On a non-archimedean broyden method
OOn A Non-Archimedean Broyden Method
Xavier Dahan
Tohoku University, IEHESendai, Japan [email protected]
Tristan Vaccon
Université de Limoges; CNRS, XLIM UMR 7252Limoges, France [email protected]
ABSTRACT
Newton’s method is an ubiquitous tool to solve equations, both inthe archimedean and non-archimedean settings — for which it doesnot really differ. Broyden was the instigator of what is called “quasi-Newton methods”. These methods use an iteration step where onedoes not need to compute a complete Jacobian matrix nor its inverse.We provide an adaptation of Broyden’s method in a general non-archimedean setting, compatible with the lack of inner product, andstudy its Q and R convergence. We prove that our adapted methodconverges at least Q-linearly and R-superlinearly with R-order 2 m in dimension m . Numerical data are provided.
KEYWORDS
System of equations, Broyden’s method, Quasi-Newton, p-adicapproximation, Power series, Symbolic-numeric, p-adic algorithm
ACM Reference format:
Xavier Dahan and Tristan Vaccon. 2020. On A Non-Archimedean BroydenMethod. In
Proceedings of The 45th International Symposium on Symbolicand Algebraic Computation, Kalamata, Greece, July 2020 (ISSAC’20),
In the numerical world.
Quasi-Newton methods refer to a class ofvariants of Newton’s method for solving square nonlinear systems,with the twist that the inverse of the Jacobian matrix is “approxi-mated” by another matrix. When compared to Newton’s method,they benefit from a cheaper update at each iteration (See e.g. [10,p.49-50, 53]), but suffer from a smaller rate of convergence. Theywere mainly introduced by Broyden in [6], which has sparkednumerous improvements, generalizations, and variants (see thesurveys [10, 19]). It is now a fundamental numerical tool (that findsits way in entry level numerical analysis textbooks [8, § 10.3]). Tosome extent, this success stems from: the specificities of machineprecision arithmetic as commonly used in the numerical commu-nity, the fact that Newton’s method is usually not quadraticallyconvergent from step one, and that the arithmetic cost of an itera-tion is independent of the quality of the approximation reached. Inanother direction, variants of Broyden’s method have known dra-matic success for unconstrained optimization — the target systemis the gradient of the objective function, the zeros are then criticalpoints— where it takes advantage of the special structure of theHessian (see Sec. 7 of [10]). Another appealing feature of Broyden’s
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ISSAC’20, Kalamata, Greece © 2020 Copyright held by the owner/author(s). ...$15.00DOI: method is the possibility to design derivative-free methods gener-alizing to the multivariate case the classical secant method (whichcan be thought of as Broyden’s in dimension one). This feature is amain motivation for this work.
Non-archimedean.
It is a natural wish to transpose such a fun-damental numerical method to the non-archimedean framework,offering new tools to perform exact computations, typically for sys-tems with p -adic or power series coefficients. For this adaptation,several non-trivial difficulties have to be overcome: e.g. no innerproducts, a more difficult proof of convergence, or a management ofarithmetic at finite precision far more subtle. This article presentssatisfactory solutions for all these difficulties, which we believe canbe expanded to a broader variety of quasi-Newton methods.Bach proved in [1] that in dimension one, the secant methodcan be on an equal footing with Newton’s method in terms of com-plexity. We investigate how this comparison is less engaging insuperior dimension (see Section 6). To our opinion, this is due to theremarkable behavior of Newton’s method in the non-archimedeansetting. No inversion of the Jacobian is required at each iteration(simply a matrix multiplication, this is now classical see [5, 16, 17]).The evaluation of the Jacobian is also efficient for polynomial func-tions (in dimension m , it involves only O ( m ) evaluations, insteadof m over R , see [2]). It displays also true quadratic behavior fromstep one which, when combined with the natural use of finite pre-cision arithmetic (against machine precision over R ), offers a ratiocost/precision gained that is hard to match.And indeed, our results show that for large dimension m andpolynomials as input, there is little hope for Broyden to outperformNewton, although it depends on the order of superlinear conver-gence of Broyden’s method. In this respect more investigation isnecessary, but for now the interest lies more in the theoretical ad-vances and in the situations mentioned in “ Motivations ” thereafter.
Relaxed arithmetic.
Since the cost of one iteration of Broyden’smethod involves m instead of m ω for Newton, we should mentionthe relaxed framework (a.k.a online [11]) which show essentiallythe same decrease of complexity, while maintaining quadratic con-vergence. It has been implemented efficiently for power series [23],and for p -adic numbers [3]. In case of a smaller m and a largerprecision of approximation required, FFT trading [24] has to bementioned. These techniques are however unlikely to be suitedto the Broyden iteration, since it is a priori not described by afixed-point equation, a necessity for the relaxed machinery. Motivations.
As explains Remark 6.4, it seems unlikely in thenon-archimedean world that with polynomials or rational fractions,a quasi-Newton method meets the standard of Newton’s method.The practical motivations concern: a r X i v : . [ c s . S C ] S e p SSAC’20, July 2020, Kalamata, Greece Xavier Dahan and Tristan Vaccon
1/ Derivative-free method: instead of starting with the Jacobianat precision one, use a divided-difference matrix. A typical applica-tion is when the function is given by a “black-box” and there is nodirect access to the Jacobian.2/ When computing the Jacobian does not allow shortcuts likein the case of rational fractions [2], evaluating it may require upto Lm operations, where L is the complexity of evaluation of theinput function. Regarding the complexity of Remark 6.4, Broyden’smethod then becomes beneficial when L (cid:38) m − m ω − .3/ While Newton’s method over general Banach spaces of infinitedimension can be made effective when the differential is effectivelyrepresentable (integral equations [15, § 5][14] are a typical exam-ple), it is in general difficult or impossible to compute it. On theother hand, Broyden’s method or its variants have the ability towork with approximations of the differential, including of finiterank , by considering a projection (as shown in [14, 15] and the ref-erences therein; the dimension of the projection is increased at eachiteration). In the non-archimedean context, ODEs with parameters,for example initial conditions, constitute a natural application. Organization of the paper.
Definitions and notations are intro-duced in Section 2. Section 3 explains how Broyden’s method canbe adapted to an ultrametric setting. In Section 4, we study the Qand R-order of convergence of Broyden’s method (see Definition2.1), presenting our main results. It is followed by Section 5, whereare introduced developments and conjectures on Q-superlinearity.Finally, in Section 6, we explain how our Broyden’s method canbe implemented with dynamical handling of the precision, and weconclude with some numerical data in Section 7.
