INTRODUCTION TO PADÉ APPROXIMANTS. E. B. Saff Center for Constructive Approximation

Transcription

1 INTRODUCTION TO PADÉ APPROXIMANTS E. B. Saff Center for Constructive Approximation

2 H. Padé ( ) Student of Hermite Histhesiswon French Academy of Sciences Prize C. Hermite ( ) Used Padé approximants to prove that e is trancendental But origins of subject go back to Cauchy, Jacobi, Frobenius. Historical Reference: C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, (Berlin, 1991)

3 Why Padé? 1) Convergence Acceleration [e.g. ɛ-algorithm] 2) Numerical Solutions to Partial Differential Equations [exp(at) Q(At) 1 P (At)] 3) Analytic Continuation of Power Series [regions of convergence beyond a disk] 4) Includes Study of Orthogonal Polys on Interval [Padé denominators for Markov functions are orthogonal] 5) Finding Zeros/Roots, Poles/Singularities [usezerosandpolesofpadé approximants to predict - e.g. QD algorithm]

4 Padé Approximants (PA) generalize Taylor Polynomials Given f(z) = k=0 c k z k Taylor poly P m (z) = m k=0 c k z k Then f(z) P m (z) = k=m+1 c k z k Equivalently, f(z) P m (z) =O ( z m+1) P m (0) = f(0) P m(0) = f (0) P (m) (0) = f (m) (0).

5 Idea of PA: Given m, n Rational function R = P/Q deg P m, deg Q n Choose P, Q so that l as large as possible. (f R)(z) =O ( z l), How large can we expect l to be? P has m +1 parameters Q has n +1 parameters P/Q has 1 parameter So total of m + n +1 parameters Expect: ( ) f P (z) =O ( z m+n+1). Q

6 NOT ALWAYS POSSIBLE Ex: m = n =1, f(z) =1+z 2 + z 4 +. Want R(z) = P (z) Q(z) = az + b cz + d. (1) R(z) =1+z 2 + O ( z 3). But R is either identically constant or one-to-one. From (1), neither is possible [ R (0) = 0 ]. Idea: Linearize by requiring Qf P = O ( z m+n+1) P k := all polynomials of degree k.

7 DEF Let f(z) = 0 c k z k be a formal power series, and m, n nonnegative integers. A Padé form (PF) of type (m, n) isapair(p, Q) such that P = Q 0and m k=0 p k z k P m, Q = n k=0 q k z k P n, (2) Qf P = O ( z m+n+1) as z 0. Proposition Padé formsoftype(m, n) always exist. Proof. (2) is a system of m+n+1 homogeneous equations in m + n + 2 unknowns: (3) n j=0 c k j q j p k =0, 0 k m (4) n j=0 c k j q j =0, k = m +1,...,m+ n. c m,n := ( c m+i j ) n i,j=1 Toeplitz matrix

8 THM Every PF of type (m, n) forf(z) yields the same rational function. Proof. (P, Q) and ( ˆP, ˆQ ) are PF s. Qf P = O ( z m+n+1) ˆQf ˆP = O ( z m+n+1) so ˆQP + ˆPQ = O ( z m+n+1) P m+n. Thus ˆPQ ˆQP ˆP/ˆQ P/Q. DEF The uniquely determined rational P/Q is called the Padé Approximant (PA) of type (m, n) forf(z), and is denoted by [m/n] f (z) or r m,n (f; z).

9 Remark In reduced form [m/n] f (z) =p m,n (z)/q m,n (z), where we (often) normalize so that q m,n (0) = 1, p m,n (0) = c 0, p m,n and q m,n relatively prime. Padé Table for f [0/0] [0/1] [0/2] [1/0] [1/1] [1/2] Taylor polys [2/0] [2/1] [2/2] Equal entries occur in square blocks.

10 Ex: f(z) =1+z 2 + z 4 + z 6 + ( = 1 ) 1 z 2 Block structure [0/0] = [0/1] [0/2] = [1/0] = [1/1] [1/2] = [2/0] = [2/1] [ ] = [3/0] = [3/1] [ ] = [4/0] = [4/1] [ ] = THM Let p/q be a reduced PA for f(z), with c 0 0. Suppose and m = exact deg of p n = exact deg of q qf p = O ( z m+n+k+1) exactly.

11 Then (a) k 0 (b) [µ/ν] f = p/q iff m µ m + k, n ν n + k. See: W. B. Gragg, The Padé Table and its Relation to Certain Algorithms of Numerical Analysis, SIAM Review (1972), DEF APadé approximant is said to be normal if it appears exactly once in table. We say f is normal if every entry in its Padé table is normal. Ex: f(z) =e z is normal.

12 Determinant Representations and Frobenius Identities. f(z) = k=0 c k z k, u m,n (z) := v m,n (z) := f m (z) := m k=0 c k z k P m f m (z) zf m 1 (z) z n f m n (z) c m+1 c m c m n+1 c m+2 c m+1 c m n+2 c m+n c m+n 1 c m 1 z z n c m+1 c m c m n+1 c m+2 c m+1 c m n+2 c m+n c m+n 1 c m Note: u m,n (z) P m, v m,n (z) P n. THM f(z)v m,n (z) u m,n (z) = O ( z m+n+1).

