Asymptotic Solutions of Polynomial Equations with Exp-Log Coefficients
aa r X i v : . [ c s . S C ] A p r ASYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONSWITH EXP-LOG COEFFICIENTS
ADAM STRZEBOŃSKI
Abstract.
We present an algorithm for computing asymptotic approxima-tions of roots of polynomials with exp-log function coefficients. The real andimaginary parts of the approximations are given as explicit exp-log expressions.We provide a method for deciding which approximations correspond to realroots. We report on implementation of the algorithm and present empiricaldata. Introduction
Definition 1.
The set of exp-log functions is the smallest set of partial functions R → R containing exp , log , the identity function and the constant functions, closedunder addition, multiplication and composition of functions.The domain D ( f ) of an exp-log function f is determined as follows:(1) the domain of exp , the identity function and the constant functions is R and the domain of log is R + ,(2) D ( f + g ) = D ( f g ) = D ( f ) ∩ D ( g ) ,(3) D ( f ( g )) = g − ( D ( f )) .In particular, D ( f ) is an open set and f is C ∞ in D ( f ) . Remark . The multiplicative inverse function R \ { } ∋ x → /x = x exp( − log( x )) ∈ R and the real exponent power functions R + ∋ x → x r = exp( r log( x )) ∈ R for r ∈ R , are exp-log functions.The domain of an exp-log function consists of a finite number of open, possiblyunbounded, intervals and an exp-log function has a finite number of real roots.An algorithm computing domains and isolating intervals for real roots of exp-logfunctions is given in [11, 12].We say that a partial function f : R → C is defined near infinity if D ( f ) ⊇ ( c, ∞ ) for some c ∈ R . Definition 3. A Hardy field [1] is a set of germs at infinity of real-valued functionsthat is closed under differentiation and forms a field under addition and multipli-cation.
Theorem 4. [4, 3]
The germs at infinity of exp-log functions defined near infinityform a Hardy field.
Let P ( x, y ) = a n ( x ) y n + . . . + a ( x ) where, for ≤ i ≤ n , a i ( x ) = u i ( x ) + ıv i ( x ) and u i and v i are exp-log functions defined near infinity ( ı denotes the imaginaryunit). Problem 5.
Describe the asymptotic behaviour of roots of P in y as x tends toinfinity.The following theorem [8, 9] shows that the problem is well posed, that is theroots of P in y are C ∞ functions in x defined near infinity. Theorem 6. If H is a Hardy field, then there exists a Hardy field K ⊇ H suchthat K [ ı ] is algebraically closed. Corollary 7.
There exists a Hardy field K such that P has n roots in y (countedwith multiplicities) in K [ ı ] . Definition 8.
Let f : R → C be a partial function defined near infinity. Wesay that partial functions f , . . . , f m : R → C defined near infinity form an m -term asymptotic approximation of f if, for ≤ i < m , lim x →∞ f i +1 ( x ) f i ( x ) = 0 and lim x →∞ f ( x ) − P mi =1 f i ( x ) f m ( x ) = 0 .In this paper we present an algorithm which computes asymptotic approxima-tions of roots of P in y . The approximations are given as exp-log expressions. Thealgorithm makes use of the theory of “most rapidly varying” subexpressions devel-oped in [2] to compute limits of exp-log functions. In fact our algorithm applies inthe more general case of MrvH fields. The algorithm is based on a Newton polygontechnique [6, 14, 13] extended to “series” with arbitrary real exponents.Algorithms given in [10, 13] solve the problem of finding asymptotic solutions ofpolynomial equations in more general settings. We chose to extend the algorithm of[2] because we find it simpler to implement and we can give a direct and elementaryproof that the computed expressions satisfy our (weaker) requirements.
Example 9.
Let P ( x, y ) = y − exp( x ) y − log( x ) . One-term asymptotic approxi-mations of roots of P in y computed with our algorithm are r = − exp( − x ) log( x ) , r = − exp( − x ) − / , r = exp( − x ) − / , r = − ı exp( − x ) − / , r = ı exp( − x ) − / .Let us estimate the relative error ε i = | r i − r ∗ i r i | of the approximations, where r ∗ i isthe exact root closest to r i . Using the bound | y − y ∗ | ≤ | P ( x, y ) ∂P∂y ( x, y ) | on the distance from y to the closest root y ∗ of P ( x, y ) , after simplifications validfor x > , we get ε ≤ x ) exp(5 x ) − x ) , and for i = 1 , ε i ≤ log( x ) exp( − x ) . Bothbounds tend to zero as x tends to infinity and are decreasing for x > exp( W ( )) =1 . . . . , where W is the principal branch of the Lambert W function. Evaluatingthe bounds at x = 10 we get log ε ≤ − . and log ε i ≤ − . for i = 1 .For x = 1000 we get log ε ≤ − and log ε i ≤ − for i = 1 . This showsthat we can obtain approximations of roots of P (1000 , y ) to digits of precisionby evaluating the asymptotic approximations. The evaluation takes ms. Forcomparison, direct computation of roots of P (1000 , y ) to digits of precisiontakes ms. Figure 1.1 shows the asymptotic approximations of the real roots of P ( x, y ) (dashed curves) and the exact roots (solid green curves). SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS3
Figure 1.1.
Real roots from Example 9 - - Most rapidly varying subexpressions
In this section we give a very brief summary of terminology and facts necessaryfor formulating Algorithm 14. We will use the algorithm to compute approximationsof coefficients of P in terms of a “most rapidly varying” subexpression present inthe coefficients. The algorithm is based on the algorithm MrvLimit described in[2]. For a more detailed introduction and proofs of the stated facts see [2]. Definition 10.