Throughout the paper, K refers to a complete, discrete valuationfield, val : K (cid:16) Z ∪ { + ∞} to its valuation, O K its ring of integersand π a uniformizer. For k ∈ N , we write O ( π k ) for π k O K . Let m ∈ Z ≥ . We are interested in computing an approximationof a non-singular zero x (cid:63) of f : K m → K m through an iterativesequence of approximations, ( x n ) n ∈ N ∈ ( K m ) N . Note that all ourvectors are column-vectors. For any x ∈ K m where it is well-defined, we denote by f (cid:48) ( x ) ∈ M m ( K ) the Jacobian matrix of f at x . We will use the following notations (borrowed from [13]): f n = f ( x n ) , y n = f n + − f n , s n = x n + − x n (1)We denote by ( e , . . . , e m ) the canonical basis of K m . In K m , O ( π k ) means O ( π k ) e + · · · + O ( π k ) e m . Newton’s iteration produces a sequence ( x n ) n ∈ N given by: x n + = x n − f (cid:48) ( x n ) − · f ( x n ) . (N)For quasi-Newton methods, the iteration is given by: x n + = x n − B − n · f ( x n ) , (⇒ s n = − B − n · f n ) (QN)with B n presumably not far from f (cid:48) ( x n ) . More precisely, it is ageneralization of the design of the secant method over K whereone approximates f (cid:48) ( x n ) by f ( x n )− f ( x n − ) x n − x n − . In quasi-Newton, it isthus required that: B n · ( x n − x n − ) = f ( x n ) − f ( x n − ) (⇒ B n · s n − = y n − ) (2) Discrete valuation is only needed in Section 6. For the rest complete and ultrametricis enough.
By this condition alone, B n is obviously underdetermined. To miti-gate this issue, B n is taken as a one-dimensional modification of B n − satisfying (2). Concretely, a sequence ( u n ) n ∈ N ∈ ( K m ) N isintroduced such that: B n = B n − + ( y n − − B n − s n − ) · u n − t . (3)1 = u n − t · s n − . (4)In Broyden’s method over R , u n − is defined by: u n − = s n − s n − t · s n − . (5)The computation of the inverse of B n can then be done using theSherman-Morrison formula (see [22]): B − n = B − n − + ( s n − − B − n − y n − ) · s n − t B − n − s n − t B − n − y n − . (6)This formula gives rise to the so-called “good Broyden’s method”.Using [22] provides the following alternative formulae: B n = B n − + f n · u n − t . (7) B − n = B − n − − B − n − f n · u n − t B − n − u n − t B − n − y n − . (8) We recall some notions on convergence of sequences commonlyused in the analysis of the behavior of Broyden’s method.Definition 2.1 ([20] Chapter 9).
A sequence ( x k ) k ∈ N ∈ ( K m ) N has Q-order of convergence µ ∈ R > to a limit x (cid:63) ∈ K m , if: ∃ r ∈ R + , ∀ k large enough, (cid:107) x k + − x (cid:63) (cid:107)(cid:107) x k − x (cid:63) (cid:107) µ ≤ r . If we can take µ = and r < in the previous inequality, we saythat ( x k ) k ∈ N has Q-linear convergence. For µ = , we say it hasQ-quadratic convergence. The sequence is said to have Q-superlinearconvergence if lim k → + ∞ (cid:107) x k + − x (cid:63) (cid:107)(cid:107) x k − x (cid:63) (cid:107) = . It is said to have R-order of convergence µ ∈ R ≥ if lim sup (cid:107) x k − x (cid:63) (cid:107) / µ k < . Remark . For both Q and R, we write has convergence µ to mean has convergence at least µ . Broyden’s method satisfies the following convergence results:Theorem 2.3.
Over R m , under usual regularity assumptions, Broy-den’s method defined by Eq. (5) converges locally Q-superlinearly[7], exactly in m steps for linear systems, and with R-order at least m > [13]. Unfortunately, for general K , Eq. (5) is not a good fit. Indeed, thequadratic form x (cid:55)→ x t x can be isotropic over K m , i.e. there canbe an s n (cid:44) s nt · s n =
0. This is the case, for example if s n = ( X , X ) in F (cid:74) X (cid:75) . Consequently, (5) has to be modified. Tryingto seek for another quadratic form that would not be isotropic ispointless, since for example there is none over Q mp for m ≥ R-convergence is a weaker notion, aimed at sequences not monotonically decreasing. By locally, we mean that for any x and B in small enough balls around x (cid:63) and f (cid:48) ( x (cid:63) ) , the following convergence property is satisfied. n A Non-Archimedean Broyden Method ISSAC’20, July 2020, Kalamata, Greece Remark . In the sequel, all the B i ’s will be invertible matrices.Consequently, s n + = f ( x n ) = . We thereforeadopt the convention that if for some x n , we have f ( x n ) = , thenthe sequences ( x v ) v ≥ n and ( B v ) v ≥ n will be constant, and this casedoes not require any further development. We use the following natural (non-normalized) norm on K definedfrom its valuation: for any x ∈ K , (cid:107) x (cid:107) = − val ( x ) , except for K = Q p , where we take the more natural p − val ( x ) over Q p . Our norm on K can naturally be extended to K m : for any x = ( x , . . . , x m ) ∈ K m , (cid:107) x (cid:107) = max i | x i | . We denote by val ( x ) the minimal valuationamong the val ( x i ) ’s. It defines the norm of x . Lemma 3.1.
Let (cid:103) · (cid:103) be the norm on M m ( K ) induced by (cid:107) · (cid:107) .Let us abuse notations by denoting with (cid:107) · (cid:107) the max-norm on thecoefficients of the matrices of M m ( K ) . Then (cid:103) · (cid:103) = (cid:107) · (cid:107) . Proof. Let A ∈ M n ( K ) . If x ∈ K m is such that (cid:107) x (cid:107) ≤ , then byultrametricity, it is clear that (cid:107) Ax (cid:107) ≤ (cid:107) A (cid:107) , hence (cid:103) A (cid:103) ≤ (cid:107) A (cid:107) . If i ∈ N is such that (cid:107) A (cid:107) is obtained with a coefficient on the columnof index i , then (cid:107) Ae i (cid:107) = (cid:107) A (cid:107) , whence the equality. (cid:3) Consequently, the max -norm on the coefficients of a matrix is amatrix norm. For rank-one matrices, the computation of the normcan be made easy using the following corollary of Lemma 3.1.Corollary 3.2.
Let a , b ∈ K m be two vectors. Then (cid:107) a t · b (cid:107) = (cid:107) a (cid:107) · (cid:107) b (cid:107) . (9) For the sequence ( x n ) n ∈ N to be well defined, the sequence ( u n ) n ∈ N must satisfy Eqs (3)-(4) and also: s nt B − n y n (cid:44) , (10)to ensure Eq. (6) makes sense. Many different u n ’s can satisfy thoseconditions. Over R , Broyden’s choice of u n defined by (5) can becharacterized by minimizing the Frobenius norm of B n + − B n . Wecan proceed similarly over K . Lemma 3.3. If B n + satisfies (2) , then: (cid:107) B n + − B n (cid:107) ≥ (cid:107) y n − B n s n (cid:107)(cid:107) s n (cid:107) . (11)Proof. It is clear as in this case, ( B n + − B n ) s n = y n − B n s n . (cid:3) This inequality can become an equality with a suitable choice of u n as shown in the following lemma.Lemma 3.4. Let l be such that val ( s n , l ) = val ( s n ) . Then u n = s − n , l e l satisfies (4) and reaches the bound in (11) . Nevertheless, this is not enough to have B n invertible in general,as we can see from the Sherman-Morrison formula (8): Over R , it is of course denoted by (cid:107) · (cid:107) ∞ , but when based on a non-archimedeanabsolute value, this notation is not used since it is implicitly unambiguous: othernorms such as the (cid:107) · (cid:107) p are mostly useless. Lemma 3.5. B n defined by Eq. (3) is invertible if and only if u n − t B − n − y n − (cid:44) . (12)The next lemma shows how to choose l , up to the condition ( B − n − y n − ) l (cid:44)
0, which actually never occurs after Corollary 4.3.Lemma 3.6.
