of the form 0) is a y 3 + a y 4 e a y + a y e Gθ, θ, y, y, y 3, y 4 ) = y sinθ ) y sinθ ) y 3 cosθ ) y 4 cosθ ) Since θ i only appears in f as arguments of the sine and cosine functions, we can compute solutions of f = 0 by using the solutions of a related polynomial system In particular, consider the polynomial system g : C 4 C 4 obtained by replacing sinθ i ) and cosθ i ) with s i and c i, respectively, and adding the Pythagorean identities, namely gs, s, c, c ) = a c + a c e a s + a s e s + c s + c Given a solution of g = 0, solutions of f = 0 are generated using either the arcsin or arccos functions Moreover, it is easy to verify that, for general a i, e i C, g = 0 has two solutions and thus f = 0 has two π-periodic families of solutions Consider the inverse kinematics problem with a = 3, a =, and E =, 35) We used Bertini to numerically approximate the two solutions of g = 0 For demonstration, consider the two digit rational approximations of the solutions X = 00 65, 77, 76, 64) and X = 95, 3, 30, 95) 00 3

14 k βg, NG kz )) βg, NG kz )) Table : Newton residuals for G The certified upper bounds for αg, X i ) computed by alphacertified using exact rational arithmetic and rounded to four digits are and 00788, respectively Hence, X and X are both approximate solutions of g = 0 Furthermore, alphacertified certified that the associated solutions are distinct and real We now consider two corresponding approximations to solutions of G = 0 namely Z = 07, 6, 065, 077, 076, 064) and Z = 874, 034, 095, 03, 030, 095) The upper bounds for αg, Z i ) computed by alphacertified using 96-bit floating point arithmetic and rounded to four digits are 065 and 0355, respectively In order to reduce the effect of roundoff errors, we also used 04-bit floating point arithmetic and obtained the same four digit value Hence, alphacertified has soft certified that Y and Y are both approximate solutions of G = 0 Furthermore, alphacertified has soft certified that the associated solutions are distinct and real Table lists the Newton residuals computed by alphacertified using 4096-bit precision which demonstrates the quadratic convergence of Newton s method By using Euler s formula, we could alternatively use the polynomial-exponential system G : C 6 C 6 of the form 0) where G θ, θ, x, x, y, y ) = and i = Consider the two points a x + a x e + ie a y + a y e ie x y x y y expiθ ) y expiθ ) W = 07, 6, i, i, i, i) and W = 874, 034, i, i, i, i) The upper bounds for αg, W i ) computed by alphacertified using both 96-bit and 04-bit floating point arithmetic and rounded to four digits are 049 and 04, respectively In particular, alphacertified soft certified that W and W are both approximate solutions of G = 0 with distinct associated solutions Finally, consider the polynomial system obtained by replacing the sine and cosine functions in f with a third and second degree truncated Taylor series approximation, respectively, centered 4

15 at the origin, namely f p θ, θ ) = [ ] a + θ/) + a + θ/) e a θ + θ/6) 3 + a θ + θ/6) 3 e The system of equations f p = 0 has six solutions and yield six solutions of f = 0 upon deforming f p to f These six solutions split into two groups of three based on the values of sinθ i ) and cosθ i ) corresponding to the two families of solutions of f = 0 5 A compliant mechanism In [], Su and McCarthy study a polynomial-exponential system modeling a compliant four-bar linkage displayed in Figure 4 of [] Upon solving a related polynomial system and applying Newton s method, they conclude based on the numerical results that a specific compliant fourbar linkage has two stable configurations We will first use alphacertified to certify that their numerical approximations of the two stable configurations are indeed approximate solutions Afterwards, we will use the approaches of Section 3 to recompute these two stable configurations The polynomial-exponential system f : C 5 C 5 modeling a compliant four-bar linkage is where fα, θ, θ, ν, ν ) = Rα)W W ) + G + r csθ ) G r csθ ) Rα)W W )ν + r csθ ) r csθ )ν k α α 0 θ + θ 0 )ν ) + k α α 0 θ + θ 0 )ν ν ) [ ] cosα) sinα) Rα) = sinα) cosα) and csθ) = [ cosθ) sinθ) We note that each of the first two lines in f consists two functions Additionally, f is not algebraic since X, sinx), and cosx) all appear in f, where X = α, θ, θ The values for the specific linkage under consider are [ ] [ ] [ ] [ ] W =, W =, G =, G 0 =, r 0 = r = 50, k = 950, k = 5849, θ 0 = 4486, θ 0 = 095, and α 0 = 069 with numerical approximations for the stable configurations A = 06933, , , 06074, ) and A = 56473, 03930, , , ) The polynomial-exponential system G : C C of the form 0) is Gα, θ, θ, ν, ν, y,, y 6 ) = Ry, y )W W ) + G + r csy 3, y 4) G r csy 5, y 6) Ry, y )W W )ν + r csy 3, y 4) r csy 5, y 6)ν k α α 0 θ + θ 0 )ν ) + k α α 0 θ + θ 0 )ν ν ) y sinα) y cosα) y 3 sinθ ) y 4 cosθ ) y 5 sinθ ) y 6 cosθ ) ] 5