The set of exp-log expressions with coefficients in a computablefield C ⊆ R is defined recursively as follows:(1) elements of C and the variable x are exp-log expressions,(2) if f and g are exp-log expressions, so are f + g , f · g and fg ,(3) if f is an exp-log expression and c ∈ C , then exp( f ) , log( f ) , and f c areexp-log expressions.Each exp-log expression represents an exp-log function, however the same func-tion may be represented by many different expressions. In the following, when werefer to the domain, point values, and limits of an exp-log expression, we mean thedomain, point values, and limits of the corresponding exp-log function. Definition 11.
Let E ∞ ( x ) be the set of exp-log expressions f such that, for some c ∈ R , ( c, ∞ ) ⊆ D ( f ) and either f = 0 as an expression or f is nonzero on ( c, ∞ ) . Remark . Note that we exclude from E ∞ ( x ) expressions that are identically zeroin a neighbourhood of infinity, but are not explicitly zero e.g. f = exp(log(( x − c ) ) / − x + c . The algorithm ExpLogRootIsolation of [11] can be used to checkwhether a given exp-log expression is defined near infinity and to detect expres-sions that are identically zero in a neighbourhood of infinity and replace them
ADAM STRZEBOŃSKI with explicit zeros.
ExpLogRootIsolation requires a zero test algorithm for elemen-tary constants. Termination of the currently known zero test algorithm relies onSchanuel’s conjecture [7, 12].The germs at infinity of functions represented by elements of E ∞ ( x ) form theHardy field of exp-log functions defined near infinity. Theorem 13. If f and g are nonzero elements of a Hardy field, then the limit lim x →∞ log | f ( x ) | log | g ( x ) | exists (in R = R ∪ {−∞ , ∞} ). Moreover, if lim x →∞ g ( x ) = 0 and lim x →∞ log | f ( x ) | log | g ( x ) | = 0 then, for any e > , lim x →∞ f ( x ) g ( x ) e = 0 The theorem follows from the results in section 3.1.2 of [2].Following [2], we say that g is more rapidly varying than f , or g is in a highercomparability class than f , if lim x →∞ log | f ( x ) | log | g ( x ) | = 0 and we denote it f ≺ g . We say that f and g have the same order of variation, or f and g are in the same comparability class, if lim x →∞ log | f ( x ) | log | g ( x ) | ∈ R \ { } and we denote it f ≍ g . We will also use f (cid:22) g to denote f ≺ g ∨ f ≍ g . ω is a most rapidly varying subexpression of f if ω is a subexpression of f and nosubexpression of f is more rapidly varying than ω . Let mrv ( f ) be the set of mostrapidly varying subexpressions of f . We will write mrv ( f ) ≺ g (resp. mrv ( f ) ≍ g )if for all ω ∈ mrv ( f ) , ω ≺ g (resp. ω ≍ g ). Let mrv ( f , . . . , f n ) be the set of ω such that ω is a subexpression of some f i and no subexpression of any f j is morerapidly varying than ω , that is mrv ( f , . . . , f n ) the max of mrv ( f ) , . . . , mrv ( f n ) (as in Algorithm 3.12 of [2]).To prove termination of our algorithm we use the notion of size of an exp-logexpression defined in [2], section 3.4.1. For an exp-log expression f , let S ( f ) be theset of subexpressions of f defined by the following conditions.(1) If f does not contain the variable x , then S ( f ) = ∅ .(2) If f = x , then S ( f ) = { x } .(3) If f = g + h , f = gh , or f = gh , then S ( f ) = S ( g ) ∪ S ( h ) .(4) If f = g c then S ( f ) = S ( g ) .(5) If f = exp g or f = log g then S ( f ) = { f } ∪ S ( g ) .Then Size ( f ) is defined as the cardinality of S ( f ) .Let exp k (resp. log k ) denote k times iterated exponential (resp. logarithm), andfor f ∈ E ∞ ( x ) let f ↑ k (resp. f ↓ k ) denote f with x replaced with exp k ( x ) (resp. log k ( x ) ). The following algorithm computes approximations of elements of a finitesubset of E ∞ ( x ) in terms of their most rapidly varying subexpression. SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS5
Algorithm 14. (MrvApprox)Input: a , . . . , a n ∈ E ∞ ( x ) such that P ni =0 Size ( a i ) > .Output: ω ∈ E ∞ ( x ) , b , . . . , b n ∈ E ∞ ( x ) , e , . . . , e n ∈ R ∪ {∞} , d > , and k ∈ Z ≥ such that • ω > and lim x →∞ ω = 0 , • mrv ( b , . . . , b n ) ≺ ω , • if a i = 0 then b i = 0 and e i = ∞ , • if a i = 0 then e i ∈ R and lim x →∞ ω − ( e i + d ) ( a ↑ ki − b i ω e i ) = 0 , • P ni =0 Size ( b i ) < P ni =0 Size ( a i ) . The algorithm proceeds in a very similar manner to the algorithm MrvLimitdescribed in [2]. First, it finds the set
Ω = mrv ( a , . . . , a n ) . If x ∈ Ω the algorithmreplaces x with exp( x ) in a , . . . , a n and recomputes Ω until x / ∈ Ω . k is the numberof replacements performed in this step. Then the algorithm picks ω such that ω or /ω belongs to Ω , ω > , and lim x →∞ ω = 0 , and rewrites all elements of Ω in terms of ω . If a ↑ ki contains a subexpression in the same comparability class as ω , let b i ω e i be the first term of Series ( a ↑ ki , ω ) (as in section 3.3.3 of [2]), and let d i > be the difference between the exponents of ω in the second and in the firstterm of the series ( d i = ∞ if a ↑ ki = b i ω e i ). If a ↑ ki does not contain subexpressionsin the same comparability class as ω , then b i = a ↑ ki , e i = 0 , and d i = ∞ . Inboth cases mrv ( b i ) ≺ ω . Pick < d < min ≤ i ≤ n d i . Then ω − ( e i + d ) ( a ↑ ki − b i ω e i ) iseither or a power series in ω with positive exponents and coefficients in a lowercomparability class than ω , hence lim x →∞ ω − ( e i + d ) ( a ↑ ki − b i ω e i ) = 0 . Section 3.4.1of [2] proves that Size ( b i ) ≤ Size ( a i ) , with the strict inequality if a ↑ ki containsa subexpression in the same comparability class as ω . This shows that the lastrequirement is satisfied. 3. Root continuity
To prove correctness of our algorithm we need a polynomial root continuitylemma that does not assume fixed degree of the polynomial. The lemma is verysimilar to Theorem 1 of [15], except that our version provides explicit bounds.
