Effective analytic functions Joris van der Hoeven D pt. de Math matiques (b t. 425) Universit Paris-Sud Orsay Cedex France

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1 Effective analytic functions Joris van der Hoeven D pt. de Math matiques (b t. 425) Universit Paris-Sud 945 Orsay Cedex France vdhoeven@texmacs.org August, 23 One approach for computations with special functions in computer algebra is the systematic use of analytic functions whenever possible. This naturally leads to problems of how to answer questions about analytic functions in a fully eective way. Such questions comprise the determination of the radius of convergence or the evaluation of the analytic continuation of the function at the endpoint of a broken like path. In this paper, we propose a rst denition for the notion of an eective analytic function and we show how to eectively solve several types of dierential equations in this context. We will limit ourselves to functions in one variable. Keywords: Complex analysis, computer algebra, algorithm, majorant method. A.M.S. subject classification: 3-4, 34M99.. Introduction An important problem in computer algebra is how to compute with special functions which are more complicated than polynomials. A systematic approach to this problem is to recognize that most interesting special functions are analytic, so they are completely determined by their power series expansions at a point. Of course, the concept of special function is a bit vague. One may for instance study expressions which are built up from a nite number of classical special functions, like exp, log or erf. But one may also study larger classes of special functions, such as the solutions to systems of algebraic dierential equations with constant coecients. Let us assume that we have xed a class F of such special functions or expressions for the sequel of this introduction. In order to develop a satisfactory computational theory for the functions or expressions in F, one may distinguish the following main subproblems: How to test whether f 2 F locally represents the zero power series? How to evaluate f 2 F safely and eciently at any point where f may be dened? The rst subproblem is called the zero-test problem and it has been studied before in several works [Ris75, DL84, DL89, Kho9, Sha89, Sha93, PG97, SvdH, vdh2b]. For large classes of special functions, it turns out that the zero-test problem for power series can be reduced to the zero-test problem for constants (see also [vdhb] for a discussion of this latter problem).

2 2 Effective analytic functions In this paper, we will focus on the second subproblem, while leaving aside the eciency considerations and restricting our attention to functions in one variable. By safe evaluation of f 2 F at z, we mean that we want an algorithm which computes an approximation f ~ 2 ( + i ) 2 for f (z) with jf ~? f (z)j < " for any " 2 2. If such an algorithm exists, then we say that f (z) is an eective complex number. By any point where the expression may be dened, we mean that we do not merely plan to study f near the point where a power series expansion was given, but that we also want to study all possible analytic continuations of f. In other words, the aim of this paper is to develop an eective complex analysis for computations with analytic functions in a class F which we wish to be as large as possible. Such computations mainly consist of safe evaluations, bound computations for convergence radii and absolute values of functions on closed disks, and analytic continuation. Part of the philosophy behind this paper also occurs in [BB85, CC9], but without the joint emphasis on eectiveness at all stages and usefulness from the implementation point of view. In previous papers [vdh99, vdha], we have also studied in detail the fast and safe evaluation of holonomic functions. When studying solutions to non-linear dierential equations, one must carefully avoid undecidable problems: Theorem. [DL89] Given a power series f = P f n z n with rational coecients, which is the unique solution of an algebraic dierential equation P (z; f ; ; f (l) ) = ; with rational coecients and rational initial conditions, one cannot in general decide whether the radius of convergence (f ) of f is < or >. What we will show in this paper, is that whenever we know that an analytic function f = P f n z n as in the above theorem may be continued analytically along an eective broken line path, then the value f () of f at the endpoint of is eective (theorem ). We will also show that we may eectively solve any monic linear dierential equation, without introducing singularities which were not already present in the coecients (theorem ). In order to prove these results, we will carefully introduce the concept of an eective analytic function in section 2. The idea is that such a function f is given locally at the origin and that we require algorithms for Computing coecients of the power series expansion; Computing a lower bound for the radius of convergence; Computing an upper bound for jf j on any closed disk of radius <. Analytic continuation. But we will also require additional conditions, which will ensure that the computed bounds are good enough from a more global point of view. In section 3, we will show that all analytic functions, which are constructed from the eective complex numbers and the identity function using +;?; ; /; d dz ;R ;exp and log, are eective. In section 4, we will study the resolution of dierential equations. It is convenient to specify the actual algorithms for computations with eective analytic functions in an object oriented language with abstract data types (like C++). Eective types and functions will be indicated through the use of a sans serif font. We will not detail memory management issues and assume that the garbage collector takes care of this. The Columbus program is a concrete implementation of some of the ideas in this paper [vdh2a], although this program works with double precision instead of eective complex numbers.