Let l be the smallest index such that val ( s n , l ) = val ( s n ) . If (cid:16) B − n − y n − (cid:17) l (cid:44) , then u n = s − n , l e l (13) satisfies Eq. (4) , reaches the bound in Eq. (11) and satisfies Eq. (12) . Let E and F be two finite-dimensional normed vector spaces over K We denote by L ( E , F ) the space of K -linear mappings from E to F .Definition 4.1. Let U be an open subset of E . A function f : U → F is strictly differentiable at x ∈ U if there exists an f (cid:48) ( x ) ∈ L ( E , F ) satisfying the following property: for all ε > , there exists aneighborhood U x , ε ⊂ U of x , on which for any y , z ∈ U x , ϵ : (cid:107) f ( z ) − f ( y ) − f (cid:48) ( x ) · ( z − y )(cid:107) F ≤ ε · (cid:107) z − y (cid:107) E . (14)Note that both z and y can vary. This property is natural inthe ultrametric context (see 3.1.3 of [9]), as the counterpart ofFréchet differentiability over R does not provide meaningful localinformation. Polynomials and converging power series satisfy strictdifferentiability everywhere they are defined.We can then adapt Theorem 3.2 of [7] in our ultrametric setting.Theorem 4.2. Let f : K m → K m and x (cid:63) ∈ U be such that f is strictly differentiable at x (cid:63) , f (cid:48) ( x (cid:63) ) is invertible and f ( x (cid:63) ) = . Then any quasi-Newton method whose choice of u n yields for all n , (cid:107) u n (cid:107) = (cid:107) s n (cid:107) − (which includes Broyden’s choice of Eq. (13) ), islocally Q -linearly converging to x (cid:63) with ratio r for any r ∈ ( , ) . Proof. Let r ∈ ( , ) . Let the constants γ , δ , and λ be satisfying: γ ≥ (cid:107) f (cid:48) ( x (cid:63) ) − (cid:107) , < δ ≤ rγ ( + r )( − r ) , < λ ≤ δ ( − r ) . (15)Let η > x (cid:63) and such thaton the ball B ( x (cid:63) , η ) , (cid:107) f ( z ) − f ( y ) − f (cid:48) ( x (cid:63) ) · ( z − y )(cid:107) ≤ λ · (cid:107) z − y (cid:107) . We restrict further η so as to have: η ≤ δ ( − r ) . Let us assume that (cid:107) B − f (cid:48) ( x (cid:63) )(cid:107) ≤ δ , (cid:107) x − x (cid:63) (cid:107) < η . We have from the condition on δ that δ γ ( + r )( − r ) ≤ r . Since3 − r > , then 2 δ γ ( + r ) ≤ r . Consequently,11 − δγ ≤ + r , the denominator being non zero because δ < ( γ ) − . Since (cid:107) f (cid:48) ( x (cid:63) ) − (cid:107) ≤ γ and (cid:107) B − f (cid:48) ( x (cid:63) )(cid:107) < δ , the BanachPerturbation Lemma ([20] page 45) in the Banach algebra M m ( K ) implies that B is invertible and: (cid:107) B − (cid:107) ≤ γ − γδ ≤ ( + r ) γ . We can now estimate what happens to x = x − B − f ( x ) . SSAC’20, July 2020, Kalamata, Greece Xavier Dahan and Tristan Vaccon (cid:107) x − x (cid:63) (cid:107) = (cid:107) x − x (cid:63) − B − f ( x )(cid:107) , (16) = (cid:107) − B − (cid:0) f ( x ) − f ( x (cid:63) ) − f (cid:48) ( x (cid:63) ) · ( x − x (cid:63) ) (cid:1) − B − (cid:0) f (cid:48) ( x (cid:63) )( x − x (cid:63) ) − B ( x − x (cid:63) ) (cid:1) (cid:107) , = (cid:107) − B − (cid:0) f ( x ) − f ( x (cid:63) ) − f (cid:48) ( x (cid:63) ) · ( x − x (cid:63) ) (cid:1) − B − (cid:0) ( f (cid:48) ( x (cid:63) ) − B )( x − x (cid:63) ) (cid:1) (cid:107) , ≤ (cid:107) B − (cid:107) (cid:0) λ (cid:107) x − x (cid:63) (cid:107) + δ (cid:107) x − x (cid:63) (cid:107) (cid:1) , ≤ (cid:107) B − (cid:107)( λ + δ )(cid:107) x − x (cid:63) (cid:107) , ≤ γ ( + r )( δ ( − r ) + δ )(cid:107) x − x (cid:63) (cid:107) , ≤ γ ( + r ) δ ( − r )(cid:107) x − x (cid:63) (cid:107) by Eq. (15) (middle) ≤ r (cid:107) x − x (cid:63) (cid:107) . (17)Consequently, (cid:107) x − x (cid:63) (cid:107) ≤ r (cid:107) x − x (cid:63) (cid:107) and (cid:107) x − x (cid:63) (cid:107) ≤ rη < η , i.e. x ∈ B ( x (cid:63) , η ) . Eq. (3) defines B by B = B − ( y − B s ) · u t for some u verifying (cid:107) u (cid:107) = (cid:107) s (cid:107) − (see Eqs. (4), Corollary 3.2). Then: (cid:107) B − B (cid:107) = (cid:107) f ( x ) − f ( x ) − B ( x − x )(cid:107) · (cid:107) x − x (cid:107) − . Therefore, (cid:107) B − f (cid:48) ( x (cid:63) )(cid:107) ≤ max (cid:16) (cid:107) B − f (cid:48) ( x (cid:63) )(cid:107) , (18) (cid:107) f ( x ) − f ( x ) − B ( x − x )(cid:107)(cid:107) x − x (cid:107) − (cid:17) , ≤ max (cid:16) (cid:107) B − f (cid:48) ( x (cid:63) )(cid:107) , (cid:107) (cid:0) B − f (cid:48) ( x (cid:63) ) (cid:1) ( x − x )(cid:107)(cid:107) x − x (cid:107) − , (cid:107) f ( x ) − f ( x ) − f (cid:48) ( x (cid:63) )( x − x )(cid:107)(cid:107) x − x (cid:107) − (cid:17) , ≤ max ( δ , λ ) ≤ δ . We can then carry on and prove by induction that for all k ,(i) (cid:107) x k − x (cid:63) (cid:107) ≤ r k (cid:107) x − x (cid:63) (cid:107) , and (ii) B k ∈ B ( f (cid:48) ( x (cid:63) ) , δ ) . (19)Heredity for Inequality (19)-(i) comes from: a same use of theBanach Perturbation Lemma on B k so that B k is invertible; that (cid:107) B − k (cid:107) ≤ ( + r ) γ and by repeating the computations (16) to (17): (cid:107) x k + − x (cid:63) (cid:107) ≤ (cid:107) B k (cid:107) − ( λ + δ )(cid:107) x k − x (cid:63) (cid:107) , ≤ ( + r ) γδ ( − r )(cid:107) x k − x (cid:63) (cid:107) , ≤ r (cid:107) x k − x (cid:63) (cid:107) . We can deal with (19)-(ii) using a similar computation as (18): (cid:107) B k + − f (cid:48) ( x (cid:63) )(cid:107) ≤ max (cid:0) (cid:107) B k − f (cid:48) ( x (cid:63) )(cid:107) , (20) (cid:107) f ( x k + ) − f ( x k ) − B k ( x k + − x k )(cid:107)(cid:107) x k + − x k (cid:107) − (cid:1) ≤ max (cid:0) (cid:107) B k − f (cid:48) ( x (cid:63) )(cid:107) , (cid:107) f ( x k + ) − f ( x k ) − f (cid:48) ( x (cid:63) )( x k + − x k )(cid:107)(cid:107) x k + − x k (cid:107) − (cid:1) , ≤ max ( δ , λ ) ≤ δ . (cid:3) Corollary 4.3.