Lemma 15.
Let p = a n z n + . . . + a = a n ( z − r ) · . . . · ( z − r n ) ∈ C [ z ] where n ≥ and a n = 0 . Let Γ = max ≤ i ≤ n | r i | and ∆ = min r i = r j | r i − r j | ( ∆ = ∞ if r = . . . = r n ).Suppose that m ≥ n , < ǫ < min(1 , , ∆2 ) , and < δ < | a n | (1 − ǫ ) ǫ m + n − ǫ m +1 .Then for every q = b m z m + . . . + b ∈ C [ z ] such that b m = 0 , for ≤ i ≤ n , | b i − a i | < δ , and, for n + 1 ≤ i ≤ m , | b i | < δ , wehave q = b m ( z − s ) · . . . · ( z − s m ) for ≤ i ≤ n , | s i − r i | < ǫ , and for n + 1 ≤ i ≤ m , | s i | > /ǫ .Proof. Let C = { c : | c | = 1 /ǫ } , D = { c : | c | ≤ /ǫ } , and, for ≤ i ≤ n , let C i = { c : | c − r i | = ǫ } , and D i = { c : | c − r i | ≤ ǫ } . Then, for ≤ i, j ≤ n , D i and D j are either identical or disjoint, D i is contained in the interior of D , D i contains ADAM STRZEBOŃSKI exactly one of the distinct roots of p , and D contains all roots of p . If c ∈ C i forsome ≤ i ≤ n , then | p ( c ) | ≥ | a n | ǫ n and | q ( c ) − p ( c ) | ≤ n X k =0 | b k − a k || z | k + m X k = n +1 | b k || z | k ≤ δ m X k =0 ǫ k We have δ m X k =0 ǫ k < | a n | (1 − ǫ ) ǫ m + n − ǫ m +1 − /ǫ m +1 − /ǫ = | a n | ǫ n Hence | q ( c ) − p ( c ) | < | p ( c ) | . By Rouche’s theorem, for ≤ i ≤ n , the numberof roots of q in D i equals the number of roots of p in D i , which concludes theproof. (cid:3) The main algorithm
Let E C ∞ ( x ) = { u + ıv : u, v ∈ E ∞ ( x ) } . This section presents the main algorithmcomputing asymptotic approximations of roots of polynomials P ∈ E C ∞ ( x )[ y ] .Let us first describe a straightforward generalization of Algorithm 14 to inputs in E C ∞ ( x ) . We extend mrv and Size to E C ∞ ( x ) by defining mrv ( u + ıv , . . . , u n + ıv n ) = mrv ( u , v , . . . , u n , v n ) and Size ( u + ıv ) = Size ( u )+ Size ( v ) . We say that u + ıv ≺ ω (resp. u + ıv ≍ ω ) if | u + ıv | = ( u + v ) / ≺ ω (resp. | u + ıv | ≍ ω ). If a = u + ıv then a ↑ k := u ↑ k + ıv ↑ k and a ↓ k := u ↓ k + ıv ↓ k . Algorithm 16. (MrvApproxC)Input: a , . . . , a n ∈ E C ∞ ( x ) such that P ni =0 Size ( a i ) > .Output: ω ∈ E ∞ ( x ) , b , . . . , b n ∈ E C ∞ ( x ) , e , . . . , e n ∈ R ∪ {∞} , d > , and k ∈ Z ≥ such that • ω > and lim x →∞ ω = 0 , • mrv ( b , . . . , b n ) ≺ ω , • if a i = 0 then b i = 0 and e i = ∞ , • if a i = 0 then e i ∈ R and lim x →∞ ω − ( e i + d ) ( a ↑ ki − b i ω e i ) = 0 , • P ni =0 Size ( b i ) < P ni =0 Size ( a i ) . (1) Let a i = u i + ıv i . Call Algorithm 14 with u , v , . . . , u n , v n as input, ob-taining ω ∈ E ∞ ( x ) , ˆ u , ˆ v , . . . , ˆ u n , ˆ v n ∈ E ∞ ( x ) , e u, , e v, , . . . , e u,n e v,n ∈ R , ˆ d > , and k ∈ Z ≥ . (2) For ≤ i ≤ n , put e i := min( e u,i , e v,i ) and b i := ˆ u i e u,i < e v,i ˆ u i + ı ˆ v i e u,i = e v,i ı ˆ v i e u,i > e v,i (3) Pick d such that < d < ˆ d and d < | e u,i − e v,i | for all i such that e u,i = e v,i . (4) Return ω , b , . . . , b n , e , . . . , e n , d , and k .Proof. To prove that the output of Algorithm 16 satisfies the required conditionswe need to prove that if a i = 0 then lim x →∞ ω − ( e i + d ) ( a ↑ ki − b i ω e i ) = 0 The other conditions follow directly from the definitions and the properties of theoutput of Algorithm 14.
SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS7
Suppose that e u,i < e v,i . Then e i = e u,i and ω − ( e i + d ) ( a ↑ ki − b i ω e i ) = ω − ( e u,i + d ) ( u ↑ ki − ˆ u i ω e u,i ) + ıω − ( e u,i + d ) v ↑ ki We have lim x →∞ ω − ( e u,i + d ) ( u ↑ ki − ˆ u i ω e u,i ) =lim x →∞ ω ˆ d − d ω − ( e u,i + ˆ d ) ( u ↑ ki − ˆ u i ω e u,i ) = 0 and lim x →∞ ω − ( e u,i + d ) v ↑ ki =lim x →∞ ω ( e v,i − e u,i )+( ˆ d − d ) ω − ( e v,i + ˆ d ) ( v ↑ ki − ˆ v i ω e v,i )+ˆ v i ω ( e v,i − e u,i ) − d = 0 since ( e v,i − e u,i ) + ( ˆ d − d ) > , ( e v,i − e u,i ) − d > , and ˆ v i ≺ ω . Cases e u,i = e v,i and e u,i > e v,i can be proven in a similar manner. (cid:3) Let P ( x, y ) = a n ( x ) y n + . . . + a ( x ) ∈ E C ∞ ( x )[ y ] . W.l.o.g. we may assume that a n and a are not identically zero.Suppose that P ni =0 Size ( a i ) > i.e. P ( x, y ) depends on x . Let ω ∈ E ∞ ( x ) , b , . . . , b n ∈ E C ∞ ( x ) , e , . . . , e n ∈ R ∪ {∞} , d > , and k ∈ Z ≥ be the output ofAlgorithm 16 for a , . . . , a n , and let Q ( x, y ) = a ↑ kn ( x ) y n + . . . + a ↑ k ( x ) .Let K be a Hardy field containing germs at infinity of exp-log functions definednear infinity, such that K [ ı ] is algebraically closed. Let α ∈ K [ ı ] be a root of Q .Since | α | ∈ K , the limit lim x →∞ log | α | log | ω | = γ exists. Claim . γ ∈ R . Proof.
Suppose that | γ | = ∞ . Then(4.1) x →∞ Q ( x, α ) a ↑ kn α n = lim x →∞ n X i =1 a ↑ kn − i a ↑ kn α i Since ω ≺ | α | , a ↑ kn − i a ↑ kn ≺ | α | , and so either lim x →∞ | α | = ∞ and lim x →∞ | n X i =1 a ↑ kn − i a ↑ kn α i | = 0 or lim x →∞ | α | = 0 and lim x →∞ | n X i =1 a ↑ kn − i a ↑ kn α i | = ∞ Both cases contradict equation (4.1). (cid:3)
Let I = { i : 0 ≤ i ≤ n ∧ a i = 0 } . If i ∈ I , put c i = ω − ( e i + d ) ( a ↑ ki − b i ω e i ) Then a ↑ ki = b i ω e i + c i ω e i + d and lim x →∞ c i = 0 . Put β := αω γ . Then γ = lim x →∞ log | βω γ | log | ω | = γ + lim x →∞ log | β | log | ω | ADAM STRZEBOŃSKI and hence β ≺ ω . We have Q ( x, α ) = X i ∈ I ( b i β i ω e i + γi + c i β i ω e i + d + γi ) Let µ = min i ∈ I e i + γi , let J = { i ∈ I : e i + γi = µ } , and let µ < ν < min( µ + d, min i ∈ I \ J e i + γi ) Then ω − ν Q ( x, α ) = ω µ − ν X i ∈ J b i β i + X i ∈ I \ J b i β i ω e i + γi − ν + X i ∈ I c i β i ω e i + d + γi − ν As x tends to infinity, all terms in the last two sums tend to zero, hence lim x →∞ ω µ − ν X i ∈ J b i β i = 0 Since β ≺ ω and µ − ν < , the cardinality of J must be at least . Consider thesubset A = { ( i, e i ) : i ∈ I } of R with coordinates denoted ( x , x ) . Then the line x = − γx + µ passes through the points { ( i, e i ) : i ∈ J } and all the other pointsof A lie above this line. This means that − γ is the slope of one of the segmentsthat form the lower part of the boundary of the convex hull of A .Let R γ ( x, z ) = P i ∈ J b i z i and Q γ ( x, z ) = ω − µ Q ( x, zω γ ) . We have Q γ ( x, z ) = R γ ( x, z ) + ω ν − µ ( X i ∈ I \ J b i ω e i + γi − ν z i + X i ∈ I c i ω e i + d + γi − ν z i ) Let ρ , . . . , ρ t ∈ K [ ı ] \ { } be the nonzero roots of R γ ( x, z ) listed with multiplic-ities. Claim . There exist roots β , . . . , β t ∈ K [ ı ] of Q γ ( x, z ) such that, for sufficientlylarge x , for ≤ j ≤ t , | β j − ρ j | < ω η , where η = ν − µ n +1) > . Proof.