3 Joris van der Hoeven 3 2. Effective analytic functions 2.. Eective numbers and power series A complex number z 2 C is said to be eective, if there exists an algorithm which takes a positive number < " 2 2 on input and which returns an approximation z~ 2 ( + i ) 2 for z with jz~? zj < ". We will also denote the approximation by z~ = z <". In practice, we will represent z by an abstract data structure Complex with a method approximate which implements the above approximation algorithm. The asymptotic complexity of an eective complex number z 2 Complex is the asymptotic complexity of its approximation algorithm. We have a natural subtype Real Complex of eective real numbers. Recall, however, that there is no algorithm in order to decide whether an eective complex number is real. In the sequel, we will use the notations R > = fx 2 R: x > g and R > = fx 2 R: x > g. Let R be a weakly eective ring, in the sense that all elements of R can be represented by explicit data structures and that we have algorithms for the ring operations ; ; +;? and. If we also have an eective zero test, then we say that R is an eective ring. An eective series over R is a series f 2 R[[z]], such that there exists an algorithm in order to compute the n-th coecient of f. Eective series over R are represented by instances of the abstract data type Series(R), which has a method expand: N! R[z], which computes the truncation f + + f n? z n? 2 R[z] of f at order n as a function of n 2 N. The asymptotic complexity of f is the asymptotic complexity of this expansion algorithm. In particular, we have an algorithm to compute the n-th coecient f n of an eective series. A survey of ecient methods to compute with eective series can be found in [vdh2c]. When f is an eective series in Series(Complex), then we denote by f 2 C[[z]] the actual series which is represented by f. If f is the germ of analytic function, then we will denote by (f ) the radius of convergence of f and by jf j r the maximum of jf j on the closed disk B r = fz 2 C: jzj 6 rg of radius r, for each r < (f ) Eective germs An eective germ of an analytic function f at is an abstract data structure Germ which inherits from Series(Complex) and with the following additional methods: A method radius: ()! Real [ f + g which returns a lower bound (f ) > for (f ). This bound is called the eective radius of convergence of f. A method norm: Real! Real which, given < r < (f ), returns an upper bound jf j r for jf j r. We assume that jf j r is increasing in r. Remark 2. For the sake of simplicity, we have reused the notations (f ) and jf j r in order to denote the applications of the methods radius and norm to f. Clearly, one should carefully distinguish between (f ) resp. jf j r and (f ) resp. jf j r. The n-th coecient of f will always be denoted by f n = f n. Given an eective germ f, we may implement a method evaluate, which takes an eective complex number z 2 Complex with jzj < (f ) on input, and which computes its eective value f (z) 2 Complex at z. For this, we have to show how to compute arbitrarily good approximations for f (z): Algorithm evaluate-approx(f ; z; ") Input: f 2 Germ and z 2 Complex with jzj < (f ), and " 2 2 Output: an approximation f ~ for f (z) with jf ~? f (z)j < "

4 4 Effective analytic functions Step. [Compute expansion order] Let r = ((f ) + jzj)/2 and M = jf j r Let n 2 N be smallest such that M r r? jzj n jzj < " r 2 Step 2. [Approximate the series expansion] Compute f^ = f + f z + + f n? z n? 2 Complex Return f^<"/2 The correctness of this algorithm follows from Cauchy's formula: jf^? f (z)j = zn 2 p i 6 jzjn 2 p I I = M r r? jzj jwj=r f (w) (w? z) w n dw M jwj=r (r? jzj) r n dw n jzj r < " Eective paths Any (l + )-tuple (z ; ; z l ) of complex numbers determines a unique ane broken line path or ane path in C, which is denoted by z! z!! z l?! z l. If z =, then we call = z! z!! z l?! z l a broken line path or a path of length l = l. We denote by P the space of all paths and by P a the space of all ane paths. The trivial path of length is denoted by. An ane path z!! z l is said to be eective if z ; ; z l 2 Complex. We denote by AnePath the type of all eective ane paths and by Path the type of all eective paths. Another notation for the path = z! z!! z l?! z l is [z? z ; ; z l? z l? ]; the rst notation is called the usual notation; the second one is called the incremental notation. Let = [ ; ; l ] =! z!! z l?! z l and = [ ; ; l ] =! z!! z l?! z l be two paths in P. Then we dene their concatenation + by + = [ ; ; l ; ; ; l ] =! z!! z l! z l + z!! z l + z l : We also dene the reversion? of by The norm of is dened by? = [? l ; ;? ] =! z l?? z l!! z? z l!? z l : jj = j j + + j l j: Notice that j?j = jj and j + j = jj + j j. We say that is a truncation of if l 6 l and i = i for i = ; ; l. In this case, we write P and? = [ l+ ; ; l ], so that (? ) + = (however, we do not always have? = + (?)). The longest common truncation of two general paths and always exists and we denote it by M. If we restrict our attention to paths z!! z l such that z ; ; z l are in a subeld of Complex with an eective zero-test (like Q[i]), then P and M are clearly eective.