Locally, one can take definition (13) to define allthe u n ’s and all the B n ’s will still be invertible. Proof. With the assumptions of the proof of Theorem 4.2, for u n defined by (13), (cid:107) u n − (cid:107) = (cid:107) s n − (cid:107) − and (4) are satisfied, and bythe Banach Perturbation Lemma, B n defined by (3) is invertible. (cid:3) Remark . The fact that Broyden’s method has locally Q-linear convergence with ratio r for any r is not enough to prove that ithasQ-superlinear convergence. Indeed, as x k is going closer to x (cid:63) , there is no reason for B k to get closer to f (cid:48) ( x (cid:63) ) . Consequently, wecannot expect from the previous result that x k and B k enter loci ofsmaller ratio of convergence as k goes to infinity. In fact, in general, B k does not converge to f (cid:48) ( x (cid:63) ) . Finally, the next lemma, consequence of the previous theorem,will be useful in the next subsection to obtain the R-superlinearconvergence.Lemma 4.5.
Using the same notations as in the proof of Theorem4.2, if r ≤ (cid:16) γ (cid:107) f (cid:48) ( x (cid:63) )(cid:107) (cid:17) − , and (cid:107) B − f (cid:48) ( x (cid:63) )(cid:107) < δ and (cid:107) x − x (cid:63) (cid:107) < η , then for all n ∈ N , (cid:107) f n + (cid:107) ≤ (cid:107) f n (cid:107) . Proof. Let n ∈ N . We have (cid:107) s n (cid:107) ≤ r (cid:107) s n − (cid:107) . Indeed, from (cid:107) x n + − x n (cid:107) ≤ max ((cid:107) x n + − x (cid:63) (cid:107) , (cid:107) x (cid:63) − x n (cid:107)) , and (cid:107) x n + − x n (cid:107) < (cid:107) x n − x (cid:63) (cid:107) , we see that (cid:107) s n (cid:107) = (cid:107) x (cid:63) − x n (cid:107) ≤ r (cid:107) x (cid:63) − x n − (cid:107) = r (cid:107) s n − (cid:107) .Then using (QN) and the Q-linear convergence with ratio r , weget that (cid:107) f n + (cid:107) ≤ r (cid:107) B n + (cid:107)(cid:107) B − n (cid:107)(cid:107) f n (cid:107) . Using (20), the definition of δ , γ in (15), and the fact that 0 < r < , we get that (cid:107) B n + (cid:107)(cid:107) B − n (cid:107) ≤ γ (cid:107) f (cid:48) ( x (cid:63) )(cid:107) , which concludes the proof. (cid:3) We first remark that the 2 n -step convergence in the linear caseproved by Gay in [13] is still valid. Indeed, it is only a matter oflinear algebra.Theorem 4.6 (Theorem 2.2 in [13]). If f is defined by f ( x ) = Ax − b for some A ∈ GL m ( K ) , then any quasi-Newton method con-verges in at most m steps (i.e. f ( x m ) = ). With this and under a stronger differentiability assumption on f , we can obtain R-superlinearity, similarly to Theorem 3.1 of [13].The proof also follows the main steps thereof.Theorem 4.7. Let us assume that on a neighborhood U of x (cid:63) , there is a c ∈ R > such that f satisfies ∀ x , y ∈ U , (cid:107) f ( x ) − f ( y ) − f (cid:48) ( x (cid:63) ) · ( x − y )(cid:107) ≤ c (cid:107) x − y (cid:107) . (21) Then there are η , δ and Γ in R > such that if x ∈ B ( x (cid:63) , η ) and B ∈ B ( f (cid:48) ( x (cid:63) ) , δ ) , then for any w ∈ Z ≥ , (cid:107) x w + m − x (cid:63) (cid:107) ≤ Γ (cid:107) x w − x (cid:63) (cid:107) . Proof.
Step 1: Preliminaries.
Condition (21) is stronger thanstrict differentiability as stated in Theorem 4.2. From its proofand Lemma 4.5, let r ∈ ( , ) and γ ≥ (cid:107) f (cid:48) ( x (cid:63) ) − (cid:107) , as well as η and δ such that: r ≤ (cid:16) γ (cid:107) f (cid:48) ( x (cid:63) )(cid:107) (cid:17) − , and if x ∈ B ( x (cid:63) , η ) and B ∈ B ( f (cid:48) ( x (cid:63) ) , δ ) , the sequences ( x n ) n ∈ N and ( B n ) n ∈ N defined byBroyden’s method (using (13)) are well defined and moreover thefour following inequalities are satisfied: for any k ∈ N , (cid:107) B k − f (cid:48) ( x (cid:63) )(cid:107) ≤ δ , (cid:107) x k + − x (cid:63) (cid:107) ≤ r (cid:107) x k − x (cid:63) (cid:107) , (cid:107) B − k (cid:107) ≤ ( + r ) γ , (cid:107) f ( x k + )(cid:107) ≤ (cid:107) f ( x k )(cid:107) . Let x ∈ B ( x (cid:63) , η ) , B ∈ B ( f (cid:48) ( x (cid:63) ) , δ ) , and ( x n ) n ∈ N and ( B n ) n ∈ N be defined by Broyden’s method. Let w ∈ N and h = (cid:107) x w − x (cid:63) (cid:107) . We This condition is satisfied by polynomials or converging power series. n A Non-Archimedean Broyden Method ISSAC’20, July 2020, Kalamata, Greece must show that there is a Γ , independent of w such that (cid:107) x w + m − x (cid:63) (cid:107) ≤ Γ h . Step 2: reference to a linear map.
Let the linear affine mapˆ f ( x ) = f (cid:48) ( x (cid:63) ) (cid:0) x − x (cid:63) (cid:1) , and ˆ x = x w and ˆ B = B w . Broyden’smethod (using first (13)) applied to those data produces the se-quences ( ˆ x n ) n ∈ N and ( ˆ B n ) n ∈ N , which are constant for n ≥ m , asa result of Theorem 4.2. We define similarly ˆ s n = ˆ x n + − ˆ x n . Wehave again for all k ∈ N the four inequalities: (cid:107) ˆ B k − f (cid:48) ( x (cid:63) )(cid:107) ≤ δ , (cid:107) ˆ x k + − x (cid:63) (cid:107) ≤ r (cid:107) ˆ x k − x (cid:63) (cid:107) , (cid:107) ˆ B − k (cid:107) ≤ ( + r ) γ (cid:107) ˆ f ( x k + )(cid:107) ≤ (cid:107) ˆ f ( x k )(cid:107) . The key to the proof is that ˆ x m = x (cid:63) and ˆ x k and x w + k are not toomuch far apart. Step 3: Statement of the induction.