Let Γ( x ) be the maximum of absolute values of roots of R γ ( x, z ) and let ∆( x ) be the minimum distance between two distinct roots of R γ ( x, z ) ( ∆( x ) = ∞ if all roots of R γ ( x, z ) are equal). Put δ ( x ) = ω ( x ) ( ν − µ ) / and ǫ ( x ) = δ ( x ) / (2 n +2) .For sufficiently large x , R γ ( x, z ) has a fixed number of distinct roots in z , equalto its number of distinct roots in K [ ı ] . Γ( x ) can be bounded from above by arational function in absolute values of b i , for i ∈ J , and, since the coefficients in z of R ( x, z ) /g.c.d. ( R ( x, z ) , ∂∂z R ( x, z )) are rational functions of b i , ∆( x ) can bebounded from below by an expression constructed from b i using rational operations,square roots and absolute value (see e.g. [5], Theorem 5). Since mrv ( b i ) ≺ ω , forsufficiently large x , < ǫ < min(1 , , ∆2 ) . Let s be the degree of R γ in z . Forsufficiently large x , | b s | (1 − ǫ ) ǫ n + s − ǫ n +1 > | b s | ǫ n + s +1 > ǫ n +2 = δ The coefficients at z i , for ≤ i ≤ n , of Q γ ( x, z ) − R γ ( x, z ) have the form ω ν − µ ξ i and lim x →∞ ξ i = 0 , hence, for sufficiently large x , the absolute value of each ofthese coefficients is less than δ . By Lemma 15, there exist roots β , . . . , β t ∈ K [ ı ] of Q γ ( x, z ) such that, for sufficiently large x , for ≤ j ≤ t , | β j − ρ j | < ǫ = ω η . (cid:3) SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS9
Fix ≤ j ≤ t and let α j = β j ω γ . Then Q ( x, α j ) = ω µ Q γ ( x, β j ) = 0 . Supposethat f , . . . , f m form an m -term asymptotic approximation of ρ j such that, for ≤ i ≤ m , mrv ( f i ) ≺ ω . For ≤ i ≤ m , put g i = f i ω γ . We have α j − P mi =1 g i g m = β j − P mi =1 f i f m = β j − ρ j f m + ρ j − P mi =1 f i f m For sufficiently large x , | β j − ρ j | < ω η . Since f m ≺ ω , lim x →∞ ω η f m = 0 . Hence, lim x →∞ α j − P mi =1 g i g m = 0 , and so g , . . . , g m form an m -term asymptotic approxima-tion of α j . Since for any f ∈ K [ ı ]lim x →∞ f ( x ) = lim x →∞ f (log k ( x )) g ↓ k , . . . , g ↓ km form an m -term asymptotic approximation of the root α j (log k ( x )) of P . Let us now consider the case where we have found an exact solution ρ j ∈ E C ∞ ( x ) of R γ ( x, z ) . To simplify the description of the case let us make the following rathertechnical definition. Definition 19.
Let ω ∈ E ∞ ( x ) , b , . . . , b n ∈ E C ∞ ( x ) , e , . . . , e n ∈ R ∪ {∞} , d > ,and k ∈ Z ≥ be the result of applying Algorithm 16 to a , . . . , a n . We will call aroot α (log k ( x )) of P asymptotically small if lim x →∞ α = 0 and α ≍ ω (in otherwords, α = βω γ with γ > ).Suppose that we have found an exact solution ρ j ∈ E C ∞ ( x ) of R γ ( x, z ) of mul-tiplicity σ j . Then there exist exactly σ j roots β ,j , . . . , β σ j ,j ∈ K [ ı ] of Q γ ( x, z ) such that, for sufficiently large x , for ≤ ι ≤ σ j , | β ι,j − ρ j | < ω η . Hence, thereexist exactly σ j roots α ι,j = β ι,j ω γ of Q ( x, y ) such that, for sufficiently large x ,for ≤ ι ≤ σ j , | α ι,j − ρ j ω γ | < ω γ + η . The mapping ϕ : ζ → ρ j ω γ + ω γ ζ is abijection between the roots ζ of Q ( x, ρ j ω γ + ω γ y ) such that, for sufficiently large x , | ζ | < ω η and the roots ϕ ( ζ ) of Q ( x, y ) such that | ϕ ( ζ ) − ρ j ω γ | < ω γ + η . Since ν > µ can be chosen arbitrarily close to µ , η can be arbitrarily small. There-fore the mapping ξ → ( ρ j ω γ ) ↓ k + ( ω γ ) ↓ k ξ is a bijection between the roots ξ of P ( x, ( ρ j ω γ ) ↓ k + ( ω γ ) ↓ k y ) that are identically zero or asymptotically small and theroots α ,j (log k ( x )) , . . . , α σ j ,j (log k ( x )) of P ( x, y ) .The above discussion suggests the following procedure for finding asymptoticapproximations of roots of P . Use Algorithm 16 for a , . . . , a n , to find ω ∈ E ∞ ( x ) , b , . . . , b n ∈ E C ∞ ( x ) , e , . . . , e n ∈ R ∪ {∞} , d > , and k ∈ Z ≥ . Compute the valuesof γ such that − γ is the slope of one of the segments that form the lower part of theboundary of the convex hull of A = { ( i, e i ) : i ∈ I } . For each γ find R γ ( x, z ) andcall the procedure recursively to find asymptotic approximations of roots of R γ .Finally, obtain asymptotic approximations of roots of P by multiplying the termsof asymptotic approximations of roots of R γ by ω γ and replacing x with log k ( x ) .The following algorithm formalizes this procedure, handles the base case, and thecase where we get an exact solution with less than the requested m terms. Notation . We use the notation ⊔ for joining lists, that is ( f , . . . , f l ) ⊔ ( g , . . . , g m ) = ( f , . . . , f l , g , . . . , g m ) For a list F = ( f , . . . , f l ) of expressions in E C ∞ ( x ) let F ↓ k = ( f ↓ k , . . . , f ↓ kl ) let X F = f + . . . + f l and, for g ∈ E C ∞ ( x ) , let gF = ( gf , . . . , gf l ) Algorithm 21. (AsymptoticSolutions)Input: P ( x, y ) = a n ( x ) y n + . . . + a ( x ) ∈ E C ∞ ( x )[ y ] with a n = 0 and a = 0 , m ∈ Z > , sflag ∈ { true , false } Output: (( F , σ ) , . . . , ( F t , σ t )) such that • for ≤ i ≤ t , F i = ( f i, , . . . , f i,m i ) is an m i -term asymptotic approximationof σ i roots of P ( x, y ) in y (counted with