5 Joris van der Hoeven 5 A subdivision of a path = [ ; ; l ] is a path of the form = [ ; ; ; ;k ; ; l; l ; ; l;kl l ], where i;j 2 (; ) with i; + + i;ki = for all i. If is a subdivision of, then we write v. Given ; and with v and v, there exists a shortest path with v and v. We call the shortest common subdivision of and and we denote it by = t. Given an analytic function f at the origin, which can be continued analytically along a path 2 P, we will denote by f () the value of f at the endpoint of the path and by f + the analytic function at zero, such that f + (") = f ( + ") 3 f ( + ["]) for all suciently small ". If = [ ; ; l ], then we will also write f + = f + ; ;+ l. The domain Dom f of f is the set of all paths along which f can be continued analytically Eective analytic functions A quasi-eective analytic function is an instance f of the abstract data type AnFunc, which inherits from Germ, and with the following additional method: continue: Complex! AnFunc computes f +z for all z with jzj < (f ). We will denote by f the analytic function which is represented by f. The domain Dom f Dom f of f is the set of all paths [ ; ; l ] 2 Path with j i j < (f + ; ;+ i? ) for all i 2 f; ; ng. The functions evaluate and continue may be extended to the class Path, for all paths which are in the domain of f. Remark 3. Notice that, similarly as in remark 2, we have reused the notation f +z in order to denote the application of the method f! continue to z. So f +z is again a quasi-eective analytic function, which should be distinguished from f +z = f +z. Given = [ ; ; l ], we will also denote f () = f + ; ;+ l? ( l ) and f + = f + ; ;+ l. Let f and g be two quasi-eective analytic functions. We will write f = g as soon as f = g. We say that f and g are strongly equal, and we write f g, if Dom f = Dom g and (f + ) = (g + ) and jf + j r = jg + j r for all 2 Dom f and r < (f + ). We say that g is better than f, if Dom f Dom g, and (f + ) 6 (g + ) and jf + j r > jg + j r for all 2 Dom f and r < (f + ). We say that a quasi-eective analytic function f satises the homotopy condition, if EA. If ; + [ ]; + [ ; 2 ]; + [ + 2 ] 2 Dom f, then f +;+( + 2 ) f +;+ ;+ 2. Here we understand that f def +;+( + 2 ) = f + if + 2 =. Let f be a quasi-eective analytic function and consider the functions (f + ) and (; r) jf + j r. We say that f satises the local continuity condition, if there exist continuous functions R: B (f ) = fz 2 C: jzj < (f )g! R > ; N : f(; r) 2 B (f ) R > : r < R()g! R > ; such that (f + ) = R() and jf + j r = N (; r) for all 2 B (f ) \ Complex and r 2 Real with 6 r < (f + ). We say that f satises the continuity condition, if EA2. f + satises the local continuity condition for each 2 Dom f. We say that f is an eective analytic function if it both satises the homotopy condition and the continuity condition. In what follows, we will always assume that instances of the type AnFunc satisfy the conditions EA and EA2.