More concretely, we prove byinduction on j that there exist γ , j and γ , j , independent of w , suchthat for 0 ≤ j ≤ m , we have the two inequalities: (cid:107) B w + j − ˆ B j (cid:107) · (cid:107) f w + j (cid:107) ≤ γ , j h , ( E , j ) (cid:107) x w + j − ˆ x j (cid:107) ≤ γ , j h . ( E , j ) Step 4: Base case.
Since B w = ˆ B and x w = ˆ x , ( E , ) and ( E , ) are clear, with γ , = γ , = . Now, let us assume that ( E , k ) and ( E , k ) are true for a given k such that 0 ≤ k < m . Step 5:
We first prove ( E , k + ) . One part of the inequality (22) isobtained thanks to: B − w + k − ˆ B − k = B − w + k ( ˆ B k − B w + k ) ˆ B − k . (cid:107) s w + k − ˆ s k (cid:107) = (cid:107) B − w + k f w + k − ˆ B − k ˆ f ( ˆ x k )(cid:107)≤ max (cid:16) (cid:107) B − w + k (cid:107) · (cid:107) ˆ B − k (cid:107) · (cid:107) B w + k − ˆ B k (cid:107) · (cid:107) f w + k (cid:107) , (22) (cid:107) ˆ B − k (cid:107) · (cid:107) f w + k − ˆ f ( ˆ x k )(cid:107) (cid:17) ≤ (cid:107) ˆ B − k (cid:107) max (cid:16) (cid:107) B − w + k (cid:107) · (cid:107) B w + k − ˆ B k (cid:107) · (cid:107) f w + k (cid:107) , (cid:107) f w + k − ˆ f ( x w + k )(cid:107) , (cid:107) ˆ f ( x w + k ) − ˆ f ( ˆ x k )(cid:107) (cid:17) (23)The first term on the r.h.s. of (23) is upper-bounded by ( + r ) γ γ , k h using ( E , k ) and (cid:107) B − w + k (cid:107) ≤ ( + r ) γ .For the second term of (23), using (21): (cid:107) f w + k − f ( x (cid:63) ) − f (cid:48) ( x (cid:63) ) · ( x w + k − x (cid:63) )(cid:107) ≤ c (cid:107) x w + k − x (cid:63) (cid:107) and (cid:107) x w + k − x (cid:63) (cid:107) ≤ (cid:107) x w − x (cid:63) (cid:107) = h , it is upper-bounded by c h . Finally, the last term is equal to f (cid:48) ( x (cid:63) )( x w + k − ˆ x k ) whose norm isupper-bounded by (cid:107) f (cid:48) ( x (cid:63) )(cid:107) γ , k h thanks to ( E , k ) . This is enoughto define γ , k such that (cid:107) s w + k − ˆ s k (cid:107) ≤ γ , k h (‡) . Consequently,with γ , k + = max ( γ , k , γ , k ) , we do have (cid:107) x w + k + − ˆ x k + (cid:107) ≤ γ , k + h , and ( E , k + ) is satisfied. Step 6.0:
We now prove ( E , k + ) . We first deal with some pre-liminary cases. If s w + k = , (that is x w + k + = x w + k ) then theproperty (2) s w + k = − B − w + k f w + k implies that f w + k =
0, and theproperty B w + k + s w + k = y w + k implies that f w + k = f w + k + = ( E , k + ) is satisfied with γ , k + = . If ˆ s k = , then similarlyˆ f ( ˆ x w + k ) = ˆ f ( ˆ x w + k + ) =
0. Therefore, as we have seen before, (cid:107) f w + k + (cid:107) = (cid:107) f w + k + − ˆ f ( x w + k + ) + ˆ f ( x w + k + ) − ˆ f ( ˆ x k + )(cid:107) , ≤ max (cid:0) c , (cid:107) f (cid:48) ( x (cid:63) )(cid:107) γ , k + (cid:1) h . Then, using that (cid:107) B w + k + − ˆ B k + (cid:107) ≤ max ((cid:107) B w + k + − f (cid:48) ( x (cid:63) )(cid:107) , (cid:107) ˆ B k + − f (cid:48) ( x (cid:63) )(cid:107)) ≤ δ , ( E , k + ) is satisfied with: γ , k + = δh max (cid:0) c , (cid:107) f (cid:48) ( x (cid:63) )(cid:107) γ , k + (cid:1) . Step 6.1 :
We can now assume that both s k and ˆ s k are non zero.To prove that there is a γ , k + (independent of w ) such that ( E , k + ) holds, then in view of the fact that (cid:107) f w + k + (cid:107) ≤ (cid:107) f w + k (cid:107) (Lemma 4.5)of ( E , k ) and of the definition (Eq. (3)) of B k + and ˆ B k + , it is enoughto prove that there is some γ , k + (independent of w ) such that: (cid:107) ( y w + k − B w + k s w + k ) u w + kt − (cid:16) ˆ y k − ˆ B k ˆ s k (cid:17) ˆ u kt (cid:107) · (cid:107) f w + k + (cid:107) ≤ γ , k + h . (24)Using that (cid:107) f w + k + (cid:107) ≤ (cid:107) f w + k (cid:107) (by Lemma 4.5), we obtain: (cid:107) f w + k + (cid:107) · (cid:107) ( y w + k − B w + k s w + k ) u w + kt − (cid:16) ˆ y k − ˆ B k ˆ s k (cid:17) ˆ u kt (cid:107)≤(cid:107) f w + k (cid:107) max (cid:0) (cid:107) y w + k − f (cid:48) ( x (cid:63) ) s w + k (cid:107) · (cid:107) u w + kt (cid:107) , (cid:107)( f (cid:48) ( x (cid:63) ) − B w + k ) s w + k u w + kt − ( f (cid:48) ( x (cid:63) ) − ˆ B k ) ˆ s k ˆ u kt (cid:107) (cid:17) ≤(cid:107) f w + k (cid:107) max (cid:0) (cid:107) y w + k − f (cid:48) ( x (cid:63) ) s w + k (cid:107) · (cid:107) u w + kt (cid:107) , (25) (cid:107)( f (cid:48) ( x (cid:63) ) − ˆ B k )( s w + k u w + kt − ˆ s k ˆ u kt )(cid:107) , (26) (cid:107)( B w + k − ˆ B k ) s w + k u w + kt (cid:107) (cid:17) . (27) Step 6.2:
From f w + k = − B w + k s w + k , we have (cid:107) f w + k (cid:107) ≤ (cid:107) s w + k (cid:107) · max ((cid:107) B w + k − f (cid:48) ( x (cid:63) )(cid:107) , (cid:107) f (cid:48) ( x (cid:63) )(cid:107)) ≤ (cid:107) s w + k (cid:107) · max ( δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107)) (•) .Otoh by (21), (cid:107) y w + k − f (cid:48) ( x (cid:63) ) s w + k (cid:107) ≤ c (cid:107) s w + k (cid:107) . It follows thatthe first term (25) can be upper-bounded in the following way:(25) ≤ c (cid:107) s w + k (cid:107) (cid:107) u w + kt (cid:107) max ( δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107)) ≤ c h max ( δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107)) , the rightmost inequality being obtained from (cid:107) u w + kt (cid:107) = (cid:107) s w + k (cid:107) − and (cid:107) s w + k (cid:107) ≤ max ((cid:107) x w + k + − x (cid:63) (cid:107) , (cid:107) x w + k − x (cid:63) (cid:107)) = (cid:107) x w + k − x (cid:63) (cid:107) ≤ (cid:107) x w − x (cid:63) (cid:107) = h . Step 6.3:
The third one (27) can be upper-bounded using ( E , k ) :(27) ≤ (cid:107) f w + k (cid:107)(cid:107)( B w + k − ˆ B k ) s w + k u w + kt (cid:107) ≤ γ , k h . Step 6.4:
For the second one (26), observe that: s w + k u w + kt − ˆ s k ˆ u kt = ( s w + k − ˆ s k ) u w + kt − ˆ s k ( u w + kt − ˆ u kt ) . (28)The first term is easy to manage using the previous inequality (•) on (cid:107) f w + k (cid:107) , the inequality (‡) on (cid:107) s w + k − ˆ s k (cid:107) and (cid:107) s w + k (cid:107)(cid:107) u w + kt (cid:107) = (cid:107) f w + k (cid:107) · (cid:107)( s w + k − ˆ s k ) u w + kt (cid:107) ≤ max ( δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107)) γ , k h . (29)The second one of Eq. (28) is a little bit trickier. Define as in (13), u w + k = s − w + k , l e l and ˆ u k = ˆ s − k , ˆ l e ˆ l for some given l and ˆ l . If l = ˆ l , we have: (the last inequality below follows from (‡) ). (cid:107) u w + k − ˆ u k (cid:107) = | s − w + k , l − ˆ s − k , l | = | s w + k , l − ˆ s k , l || s w + k , l | · | ˆ s k , l | = | s w + k , l − ˆ s k , l |(cid:107) s w + k (cid:107) · (cid:107) ˆ s k (cid:107)≤ (cid:107) s w + k − ˆ s k (cid:107)(cid:107) s w + k (cid:107) · (cid:107) ˆ s k (cid:107) ≤ γ , k h (cid:107) s w + k (cid:107) · (cid:107) ˆ s k (cid:107) . From this and from (cid:107) f w + k (cid:107) = (cid:107) B w + k (cid:107) · (cid:107) s w + k (cid:107) we get: (cid:107) f w + k (cid:107) · (cid:107) u w + k − ˆ u k (cid:107) · (cid:107) ˆ s k (cid:107) ≤ γ , k max (cid:0) δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107) (cid:1) h . (30)If l (cid:44) ˆ l , then either (cid:107) s w + k − ˆ s k (cid:107) = (cid:107) s w + k (cid:107) , if (cid:107) ˆ s k (cid:107) ≤ (cid:107) s w + k (cid:107) , or (cid:107) s w + k − ˆ s k (cid:107) = (cid:107) ˆ s k (cid:107) , if (cid:107) s w + k (cid:107) ≤ (cid:107) ˆ s k (cid:107) . In the first case, we have (cid:107) u w + k − ˆ u k (cid:107) = (cid:107) ˆ s k (cid:107) − , SSAC’20, July 2020, Kalamata, Greece Xavier Dahan and Tristan Vaccon and then, the second term of (28) multiplied by (cid:107) f w + k (cid:107) verifies: (cid:107) f w + k (cid:107) · (cid:107) u w + k − ˆ u k (cid:107) · (cid:107) ˆ s k (cid:107) ≤ max (cid:0) δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107) (cid:1) (cid:107) s w + k (cid:107)≤ max (cid:0) δ , (cid:107) f (cid:48) ( x (cid:63) )(cid:107) (cid:1) γ , k h . (31)The second case follows with the same computation. Eqs (31) (30) (29)prove together the bound on the expression (26) in (28). In turn withthe bounds on the terms (25) and (27), prove (24). This concludesthe proof of ( E , k + ) , and finally the induction. Step 7:
Consequently, (cid:107) x w + m − ˆ x m (cid:107) ≤ γ , m h . Thanks toTheorem 4.2, ˆ x m = x (cid:63) , and thus, we have proved that for any w , (cid:107) x w + m − x (cid:63) (cid:107) ≤ γ , m (cid:107) x w − x (cid:63) (cid:107) . (cid:3) Theorem 4.7 has for immediate consequence:Theorem 4.8.
Broyden’s method has locally R-order of conver-gence m . Proof. Let us take x and B as in the proof of the previoustheorem, and same constants and notations. For any w , (cid:107) x w + m − x (cid:63) (cid:107) ≤ Γ (cid:107) x w − x (cid:63) (cid:107) . Consequently, for 0 ≤ k < m , l ∈ N , and µ = / m , (cid:107) x lm + k − x (cid:63) (cid:107) µ − lm − k ≤ (cid:107) x k − x (cid:63) (cid:107) l µ − lm − k Γ ( l − ) µ − lm − k ≤ (cid:107) x k − x (cid:63) (cid:107) l − l − k m Γ ( l − ) − l − k m ≤ (cid:107) x k − x (cid:63) (cid:107) − k m Γ ( − − l ) − k m . For simplicity, we can assume that Γ ≥ . Thus, (cid:107) x lm + k − x (cid:63) (cid:107) µ − lm − k ≤ (cid:107) x k − x (cid:63) (cid:107) − k m Γ − k m . ≤ (cid:107) x − x (cid:63) (cid:107) − k m Γ − k m . Therefore, for (cid:107) x − x (cid:63) (cid:107) small enough, we get that for all k such that 0 ≤ k < m , (cid:107) x − x (cid:63) (cid:107) − k m Γ − k m < , and hence,lim sup s (cid:107) x s − x (cid:63) (cid:107) µ s < . From 9.2.7 of [20], we then obtain thatBroyden’s method do have locally R-order of convergence 2 m . (cid:3) A Q-order of µ implies an R-order of µ . The converse is not true.Over R , one of the most important result concerning Broyden’smethod is that it is Q-superlinear. The extension of this result tothe non-archimedean case remains an open question. : secant method In dimension one, Broyden’s method reduces to the secant method.It is known since [1] that the p -adic secant method applied onpolynomials has order Φ , the golden ratio. Its generalization to ageneral non-archimedean context is straightforward.Proposition 5.1. Let us assume that m = and on a neighborhood U of x (cid:63) , there is a c ∈ R > such that f satisfies (21) on U . Then thesecant method has locally Q-order of convergence Φ . Proof. Let us assume that we are in the same context as in theproof of Theorem 4.7, with some Q-linear convergence of ratio r < . Let us define ε k = x k − x (cid:63) for k ∈ N . For all k ∈ N , | ε k + | < | ε k | . Then by ultrametricity, | x k + − x k | = | ε k | . Also, we further assume that c | ε | < | f (cid:48) ( x (cid:63) )| so that for all k ∈ N , | f (cid:48) ( x (cid:63) ) × ( x k + − x k )| > c |( x k + − x k )| , which also implies byultrametricity and (21) that for all k ∈ N , | f ( x k + ) − f ( x k )| = | f (cid:48) ( x (cid:63) ) × ( x k + − x k )| . Similarly, | f ( x k )| = | f (cid:48) ( x (cid:63) )|| ε k | . Now, let n ∈ Z > . Broyden’s iteration is given by: x n + = x n − x n − x n − f ( x n ) − f ( x n − ) . It rewrites as: | ε n + | = | ε n − ε n f ( x n ) − ε n − f ( x n ) f ( x n ) − f ( x n − ) | = | ε n − f ( x n ) − ε n f ( x n − ) f ( x n ) − f ( x n − ) |≤ c max (cid:0) | ε n − || ε n | , | ε n − | | ε n | (cid:1) | f ( x n ) − f ( x n − )| ≤ c | f (cid:48) ( x (cid:63) )| | ε n || ε n − | . Let us write C = c | f (cid:48) ( x (cid:63) )| and v n = Cε n . Then, v n + ≤ v n v n − forany n > v n + v Φ n ≤ v − Φ n v n − ≤ (cid:32) v n v Φ n − (cid:33) − Φ , as Φ = Φ + . If we define ( Y n ) n ∈ Z ≥ by Y = v v Φ and Y n + = Y − Φ n , then v n + v Φ n ≤ Y n . Since | − Φ | < , then Y n converges to 1 . Therefore,it is bounded by some D ∈ R + , and v n + v Φ n ≤ D for all n ∈ Z ≥ . Thisconcludes the proof. (cid:3)
Over R , Broyden’s method is known to converge Q-superlinearly.The key point is that for any E ∈ M m ( R ) and s ∈ R m \ { } , (cid:107) E (cid:18) I − s · s t ( s t · s ) (cid:19) (cid:107) F = (cid:107) E (cid:107) F − (cid:18) (cid:107) Es (cid:107) (cid:107) s (cid:107) (cid:19) , (32)equation ( . ) of [10]. The minus sign is a blessing as it allows theappearance of a telescopic sum which plays a key role in provingthat (cid:107) x n + − x (cid:63) (cid:107)(cid:107) x n − x (cid:63) (cid:107) converges to zero. Unfortunately, there does notseem to be a non-archimedean analogue to this equality. Thanks toTheorem 4.7, we nevertheless believe in the following conjecture.Conjecture 5.2. In the same setting as Theorem 4.7, Broyden’smethod has locally Q-superlinear convergence.