multiplicities), • f i, , . . . , f i,m i ∈ E C ∞ ( x ) , • either m i = m or m i < m and f i, + . . . + f i,m i is an exact root of P ofmultiplicity σ i , • if sflag = false then σ + . . . + σ t = n and F , . . . , F t are asymptotic approx-imations of all complex roots of P ( x, y ) , • if sflag = true then F , . . . , F t are asymptotic approximations of all asymp-totically small complex roots of P ( x, y ) . (1) If P does not depend on x then (a) if sflag = true return () , (b) let r , . . . , r t ∈ C be the distinct roots of P , (c) for ≤ i ≤ t , let σ i be the multiplicity of r i , (d) return ((( r ) , σ ) , . . . , (( r t ) , σ t )) . (2) Apply Algorithm 16 to a , . . . , a n , obtaining ω ∈ E ∞ ( x ) , b , . . . , b n ∈ E C ∞ ( x ) e , . . . , e n ∈ R ∪ {∞} , d > , and k ∈ Z ≥ . (3) Let I = { i : 0 ≤ i ≤ n ∧ b i = 0 } and A = { ( i, e i ) : i ∈ I } . Compute γ , . . . , γ l such that the lower part of the boundary of the convex hull of A consists of segments with slopes − γ , . . . , − γ l . (4) Set R = () . (5) For ≤ j ≤ l do: (a) if sflag = true and γ j ≤ , continue the loop with the next j , (b) compute µ = min i ∈ I e i + γ j i , J = { i ∈ I : e i + γ j i = µ } ,(c) let λ = min J and let R ( x, z ) = P i ∈ J b i z i − λ , (d) compute (( F , σ ) , . . . , ( F t , σ t )) = AsymptoticSolutions ( R ( x, z ) , m, false) where F ι = ( f ι, , . . . , f ι,m ι ) , for ≤ ι ≤ t , (e) For ≤ ι ≤ t do: (i) put G = ( ω γ j F ) ↓ k , (ii) if m ι = m , set R = R ⊔ (( G, σ ι )) and continue the loop with thenext ι , (iii) if m ι < m , put r = P G , (iv) compute P ( x, r + ( ω γ j ) ↓ k y ) = a r,n ( x ) y n + . . . + a r, ( x ) , with a r,i ∈ E C ∞ ( x ) , and let λ = min { i : a r,i = 0 } , SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS11 (v) if λ > , set R = R ⊔ (( G, λ )) , and if λ = σ ι , continue the loopwith the next ι , (vi) put P r ( x, y ) = a r,n ( x ) y n − λ + . . . + a r,λ ( x ) , (vii) compute (( G , τ ) , . . . , ( G s , τ s )) = AsymptoticSolutions ( P r ( x, y ) , m − m ι , true) (viii) for ≤ κ ≤ s , (ix) set R = R⊔ (( G ⊔ ( ω γ ) ↓ k G , τ ) , . . . , ( G ⊔ ( ω γ ) ↓ k G s , τ s )) (6) Return R .Proof. For a proof of termination of the algorithm, put
Σ = P ni =0 Size ( a i ) and letus use pairs ( m, Σ) ∈ Z > × Z ≥ as a metric. Note that Z > × Z ≥ with the lexico-graphic order does not admit infinite strictly decreasing sequences. The recursivecalls to Algorithm 21 in step d ) have the same value of m and a strictly lowervalue of Σ , because according to the specification of Algorithm 16, P ni =0 Size ( b i ) < P ni =0 Size ( a i ) . The recursive calls to Algorithm 21 in step e )( vii ) have a strictlylower value of m . This shows that the algorithm terminates.Correctness of the algorithm follows from the discussion earlier in this section. (cid:3) Example 22.
Find one-term asymptotic approximations of roots of P = y − exp( x ) y + x exp( πx ) y + log( x ) y − x Applying Algorithm 16 to the coefficients of P we get ω = exp( − x ) , ( b , b , b , b , b , b ) = ( − x , log( x ) , , x, − , e , e , e , e , e , e ) = (0 , , ∞ , − π, − , d = ∞ , and k = 0 (the coefficients and exponents can be read off P written interms of ω : y − ω − y + xω − π y + log( x ) y − x ; in this case a ↑ ki = b i ω e i hence d = ∞ ). The set A and the lower part of the boundary of the convex hull of A areshown in Figure 4.1. We obtain γ = π and γ = − π .For γ we get R = xy − x . In the recursive call to Algorithm 21 applyingAlgorithm 16 to the coefficients of R yields ω = exp( − x ) , ( b , , b , , b , , b , ) = ( − , , , e , , e , , e , , e , ) = ( − , ∞ , ∞ , − d = ∞ , and k = 1 . The lower part of the boundary of the convex hull of A consists of one segment and we get γ , = − and the corresponding poly-nomial R , = y − . The recursive call to Algorithm 21 returns simple roots , − − ı √ , − ı √ of R , . We have ( ω γ , ) ↓ = √ x hence Algorithm 21 for R returns √ x, − − ı √ √ x, − ı √ √ x Figure 4.1.
The Newton polygon of A - - - - - - all with multiplicity . Since ( ω γ ) ↓ = exp( − π x ) , we add √ x exp( − π x ) , − − ı √ √ x exp( − π x ) , − ı √ √ x exp( − π x ) to R .For γ we get R = y + x . In the recursive call to Algorithm 21 applying Algo-rithm 16 to the coefficients of R yields ω = exp( − x ) , ( b , , b , , b , ) = (1 , , , ( e , , e , , e , ) = ( − , ∞ , , d = ∞ , and k = 1 . The lower part of the bound-ary of the convex hull of A consists of one segment and we get γ , = − andthe corresponding polynomial R , = y + 1 . The recursive call to Algorithm 21returns simple roots − ı, ı of R , . We have ( ω γ , ) ↓ = √ x hence Algorithm 21 for R returns − ı √ x, ı √ x , both with multiplicity . Since ( ω γ ) ↓ = exp( π x ) , we add − ı √ x exp( π x ) , ı √ x exp( π x ) to R .Finally, the algorithm returns one-term asymptotic approximations R =( √ x exp( − π x ) , − − ı √ √ x exp( − π x ) , − ı √ √ x exp( − π x ) , − ı √ x exp( π x ) , ı √ x exp( π x )) all with multiplicity . 5. Real roots
Imaginary part of an non-real root may be asymptotically smaller than the realpart, hence real asymptotic approximations can correspond to non-real roots.
Example 23.