6 6 Effective analytic functions Remark 4. In fact, there are several alternatives for the denition of eective analytic functions, by changing the homotopy and continuity conditions. The future will learn us which conditions are best suited for complex computations. Nevertheless, there is no doubt that the spirit of the denition should be preserved: quasi-eectiveness plus additional conditions which will allow us to prove global properties Analytic continuation along subdivided paths The extended domain Dom ] f of an eective analytic function f is the set of all paths, such that there exists a subdivision of with 2 Dom f. For a tuple f = (f ; ; f n ) of eective analytic functions, we also dene Dom f = Dom f \ \ Dom f n and similarly for Dom ] f and Dom f. We say that an eective analytic function f (or a tuple of such functions) is faithful, if Dom ] f =Dom ] f \ Path. We say that a subset P of Path is eective, if there exists an algorithm to decide whether a given eective path belongs to Path. Now let us choose constants ; 2 (; ) \ Real with < and consider an eective analytic function f. Then the following algorithm may be used in order to evaluate f at any path 2 Dom ] f: Algorithm evaluate-subdiv(f ; ) Input: f 2 AnFunc and 2 Path, such that 2 Dom ] f Output: f () Step. [Handle trivial cases] If =, then return f () Write = [] + Compute an approximation ~ 2 2 of =? + 2 If ~ < then return evaluate-subdiv(f + ; ) Step 2. [Subdivide path] Let = j(f )j jj Return evaluate-subdiv(f + ; [? ] + ) (f ) with j~? j <? 2 (f ). We notice that ~ < implies < (f ) and ~ > implies > (f ). The correctness proof of this algorithm relies on three lemmas: Lemma 5. Let ; 2 2 Dom f with v 2. Then f + f +2. Proof. Let us proof the lemma by induction over the dierence d of the lengths of 2 and. If d =, then = 2 and we are done. Otherwise, let be longest, such that there exist paths and 2 and numbers ; 2 and 3 with = + [ ] + and 2 = 2 + [2 ; 3 ] +. If d >, then the induction hypothesis implies f +2 f +2 +[2 + 3 ]+ f + : Otherwise, we have = 2 and = Consequently, the homotopy condition implies that f + +[ ] f +2 +[2 ; 3 ], whence f + f +2. Remark 6. In fact, the above lemma even holds for homotopic paths ; 2 2 Dom f, but we will not need this in what follows. Lemma 7. If ; ; 2 2 Dom f are such that v and v 2, then t 2 2 Dom f.

7 Joris van der Hoeven 7 Proof. Let t 2 = [ ; ; l]. Let us prove by induction over i that i = [; ; i]2dom f. If i =, then we have nothing to do. Assume now that we have proved the assertion for a given i < l and let us prove it for i +. Since t 2 is the shortest common subdivision of and 2, there exist a k 2 f; 2g and a path C k, such that v i. By lemma 5, we have (f +i ) = (f + ). Therefore, j i+ j 6 j j < (f +i ), where is such that + [ ] P k. This shows that i+ 2 Dom f, as desired. Lemma 8. Let 2 Dom f. Then there exists a > such that for any 2 Dom f, with P for some w, we have (f + ) >. Proof. Write = [ ; ; l ] and let i 2 f; ; lg. The continuity condition implies that the function 2 [; i ] \ Complex (f + ; ;+ i? ;+) is the restriction of a continuous function R on the compact set [; i ]. Consequently, there exists a lower bound i > for R on [; i ]. On the other hand, any 2 Dom f, with P for some w, is a subdivision of a path of the form [ ; ; i? ; ] with 2 [; i ]. Taking = min f ; ; l g, we conclude by lemma 5. Proof of the algorithm. The algorithm is clearly correct if it terminates. Assume that the algorithm does not terminate for = [ ; ; l ] and let = [ ; ; l ] be a subdivision of in Dom f. Let ; 2 ; be the sequence of increments, such that evaluate-subdiv is called successively for f ; f + ; f + ;+ 2 ;. Let > be the constant we nd when applying lemma 8 to. Since jj, there exists an i with ji+ j <. Now let = [ ; ; i ] and let j be such that j[; ; j]j < j j and j[ ; ; j+ ]j > j j. Then = [ ; ; j; j i??? j] 2 Dom f. By lemma 7, it follows that t 2 Dom f. Hence (f + ) = (f +( t )) >. This yields the desired contradiction, since j i+ j = (f + ) >. 3. Operations on effective analytic functions In this section we will show how to eectively construct elementary analytic functions from the constants in Complex and the identity function z, using the operations +;?; ; /; d dz ;R ; exp and log. In our specications of the corresponding concrete data types which inherit from AnFunc, we will omit the algorithms for computing the coecients of the series expansions, and refer to [vdh2c] for a detailed treatment of this matter. 3.. Basic eective analytic functions Constant eective analytic functions are implemented by the following concrete type ConstantAnFunc which derives from AnFunc (this is reected through the B symbol below): Class ConstantAnFunc B AnFunc z 2 Complex new: z~ 2 Complex z 3 z~ radius: () norm: r jzj continue: ConstantAnFunc(z) The method new is the constructor for ConstantAnFunc. In the method continue it is shown how to call the constructor. In a similar way, the following data type implements the identity function:

8 8 Effective analytic functions Class IdentityAnFunc B AnFunc z 2 Complex new: () z 3 new: z~ 2 Complex z 3 z~ radius: () norm: r jzj + r continue: IdentityAnFunc(z + ) The default constructor with zero arguments returns the identity function centered at. The other constructor with one argument z returns the identity function centered at z. The conditions EA and EA2 are trivially satised by the constant functions and the identity function. They all have domain Path The ring operations The addition of eective analytic functions is implemented as follows: Class SumAnFunc B AnFunc f ; g 2 AnFunc new: (f ~ 2 AnFunc; g~ 2 AnFunc) f 3 f ~ ; g 3 g~ radius: () min ((f ); (g)) norm: r jf j r + jgj r continue: SumAnFunc(f + ; g + ) We clearly have Dom (f + g) = Dom f \ Dom g and (f + g) f + g for all paths in Dom (f + g). Consequently, condition EA is satised by f + g. Since min is a continuous function, condition EA2 is also satised. Subtraction is implemented in the same way as addition: only the series computation changes. Multiplication is implemented as follows: Class ProductAnFunc B AnFunc f ; g 2 AnFunc new: (f ~ 2 AnFunc; g~ 2 AnFunc) f 3 f ~ ; g 3 g~ radius: () min ((f ); (g)) norm: r jf j r jgj r continue: ProductAnFunc(f + ; g + ) We again have Dom (f g) = Dom f \ Dom g and the conditions EA and EA2 are veried in a similar way as in the case of addition Dierentiation and integration In order to dierentiate an eective analytic function f, we have to be able to bound f on each disk D r with r < (f ). Fixing a number 2 (; ) \ Real, this can be done as follows: Class DerAnFunc B AnFunc f 2 AnFunc

9 Joris van der Hoeven 9 new: f ~ 2 AnFunc f 3 f ~ radius: () (f ) norm: r s (s? r) 2 jf j s, where s 3 (f ) + ((f )? r) continue: DerAnFunc(f + ) Let us show that the bound for the norm is indeed correct. Given the bound jf j s for s on D s, Cauchy's formula implies that jf n j 6 jf j s /s n for all n. Consequently, for all z 2 B r: jf (z)j = X X (n + ) f n+ z n 6 n= n= (n + ) jf j s r n s n+ = s (s? r) 2 jf j s: We have Dom f = Dom f and the fact that is a constant, which is xed once and for all, ensures that condition EA2 is again satised by f. The actual choice of is a compromise between keeping jf j s as small as possible while keeping s? r as large as possible. Bounding the value of an integral on a disk is simpler, using the formula f (z) 6 jj max jf (z)j: z2[;d] For the analytic continuation of integrals, we have to keep track of the integration constant, which can be determined using the evaluation algorithm from section 2.2. In the algorithm below, this integration constant corresponds to c. Class IntAnFunc B AnFunc f 2 AnFunc c 2 Complex new: f ~ f 3 f ~, c 3 new: (f ~ 2 AnFunc; c~2 Complex) f 3 f ~ ; c 3 c~ radius: () (f ) norm: r jcj + r jf j r continue: IntAnFunc(f + ; this()) R z The domain of f (t) dt is the same as the domain of f Inversion In order to compute the inverse /f of an eective analytic function f with f (), we should in particular show how to compute a lower bound for the norm of the smallest zero. Moreover, this computation should be continuous as a function of the path. Again, let 2 (; ) \ Real be a parameter and consider Class InvAnFunc B AnFunc f 2 AnFunc new: f ~ 2 AnFunc f 3 f ~ jf ()j radius: () min (f ); jf j (f )