One remarkable feature of Newton’s method in an ultrametriccontext is the way it can handle precision. For example, if π isa uniformizer, if we assume that (cid:107) f (cid:48) ( x (cid:63) ) − (cid:107) = , x n known atprecision O ( π n ) is enough to obtain x n + at precision O ( π n + ) . To that intent, it thus suffices to double the precision at each newiteration. Hence the working precision of Newton’s method can betaken to grow at the same rate as the rate of convergence.The handling of precision is more subtle in Broyden. This ishowever crucial to design efficient implementations. Note that inthe real numerical setting, most works using Broyden’s methodsare employing fixed finite precision arithmetic, and do not address n A Non-Archimedean Broyden Method ISSAC’20, July 2020, Kalamata, Greece precision. Additionally, the lack of a knowledge of a precise expo-nent of convergence requires special care, and the presence of adivision also complicates the matter. We explain hereafter how tocope with those issues.For simplicity, we will make the following hypotheses through-out this section, which correspond to the standard ones in theNewton-Hensel method. They are that the starting x and B arein a basin of convergence at least linear. This allows us to replaceany encountered x n by its lift ˜ x n to a higher precision (and samefor B n ). Indeed, ˜ x n will still be in the basin of convergence andthen follows the same convergence property. These liftings allowto mitigate the fact that some divisions are reducing the amount ofprecision so that only arbitrary added digits are destroyed by thedivisions. Assumption 6.1.
We assume that x and x (cid:63) are in O K , andthat (cid:107) f (cid:48) ( x (cid:63) )(cid:107) = (cid:107) f (cid:48) ( x (cid:63) ) − (cid:107) = (cid:107) B (cid:107) = (cid:107) B − (cid:107) = . We also as-sume that some ρ ≤ and ρ ≤ are given such that B ( x (cid:63) , ρ ) × B ( f (cid:48) ( x (cid:63) ) , ρ ) , is a basin of convergence at least linear and for any x ∈ B ( x (cid:63) , ρ ) , and ρ ≤ ρ , f ( x + B ( , ρ )) = f ( x ) + f (cid:48) ( x (cid:63) ) · B ( , ρ ) (see the Precision Lemma 3.16 of [9]) The assumption on B and f (cid:48) ( x (cid:63) ) states that they are unimodu-lar, which is the best one can assume regarding to conditioning andprecision. Indeed if M ∈ GL m ( K ) is unimodular ( (cid:107) M (cid:107) = (cid:107) M − (cid:107) = x ∈ K m , (cid:107) Mx (cid:107) = (cid:107) x (cid:107) . Over Q p , M ∈ M m ( Z p ) is unimodular if and only if its reduction in M m ( Z / p Z ) is invert-ible (and idem for Q (cid:74) T (cid:75) and Q ). The last assumption is there toprovide the precision on the evaluations f ( x k ) ’s. It is satisfied if f ∈ O K [ X , . . . , X m ] . Precision and complexity settings.
Let M ( N ) be a superadditiveupper-bound on the arithmetic complexity over the residue field of O K for the computation of the product of two elements in O K atprecision O ( π N ) , and L be the size of a straight-line program thatcomputes the system f . One can take M ( N ) ∈ O ˜ ( N ) .Working over K with zealous arithmetic, the ultrametric coun-terpart of interval arithmetic [9, § 2.1], the interval of integers [[ a , b [[ indicates the coefficients of an element x ∈ K representedin the computer as x = (cid:205) b − i = a x i π i , with x i ∈ O K /(cid:104) π (cid:105) . In this wayval ( x ) = a , its absolute precision is abs ( x ) = b , and its relative preci-sion is rel ( x ) = b − a . We recall the usual precision formulae, andassume in the algorithm below that it is how the software manageszealous arithmetic (as in Magma, SageMath, Pari). See loc. cit. formore details. [[ a , b [[×[[ c , d [[ = [[ a + c , min ( a + d , b + c )[[[[ a , b [[/[[ c , d [[ = [[ a − c , min ( a + d − c , b − c )[[ (P)The cost of multiplying two elements of relative precision a and b is within M ( max ( a , b )) , and to divide one by the other is in4 M ( max ( a , b )) + max ( a , b ) [25, Thm 9.4].To perform changes in the precision, we use the same notationas Magma’s function for doing so. If x has interval [[ a , b [[ , the(destructive) procedure “ChangePrec(~ x , c )” either truncates x toabsolute precision c if c ≤ b , or lifts with zero coefficients 0 π b + · · · + π c − to fit the interval [[ a , c [[ , if c > b . The non-destructivecounterpart is denoted “ChangePrec( x , c )” without ~. This an example of an adaptive method, which can also be used in Newton’s methodwhen divisions occur.
We start from an initial approximation x at precision one, for ex-ample given by a modular method. The inverse of the Jacobian atprecision one provides B − . It yields a cost of O ( m ω ) , but the com-plexity analysis of Remark 6.4 shows that it is negligible. Obtainingthese data is not always obvious [12], but is the standard hypothesisin the context of modular methods. We write v k = val ( f k ) , In an ideal situation.