Let P ( x, y ) = ( y − x exp( x ) y + exp(2 x )) + 1 . Three-term as-ymptotic approximations of roots of P in y computed with Algorithm 21 are ( x − + x − + 2 x − ) exp( x ) and ( x − x − − x − ) exp( x ) , both with multiplicity two. SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS13
The approximations are real-valued, yet P clearly does not have real roots. Com-puting more terms only adds real-valued terms of the form ax − n in the coefficient of exp( x ) . Imaginary parts would show up only in a transfinite series representation,since the imaginary parts are asymptotically smaller than x − n exp( x ) for any n .For this low degree polynomial P we can compute asymptotic approximations ofimaginary parts of roots of P , by computing R = res z ( P ( x, y + z ) , P ( x, z )) . Theroots of this polynomial of degree in y with terms, are differences of pairsof roots of P . In particular, four of the roots are equal to the imaginary parts ofroots of P , multiplied by two. One-term asymptotic approximations of roots of R computed with Algorithm 21 include two purely imaginary-valued expressions, − ιx − exp( − x ) and ιx − exp( − x ) , both of multiplicity two. This shows that, in-deed, the imaginary parts of roots of P are asymptotically smaller than x − n exp( x ) for any n .In this section we provide a method for deciding which asymptotic approxima-tions correspond to real roots.Let P ( x, y ) = a n ( x ) y n + . . . + a ( x ) ∈ E ∞ ( x )[ y ] with a n = 0 and a = 0 , andlet (( F , σ ) , . . . , ( F t , σ t )) be the output of Algorithm 21 for P , with some m > and sf lag = f alse . Note that here we assume that the coefficients of P are real-valued. For ≤ i ≤ t , F i = ( f i, , . . . , f i,m i ) . First, let us note the easy cases. Ifany of f i,j are not real-valued, then F i does not correspond to a real root. If all f i,j are real-valued and σ i = 1 then F i corresponds to a real root. If m i < m then f i, + . . . + f i,m i is equal to the exact root, hence it is evident whether the root isreal valued.The hard case is when there are real-valued asymptotic approximations with σ i > and m i = m . We will find the number of distinct real roots corresponding to eachreal-valued asymptotic approximation with multiplicity higher than one. Assumethat, possibly after reordering, the real-valued asymptotic approximations are F i ,for ≤ i ≤ s . Let r i = f i, + . . . + f i,m i , for ≤ i ≤ s . The algorithm MrvLimit of [2] contains a subprocedure which computes the sign of exp-log expressions nearinfinity. Hence, we can reorder the approximations so that, for sufficiently large x , r < . . . < r s . Put h = −∞ , h i = r i + r i +1 , for ≤ i < s , and h s = ∞ . Lemma 24. If F i corresponds to a real root α of P then, for sufficiently large x , h i − < α < h i . To prove the lemma we will use the following claim.
Claim . If F = ( f , . . . , f m f ) and G = ( g , . . . , g m g ) are asymptotic approxima-tions returned by Algorithm 21 for P and there is l ≤ min( m f , m g ) such that, forall ≤ j < l , f j = g j and f l = g l , then lim x →∞ f l g l = 1 . Proof.
The claim is true when P does not depend on x , hence, by induction, wemay assume that the claim is true for the recursive calls to Algorithm 21. If F and G were computed in different iterations of the loop in step then l = 1 , f = ( ω γ f c f ) ↓ k , g = ( ω γ g c g ) ↓ k , mrv ( c f , c g ) ≺ ω , and γ f = γ g , hence lim x →∞ | f l || g l | is either or ∞ . If F and G were computed in the same iteration of the loop instep , but in different iterations of the loop in step e ) then the claim is true bythe inductive hypothesis applied to R ( x, z ) . Finally, if F and G were computed inthe same iteration of the loop in step e ) then l > m ι , and hence neither F nor G was added in step e )( v ) . Therefore, the claim is true by the inductive hypothesisapplied to P r ( x, y ) . (cid:3) Let us now prove Lemma 24.
Proof. If m i < m then α = r i and h i − < r i < h i , because, for sufficiently large x , r < . . . < r s . Hence we can assume that m i = m . Let us prove that h i − < α .If i = 1 , then h i − = −∞ and the inequality is true. Let l be such that for all ≤ j < l f i − ,j = f i,j and f i − ,l = f i,l . We define f i − ,m i − +1 = 0 , so that such l always exist. Note that, for sufficiently large x , f i − ,l < f i,l , because r i − < r i . If lim x →∞ | f i − ,l || f i,l | ≤ put g = f i,l else put g = f i − ,l . We have α − h i − g = α − P mj =1 f i,j f i,m f i,m g + f i,l − f i − ,l g + P mj = l +1 f i,j g − P m i − j = l +1 f i − ,j g Since F i is an asymptotic approximation of α , we have lim x →∞ α − P mj =1 f i,j f i,m = 0 and, for j > l , we have lim x →∞ f i,j g = 0 and lim x →∞ f i − ,j g = 0 , therefore lim x →∞ α − h i − g = lim x →∞ f i,l − f i − ,l g If f i − ,l = 0 , then g = f i,l , for sufficiently large x , f i,l > , and lim x →∞ f i,l − f i − ,l g = , hence for sufficiently large x , α − h i − > .If f i − ,l = 0 , then the assumptions of Claim 25 are satisfied, and hence lim x →∞ f i − ,l f i,l =1 . Suppose that g = f i,l . Then lim x →∞ f i,l − f i − ,l g = 12 −
12 lim x →∞ f i − ,l f i,l = 0 Since for sufficiently large x , f i,l − f i − ,l > , lim x →∞ | f i − ,l || f i,l | ≤ , and lim x →∞ f i − ,l f i,l =1 , hence, for sufficiently large x , f i,l > . Therefore, lim x →∞ α − h i − f i,l = lim x →∞ f i,l − f i − ,l f i,l > which shows that, for sufficiently large x , h i − < α .Now suppose that g = f i − ,l . Then lim x →∞ f i,l − f i − ,l g = 12 lim x →∞ f i,l f i − ,l − = 0 Since for sufficiently large x , f i,l − f i − ,l > , and lim x →∞ | f i − ,l || f i,l | > , hence, forsufficiently large x , f i − ,l < . Therefore, lim x →∞ α − h i − f i − ,l = lim x →∞ f i,l − f i − ,l f i − ,l < which shows that, for sufficiently large x , h i − < α . The proof that, for sufficientlylarge x , α < h i is similar. (cid:3) SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS15
Let P , . . . , P k be the Sturm sequence of P in y over E ∞ ( x ) (that is coefficientsthat are identically zero near infinity are set to zero). For ≤ i ≤ s − and ≤ j ≤ k , P j ( x, h i ) ∈ E ∞ ( x ) , hence it has a constant sign θ i,j near infinity, and wecan compute θ i,j using a subprocedure of MrvLimit.