10 Effective analytic functions norm: r jf ()j? jf j r r continue: InvAnFunc(f + ) Again, the choice of is a compromise between keeping (f ) reasonably large, while keeping the bound jf j (f ) as small as possible. We have Dom ] f = f 2 Dom] f jf () g: Notice that we cannot necessarily test whether f () =. Consequently, Dom ] f necessarily eective. is not Remark 9. Instead of xing a 2 (; ) \ Real, it is also possible to compute such that (f ) = jf ()j jf ; j (f ) using a fast algorithm for nding zeros, like the secant method Exponentiation and logarithm The logarithm of an eective analytic function f can be computed using the formula log f = log f () + As to exponentiation, we use the following method: Class ExpAnFunc B AnFunc f 2 AnFunc new: f ~ 2 AnFunc f 3 f ~ radius: () (f ) norm: r exp jf j r continue: ExpAnFunc(f + ) z f (t) f (t) dt: We have Dom ] (log f ) = f 2 Dom ] f jf () g and Dom e f = Dom f. 4. Solving differential equations In this section, we will show how to eectively solve linear and algebraic dierential equations. As in the previous section, we will omit the algorithms for computing the series expansions and refer to [vdh2c]. We will use the classical majorant technique from Cauchy-Kovalevskaya in order to compute eective bounds. Given two power series f ; g 2 C[[z ; ; z n ]], we say that f is majored by g, and we write f P g, if g 2 R > [[z ; ; z n ]] and jf k ; ;k n j 6 g k ; ;k n for all k ; ; k n 2 N. If n = and f 2 AnFunc, then we write f P g if f P g. Given >, we also denote b = (? z)?. 4.. Linear dierential equations Let L ; ; L l? be eective analytic functions and consider the equation f (l) = L l? f (l?) + + L f ()

11 Joris van der Hoeven with initial conditions f () () = ; ; f (l?) () = l? in Complex. We will show that the unique solution to this equation can again be represented by an eective analytic function f, with Dom f = Dom L \ \ Dom L l?. Notice that any linear dierential equation of the form L l f (l) + + L f = g with L l () can be reduced to the above form (using division by L l, dierentiation, and linearity if g() = ). We rst notice that the coecients of f may be computed recursively using the equation f = + l? + L l? f (l?) + + L f : (2) Assume that M 2 R > and 2 R > are such that L i P M b for all i. Then the equation f^ = ^ + ^l? + M b f^(l?) + + M b f^ + R; (3) is a majorant equation [vk75, Car6] of (2) for any choices of ^; ; ^l? 2 R > and R 2 R > [[z]], such that j j 6 ^; ; j l? j 6 ^l?. Let h = b (M +)/ : We take ^i = C h (i) () for all i 2 f; ; l? g, where Now we observe that C = max fj j; ; j l? jg > max j j h () () ; ; j l? j h (l?) () M b h (l?) + + M b h P M h(m i (M +l+)/ (M ++)/ + ) (M + (l? 2) + ) b + + b P (M + ) (M + (l? ) + ) b (M +l+)/ = h (l) Therefore, we may take R = ^ + ^l? + : M b C h (l?) + + M b C h? C h 2 R > [[z]]: This choice ensures that (3) has the particularly simple solution C h. The majorant technique now implies that f P C h. From the algorithmic point of view, let = min f(l ); ; (L l? )g and assume that we want to compute a bound for jf j on B r for some r <. Let 2 (; ) \ Real be a xed constant. Then we may apply the above computation for = s? with s = r + (? r) and M = max fjl j s ; ; jl l? j s g. From the majoration f P C h, we deduce in particular that s (M +)s: jf j r 6 C s? r This leads to the following eective solution of (): Class LDEAnFunc B AnFunc L = (L ; ; L l? ) 2 Array(AnFunc) = ( ; ; l? ) 2 Array(Complex) new: (L ~ 2 Array(AnFunc); ~ 2 Array(Complex)) L 3 L ~, 3 ~