Assume an oracle provides the valuations v , v , v , . . . , v n , . . . (computed by a Broyden method at arbitrarilylarge precision). From this ideal situation, we derive the simple andcostless modifications required in reality. This analysis allows us toknow how efficient can a Broyden method be, which is noteworthyfor comparing it to Newton’s. The implementation of Iteration n ( n = B − n has interval [[ , v n [[ and is unimodular.(2) x n has interval [[ , v n + v n + [[ (non-zero entries in [[ , v n − + v n [[ ). (3) f n has interval [[ v n , v n + v n + [[ .Output: (i) B − n + with interval [[ , v n + [[ , (val ( det ( B − n )) = x n + in the interval [[ , v n + + v n + [[ (non-zero entries in [[ , v n + v n + [[ ). (iii) f n + in the interval [[ , v n + + v n + [[ .(1) ChangePrec(~ B − n , v n + ) ; [[ , v n + [[ (2) s n ← − B − n · f n ; m M ( v n + ) [[ v n , v n + v n + [[ (3) x n + ← x n + s n ; [[ , v n + v n + [[ (4) ChangePrec(~ x n + , v n + + v n + ) ; [[ , v n + + v n + [[ (5) f n + ← f ( x n + ) ; . L · M ( v n + + v n + ) [[ v n + , v n + + v n + [[ (6) f n + ← ChangePrec( f n + , v n + v n + ) ; [[ v n + , v n + + v n [[ (7) h n ← B − n · f n + ; m M ( v n + ) [[ v n + , v n + v n + [[ (8) u n ← Eq.(13) ; (negligible) [[− v n , v n + − v n [[ (9) r n ← u Tn · ChangePrec( B − n , v n ) ; m M ( v n + ) [[− v n , [[ (10) ChangePrec(~ f n + , v n ) ; [[ v n + , v n [[ (11) den ← + r n · f n + ; m M ( v n + ) [[ , v n [[ (12) Num ← h n · r n ; m M ( v n ) [[ v n + − v n , v n + [[ (13) N n ← Num / den ; 4 m M ( v n ) [[ v n + − v n , v n + [[ (14) B − n + ← B − n − N n ; [[ , v n + [[ (15) return B − n + , x n + , f n + We emphasize again that thanks to the careful changes of pre-cision undertaken, the precisions are automatically managed bythe software, would it have zealous arithmetic implemented. It isthen immediate to check that the output verifies the specifications.Moreover from the positive valuation of N n it is clear that B n + isunimodular. Thus Iteration n + Complexity of the ideal situation.
The arithmetic cost of Itera-tion n is within ( m + m ) M ( v n + ) + m M ( v n ) + L · M ( v n + + v n + ) .If we assume an exponent of convergence α > i.e. v n + ≈ αv n for “not too small” n , then the total cost to reach a precision N ≈ α (cid:96) + ≈ v (cid:96) + ( (cid:96) steps, including a 0-th one) is upper-bounded by ( m + ( m + m ) α + L ( + α ) α ) M ( N /( α − )) (33) In reality.
Using the same notations and inputs at Iteration n as in the ideal situation above, what changes in reality is thatwhile v n is known v n + and v n + are not, but are approximated by SSAC’20, July 2020, Kalamata, Greece Xavier Dahan and Tristan Vaccon αv n ≥ v n + and α v n ≥ v n + respectively, where α is fixed by theuser. Precisely, B − n and x n are known at the correct precision, but f n has an approximated interval [[ , v n + αv n [[ . To minimize theoverhead cost it induces compared to the ideal situation, once weknow v n + (Line 5) we insert some intermediate corrective stepsdenoted (5.1)-(5.5) thereafter, between Line (5) and Line (6); theyrequire no arithmetic operations.(5.1) ChangePrec(~ B − n , v n + )(5.2) ChangePrec(~ s n , v n + v n + )(5.3) Tune α if necessary using the new ratio v n + v n (5.4) ChangePrec(~ x n , v n + + αv n + )(5.5) ChangePrec(~ f n + , v n + + αv n + )Most importantly, the remaining Lines (6)-(15) are not impactedsince these computations involve now the known v n + (and notthe unknown v n + ): the intervals, and thus costs obtained arethe same as in the ideal situation. On the other hand, Lines (1)-(5) are performed as such with an overhead cost. Among them,only Lines (2), (5) have a non negligible cost. At Line (2), B − n hasapproximated interval [[ , αv n [[ , yielding a cost of m M ( αv n ) . AtLine (5) x n + has approximated interval [[ , v n ( α + α )[[ , yieldinga cost of L M ( v n ( α ( + α ))) . Thus the overhead cost “ovh n ” atIteration n is: m ( M ( αv n )− M ( v n + )) + L ( M ( v n α ( + α ))− M ( v n + + v n + )) (34)This quantity depends on the gaps αv n − v n + and α v n − v n + .These gaps increase with n , but, thanks to the tuning of Step ( . ) ,reasonably at a linear rate:Assumption 6.2. The “error gap” | αv n − v n + | = O ( n ) . Under this assumption it is easy to (crudely) bound (cid:205) (cid:96) + n = ovh n of Eq. (34) by ( L + m ) O ( N log ( N )) . Being independent on α thisis negligible in front of O ( L + m ) M ( Nα − ) for α <
2. The theorembelow wraps up the considerations made above with Eq. (33):Theorem 6.3.
If Broyden’s method has Q-order of convergence α on B ( x (cid:63) , ρ ) × B ( f (cid:48) ( x (cid:63) ) , ρ ) , then under Assumption 6.1 and 6.2, thecost of computing x (cid:63) + O ( π N ) is in O (cid:0) ( m + L ) (cid:1) M (cid:16) Nα − (cid:17) .Remark . Understanding the Q -order of convergence is a majorand notoriously difficult problem in the numerical analysis com-munity. Numerical evidence shows it deteriorates with m , and islarger than 2 / m (Theorems 4.7-4.8). Some experiments suggestthat taking α ≈ / m is not unreasonable. We then get a costin O (cid:16) ( m + L ) M (cid:16) Nα − (cid:17)(cid:17) ≈ O (cid:0) ( m + L ) M ( Nm ) (cid:1) . For comparison,denoting ω < O (( m ω + mL ) M ( N )) . Consequently, in this setting, for large m ,there is little hope that Broyden’s method can outperform New-ton’s when both are available. Remember though other worthwileapplications in the paragraph “Motivations” in Introduction. An implementation of our ultrametric Broyden method in Magma[4] with more data is available at http://xdahan.sakura.ne.jp/broyden20.html. We report the data obtained using the three families of sys-tems, derived from page 36 of [18]. The families are indexed by t ∈ π O K : • F = (cid:0) ( x − ) + ( x − ) − − tx x − t x , ( x + ) + ( x + ) − − tx (cid:1) in K [ x , x ] . • F = (cid:0) ( x − ) + ( x − ) + ( x − ) − − t − t , ( x + ) + ( x + ) + ( x + ) − − t , x + x + x − − t (cid:1) in K [ x , x , x ] . • F = (cid:0) ( x − ) + ( x − ) + ( x − ) + ( x − ) − − t − t , ( x + ) + ( x + ) + ( x + ) + ( x + ) − − t , x + x + x + x − − t , x x + x x − x x + x x + − t (cid:1) in K [ x , x , x , x ] . Valuation of f ( x k ) and numerical estimation of the order of Q-convergence for Q (cid:74) T (cid:75) are compiled in the following graphic. For K = Q p , and F p (cid:74) t (cid:75) with p =
17 we experienced the same behaviour.
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