Let c j y n j be the leading termof P j , for ≤ j ≤ k , let θ ,j be the sign near infinity of ( − n j c j , and let θ s,j bethe sign near infinity of c j . For ≤ i ≤ s − , let ν i be the number of sign changesin the sequence Θ i = ( θ i, , . . . , θ i,k ) . Criterion 26.
The number of distinct real roots of P corresponding to the asymp-totic approximation F i is equal to ν i − − ν i . Correctness of the criterion follows from Lemma 24 and Sturm’s theorem.
Example 27.
As in Example 23, let P ( x, y ) = ( y − x exp( x ) y + exp(2 x )) + 1 .One-term asymptotic approximations of roots of P in y computed with Algorithm21 are r = x − exp( x ) and r = x exp( x ) , both with multiplicity two. The Sturmsequence of P in y is P = y − x exp( x ) y + ( x + 2) exp(2 x ) y − x exp(3 x ) y +exp(4 x ) + 1 P = 4 y − x exp( x ) y + 2( x + 2) exp(2 x ) y − x exp(3 x ) P = 14 ( x −
4) exp(2 x ) y −
14 ( x − x ) exp(3 x ) y +14 ( x −
4) exp(4 x ) − P = − x −
4) exp(2 x ) y + 8 x ( x −
4) exp( x ) P = 116 ( x − x + 16) exp(4 x ) + 1 We have Θ = (1 , − , , , and Θ = (1 , , , − , . Since ν = ν = 2 , P has noreal roots near infinity (and we do not need to compute ν , since it must equal as well).Let Q ( x, y ) = ( y − x exp( x ) y + exp(2 x )) − . One-term asymptotic approxima-tions of roots of Q in y computed with Algorithm 21 are the same as for P . TheSturm sequence of Q in y is Q = y − x exp( x ) y + ( x + 2) exp(2 x ) y − x exp(3 x ) y +exp(4 x ) − Q = 4 y − x exp( x ) y + 2( x + 2) exp(2 x ) y − x exp(3 x ) Q = 14 ( x −
4) exp(2 x ) y −
14 ( x − x ) exp(3 x ) y +14 ( x −
4) exp(4 x ) + 1 Q = 16( x −
4) exp(2 x ) y − x ( x −
4) exp( x ) Q = 116 ( x − x + 16) exp(4 x ) − We have Θ = (1 , − , , − , and Θ = (1 , , , , . Since ν = 4 and ν = 0 , P has four distinct real roots near infinity. This again is sufficient to tell that r or r correspond to two real root each. And indeed, if we substitute h = ( r + r ) / x − + x ) exp( x ) into the Sturm sequence and compute the signs near infinity weget Θ = (1 , − , − , , and ν = 2 .6. Implementation and experimental results
We have implemented
AsymptoticSolutions as a part of the
Mathematica sys-tem. The implementation has been done in Wolfram Language, using elementsof the
MrvLimit algorithm, which is implemented partly in the C source code of
Mathematica and partly in Wolfram Language. The experiments have been run ona laptop computer with a . GHz Intel Core i7-4800MQ processor and GB ofRAM assigned to the Linux virtual machine.
Example 28.
We use eight exp-log expressions from examples in [2] as polynomialcoefficients. a = e x ( e /x − e − x − e /x ) a = e ex − e − x − /x − e e x a = e e ex + e − x e e ex a = e e ex e e ex − e − ex a = (3 x + 5 x ) /x a = x log( x log( x ) log(2) / log( x ) ) a = exp(4 xe − x / ( e − x + e − x x +1 )) − e x e x a = exp( xe − x e − x + e − x / ( x +1) ) e x Let P n ( x, y ) = P ni =0 a i y i . We have run the examples with n ranging from to and with varying number m of requested terms. The results are given in Table 1.For each value of m the row Time gives the computation time in seconds, the row
Iter gives the number of calls to Algorithm 21, and the row LC gives the total leafcount of the returned expressions.We can observe that for a fixed polynomial the number of recursive calls is closeto linear in the number of additional terms requested. Increasing the degree didnot necessarily lead to higher complexity, e.g. adding the degree term made thecomputation easier. A likely cause for this is that the dominating terms in thedegree polynomial were simpler than those in the degree polynomial. SYMPTOTIC SOLUTIONS OF POLYNOMIAL EQUATIONS WITH EXP-LOG COEFFICIENTS17
Table 1.
Example 28. m n
Time .
171 0 .
235 0 .
292 0 .
276 0 .
389 0 . Iter LC
56 84 124 195 192 323
Time .
231 0 .
508 1 .
18 1 .
20 1 .
02 1 . Iter
20 32 44 59 58 67 LC
136 229 498 668 588 1933
Time .
376 0 .
796 3 .
53 5 .
25 1 .
64 5 . Iter
40 63 86 124 118 137 LC
256 433 896 1303 1188 4008
Time .
968 3 .
21 26 . . .
47 20 . Iter
80 124 170 254 238 277 LC
496 837 1716 2573 2388 8158
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