12 2 Effective analytic functions radius: () min ((L ); ; (L l? )) continue: LDEAnFunc((L ;+ ; ; L l?;+ ); (this(); ; this (l?) ()) Like in C++, the keyword this stands for the current instance of the data type, which is implicit to the method. The norm method is given by Method LDEAnFunc! norm(r) s 3 r + ((this)? r) C 3 max fj j; ; j l? jg M 3 max fjl j s ; ; jl l? j s g Return C s s? r (M +)s We have proved the following theorem: Theorem. Let L ; ; L l? be eective analytic functions and let ; ; l? 2 Complex. Then there exists a faithful eective analytic solution f to () with Dom f =Dom (L ; ;L l ). In particular, if Dom (L ; ; L l ) is eective, then so is Dom f Systems of algebraic dierential equations Let us now consider a system of algebraic dierential equations d f f l A P (f ; ; f l ) P l (f ; ; f l ) A (4) with initial conditions f () = ; ; f l () = l in Complex, where P ; ; P l are polynomials in l variables with coecients in Complex. Modulo the change of variables f i = i + f ~ i P i (f ; ; f l ) = P ~ i(f ~ ; ; f ~ l ) we obtain a new system d dz f~ C B A P~ (f ~ ; ; f ~ l ) f ~ l P ~ l(f ~ ; ; f ~ l ) C A; (5) with initial conditions f ~ () = = f~ l () =. Let M > and > be such that P ~ i(z ; ; z l ) P M b (z + + z l ) for all i. Then the system of dierential equations d dz f^ f^l C B A b (f^ + + f^l) b (f^ + + f^l) C A (6) with initial conditions f^ () = = f^l () = is a majorant system of (5). The unique solution of this system therefore satises f ~ i P f^i for all i. Now the f^i really all satisfy the same rst order equation f^i = M bl (f^i )

13 Joris van der Hoeven 3 with the same initial condition f^i() =. The unique solution of this equation is f^ = = f^l =? p? 2 l M z ; l which is a power series with radius of convergence = 2 l M and positive coecients, so that jf^i (z)j 6 f^i (r) for any r <. As to the implementation, we may x =. We will denote the transformed polynomials P ~ i by P ~ i = P i;+. We will also write kp k for the smallest number M 2 R > with P (z ; ; z l ) P M b (z + + z l ). The implementation uses a class ADESystem with information about the entire system of equations and solutions and a class ADEAnFunc for each of the actual solutions f i. Class ADESystem P = (P ; ; P l ) 2 Array(Polynomial(Complex; l)) = ( ; ; l ) 2 Array(Complex) M 2 Real new: (P ~ ; ~) P 3 P ~, 3 ~, M 3 max (kp ;+ k ; ; kp l;+ k ) component: 6 i 6 l ADEAnFunc(this; i) continue: ADESystem(P ; (component()(); ; component(l)()) Class ADEAnFunc B AnFunc 2 ADESystem i 2 f; ; lg new: ( ~ ; {~) 3 ~, i 3 {~ radius: () /(2 l (! M )) p norm: r (?? 2 l r (! M ) )/l continue:! continue()! component(i) In contrast with the linear case, the domain of the solution (f ; ; f l ) to (4) is not necessarily eective. Nevertheless, the solution is faithful: Theorem. Let P ; ; P l be polynomials with coecients in Complex and let ; ; l 2 Complex. Then the system (4) admits a faithful eective analytic solution (f ; ; f l ). Proof. Let = [ ; ; l ] 2 Dom (f ; ; f l ) \ Path. Let us prove by induction over i = f; ; lg that [ ; ; i ] 2 Dom ] (f ; ; f l ). For i = we have nothing to prove, so assume that i >. For all t 2 [; ], let ;t = [ ; ; i? ; t i ] ;t = (f (;t ); ; f l(;t )) M ;t = max (kp ;+;t k ; ; kp l;+;t k )

14 4 Effective analytic functions Then M ;t is a continuous function, which admits a global maximum M ;[;] on [; ]. Now let w [ ; ; i? ] be such that 2 Dom f and let = ( i? i? )/n be such that n 2 N and j j < /(2 l M ;[;] ). Then we have = + [ ; n ; ] 2 Dom f and w [ ; ; i ]. This proves that [ ; ; i ] 2 Dom ] (f ; ; f l ) and we conclude by induction. Remark 2. In principle, it is possible to replace the algebraic dierential equations by more general non-linear dierential equations, by taking convergent power series for the P i. However, this would require the generalization of the theory in this paper to analytic functions in several variables (interesting exceptions are power series which are polynomial in all but one variable, or entire functions). One would also need to handle the transformations P i P i;+ with additional care; these transformations really correspond to the analytic continuation of the P i. 5. Conclusion and final remarks Using a careful denition of eective analytic functions, we have shown how to answer many numerical problems about analytic solutions to dierential equations. In order to generalize the present theory to analytic functions in several variables or more general analytic functions, like solutions to convolution equations, we probably have to weaken the conditions EA and EA2. Nevertheless, it is plausible that further research will lead to a more suitable denition which preserves the spirit of the present one. The Columbus program implements the approach of the present paper in the weaker setting of double precision complex numbers instead of eective complex numbers. We plan to describe this program in more details in a forthcoming paper and in particular the radar algorithm which is used to graphically represent analytic functions (see gure ). It would be nice to adapt or reimplement the Columbus program so as to permit computations with eective complex numbers when desired. Figure. Plot of a solution to f = + 4 f 3 + f with the Columbus program. The implementation of the Columbus program has also been instructive for understanding the complexity of our algorithms. For instance, the lower bound for the smallest zero in our algorithm for inversion can be extremely bad, as in the example f =? exp (n? z)