16 bound for approximation of bound for F αf, B ) αf, B ) βf, B ) βf, B ) γf, B ), γf, B ) G G Table : Values obtained for G and G at B and B where Let B i = A i, Y i ) where Ry, y ) = [ ] y y y y [ z and csw, z) = w Y = 0536, , , 095, , 06086) and Y = , , 03547, 09930, , ) The upper bounds for αg, B i ) computed by alphacertified using both 96-bit and 04-bit floating point arithmetic and rounded to four digits are 0066 and 0047, respectively In particular, alphacertified has soft certified that B and B are both approximate solutions of G = 0 Furthermore, alphacertified has soft certified that the associated solutions are distinct and real The formulation of the polynomial-exponential system can have an adverse effect on certifying solutions For example, consider the polynomial-exponential system G : C C obtained by replacing the 7 th, 9 th, and th functions of G with y + y, y 3 + y 4, and y 5 + y 6 Clearly, every solution of G = 0 must also be a solution of G = 0 Table compares the bounds for α and γ and the value of β for G and G at B and B computed by alphacertified This table shows that the bounds computed for αg, B i ) and γg, B i ) are three orders of magnitude larger than the bounds computed for αg, B i ) and γg, B i ) In particular, due to the larger bounds, alphacertified is unable to certify that B and B are approximate solutions of G = 0 If we replace B i with N G B i ), then alphacertified is able to soft certify that the resulting points are approximate solutions of G = 0 using both 96-bit and 04-bit precision We now consider solving a polynomial system obtained by replacing the sine and cosine functions with a fifth and fourth degree truncated Taylor series approximation, respectively, centered at the origin Let the polynomial system P : C C 5 consists of the first five functions in G In particular, P consists of two linear and three quadratic polynomials and thus has total degree of the polynomial Q ν defined in 3) has total degree 3 = 8 Since we are using fifth and fourth degree polynomial approximations for the sine and cosine functions, respectively, we have r i = 5 if i is odd and r i = 4 if i is even We picked random a i, b i,j C for i =,, 6 and j =,, r i and used Bertini to solve each Q ν = 0 In total, this produced numerical approximations to 356 nonsingular isolated solutions of P = 0 where P is defined in 4) The tracking of the 356 paths from P to the polynomial approximation, G p, of G produced 0 points which became the start points for the homotopy deforming G p to G This homotopy yielded 93 numerical approximations to solutions of G = 0 By using both 96-bit and 04- bit floating point arithmetic, alphacertified soft certified that each of these 93 points are indeed approximate solutions with distinct associated solutions Moreover, this computation soft certified that 65 have real associated solutions, two of which are the two stable configurations computed in [] ] 6

17 References [] DJ Bates, JD Hauenstein, AJ Sommese, and CW Wampler Bertini: Software for Numerical Algebraic Geometry Available at wwwndedu/~sommese/bertini [] L Blum, F Cucker, M Shub, and S Smale Complexity and Real Computation Springer- Verlag, New York, 998 [3] JD Hauenstein, AJ Sommese, and CW Wampler Regeneration homotopies for solving systems of polynomials Math Comp, 80, , 0 [4] JD Hauenstein and F Sottile alphacertified: certifying solutions to polynomial systems To appear in ACM Trans Math Softw, 0 [5] JD Hauenstein and F Sottile alphacertified: software for certifying solutions to polynomial systems Available at wwwmathtamuedu/~sottile/research/stories/ alphacertified [6] S Lang Real Analysis, second ed Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 983 [7] AP Morgan and AJ Sommese Coefficient-parameter polynomial continuation Appl Math Comput, 9), 3 60, 989 Errata: Appl Math Comput, 5, 07, 99 [8] M Shub and S Smale Complexity of Bézout s theorem I: Geometric aspects J Amer Math Soc, 6), , 993 [9] S Smale Newton s method estimates from data at one point The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics Laramie, Wyo, 985) Springer, New York, 986, pp [0] AJ Sommese and CW Wampler The Numerical Solution of Systems of Polynomials Arising in Engineering and Science World Scientific Press, Singapore, 005 [] B Huber and B Sturmfels A polyhedral method for solving sparse polynomial systems Math Comp, 64), , 995 [] H-J Su and JM McCarthy A polynomial homotopy formulation of the inverse static analysis of planar compliant mechanisms ASME J Mech Des, 84), , 006 [3] J Verschelde and R Cools Symbolic homotopy construction Appl Algebra Engrg Comm Comput, 4, 69 83, 993 [4] CW Wampler and AJ Sommese Numerical algebraic geometry and algebraic kinematics Acta Numerica, 0, , 0 7