15 Joris van der Hoeven 5 The problem here is that the eective radius of convergence is of the order of, while the real radius is n. Consequently, the analytic continuation from to n? will take O(n) steps instead of only. In the Columbus program, this problem has been solved by using a numerical algorithm in order to determine the radius of convergence instead of the theoretically correct one. In some cases, this leads to an exponential speedup. Theoretically correct approaches for solving the problem are to compute the smallest zero of the denominator in an appropriate radius or to use transseries-like expansions. We plan to explain these approaches in more detail in a forthcoming paper. Another trick which can be used in concrete implementations is to override the default evaluation method from section 2.5 by a more ecient one when possible, or to implement methods for the evaluation or computation of higher derivatives. Bibliography [BB85] E. Bishop and D. Bridges. Foundations of constructive analysis. Die Grundlehren der mathematische Wissenschaften. Springer, Berlin, 985. [Car6] H. Cartan. Th orie l mentaire des fonctions analytiques d'une et plusieurs variables. Hermann, 96. [CC9] D.V. Chudnovsky and G.V. Chudnovsky. Computer algebra in the service of mathematical physics and number theory (computers in mathematics, stanford, ca, 986). In Lect. Notes in Pure and Applied Math., volume 25, pages 9232, New-York, 99. Dekker. [DL84] J. Denef and L. Lipshitz. Power series solutions of algebraic dierential equations. Math. Ann., 267:23238, 984. [DL89] J. Denef and L. Lipshitz. Decision problems for dierential equations. The Journ. of Symb. Logic, 54(3):9495, 989. [Kho9] A. G. Khovanskii. Fewnomials. American Mathematical Society, Providence, RI, 99. [PG97] A. P ladan-germa. Tests eectifs de nullit dans des extensions d'anneaux di rentiels. PhD thesis, Gage, cole Polytechnique, Palaiseau, France, 997. [Ris75] R.H. Risch. Algebraic properties of elementary functions in analysis. Amer. Journ. of Math., 4():743759, 975. [Sha89] J. Shackell. A dierential-equations approach to functional equivalence. In Proc. ISSAC '89, pages 7, Portland, Oregon, A.C.M., New York, 989. ACM Press. [Sha93] J. Shackell. ero equivalence in function elds dened by dierential equations. Proc. of the A.M.S., 336():572, 993. [SvdH] J.R. Shackell and J. van der Hoeven. Complexity bounds for zero-test algorithms. Technical Report 2-63, Pr publications d'orsay, 2. [vdh99] J. van der Hoeven. Fast evaluation of holonomic functions. TCS, 2:9925, 999. [vdha] J. van der Hoeven. Fast evaluation of holonomic functions near and in singularities. JSC, 3:77743, 2. [vdhb] J. van der Hoeven. ero-testing, witness conjectures and dierential diophantine approximation. Technical Report 2-62, Pr publications d'orsay, 2. [vdh2a] Joris van der Hoeven. Columbus. Columbus.html, 222. [vdh2b] Joris van der Hoeven. A new zero-test for formal power series. In Teo Mora, editor, Proc. ISSAC '2, pages 722, Lille, France, July 22. [vdh2c] Joris van der Hoeven. Relax, but don't be too lazy. JSC, 34:479542, 22. [vdh3] J. van der Hoeven. Majorants for formal power series. Technical Report 23-5, Universit Paris-Sud, 23. [vk75] S. von Kowalevsky. ur Theorie der partiellen Dierentialgleichungen. J. Reine und Angew. Math., 8:32, 875.