13 DEF For arbitrary, but fixed polys g, h, let w m,n (z) :=g(z)u m,n (z) + h(z)v m,n (z) c m,n := det ( c m+i j ) n i,j=1 THM Between any 3 entries in the table of w m,n functions, there is a homogeneous linear relation with poly coefficients which can be computed from the coefficients c k of f. c m,n+1 w m+1,n c m+1,n w m,n+1 = c m+1,n+1 zw m,n c m+1,n w m 1,n + c m,n+1 w m,n 1 = c m,n w m,n c m,n c m+1,n w m,n+1 c m,n+1 c m+1,n+1 zw m,n 1 =(c m+1,n c m,n+1 c m,n c m+1,n+1 z)w m,n Proof. Use Sylvester s identity on determinant representation for Padé denominator v m,n. deta det A i,j;k,l =deta i;k deta j;l deta i;l deta j;k

14 Padé Approximants for the Exponential f(z) =e z = k=0 Want to find p m,n P m,q m,n P n such that (5) q m,n (z)e z p m,n (z) =O ( z m+n+1). Let D := d/dz. Then z k k! D [qe z ] = qe z + q e z = e z (I + D) q Apply D m+1 to (5) e z (I + D) m+1 q m,n +0=O (z n ) (I + D) m+1 q m,n = k m,n z n q m,n = k m,n (I + D) (m+1) z n. Recall (1 + x) (m+1) = j=0 ( 1) j( m + j m ) x j.

15 So q m,n (z)=k m,n n j=0 = k m,n n j=0 ( 1) j( m + j ) D j z n m ( 1) j( m + j m ) n! (n j)! zn j. q m,n (z) = n k=0 (m + n k)!n! ( z) k (m + n)!(n k)! k! q m,n e z p m,n = O ( z m+n+1), q m,n p m,n e z = O ( z m+n+1). So p m,n ( z) =q n,m (z), Also from p m,n (z) = m k=0 (m + n k)!m! (m + n)!(m k)! z k k!. D m+1 [q m,n e z p m,n ]=k m,n z n e z, and integration by-parts we get

16 q m,n (z)e z p m,n (z) = ( 1)n 1 (m + n)! zm+n+1 0 sn (1 s) m e sz ds. Remark For z ρ, q m,n (z) 1+ρ + ρ2 2! + ρ3 3! + = eρ. So q m,n form a normal family in C. Further, if m + n, m/n λ, Hence... coeff of z k ( 1)k (1 + λ) k k!.

17 THM (Padé) Let m j, n j Z + satisfy m j + n j, m j /n j λ as j. Then and lim j q m j,n j (z) =e z/(1+λ), lim j p m j,n j (z) =e λz/(1+λ), lim [m j/n j ](z) =e z, j locally uniformly in C. More precisely (m = m j,n= n j ) [m/n](z) e z = m!n! z m+n+1 e 2Re(z)/(1+λ) (m + n)!(m + n +1)! (1 + o(1)). COR All zeros and poles of PA s to e z go to infinity as m + n. But where are they located?

18 Zeros of p m,0 (z) = m k=0 z k /k!,m=1, 2,...,40 THM (S+Varga) For every m, n 0, the normalized Padé numerator p m,n ((n +1)z) for e z is zero-free in the parabolic region Result is sharp! P : y 2 4(x +1), x > 1.

19 THM (S+Varga) Consider any ray sequence [m j /n j ](z) wheren j /m j σ (0 σ< ). S σ := { z : arg z > cos 1 [(1 σ)/(1 + σ)] } C σ ze g(z) w σ (z) = 2 [1 + z + g(z)] (1+σ) [1 z + g(z)] where g(z) := 1+z 2 2z ( ) 1 σ 1+σ. Then 2σ (1+σ), (i) ẑ is a lim. pt. of zeros of [m j /n j ]((m j + n j )z) iff ẑ D σ := { z S σ : w σ (z) =1, z 1 }. (ii) ẑ is a lim. pt. of poles of [m j /n j ]((m j + n j )z) iff ẑ E σ := { z C \ S σ : w σ (z) =1, z 1 }.

20 More recent variations: Multi-point Padé Approx. Let B (m+n) = { x (m+n) } m+n k k=0 R, R m,n = P m,n /Q m,n, deg P m,n = m, deg Q m,n = n, interpolates e z in B (m+n). THM (Baratchart+S+Wielonsky) If B (m+n) [ ρ, ρ], m = m ν, n = n ν (m + n ), then R m,n (z) e z z C. Moreover, the zeros and poles of R m,n lie within ρ of the zeros and poles, respectively, of the Padé approximants [m/n](z) toe z. COR Conclusion holds for best uniform rational approx. to e x on any compact subinterval of R. Analogous results for best L 2 -rational approximants to e z on unit circle.

21 Introduction to Convergence Theory f(z) = k=0 c k z k [m/0] f (z) = m k=0 c k z k converges in largest open disk centered at z = 0 in which f is analytic: 1 z <R, where = lim sup R m c m 1/m. Next simplest case: [m/1] f. v m,1 (z) =det ( Assume c m+1 0. c m /c m+1. lim inf m c m+1 c m 1 z c m+1 c m 1 R ) = c m zc m+1. Then v m,1 has zero at lim sup m c m+1 c. m It s possible for ratios to have many limit points different from 1/R.

22 ALL IS NOT ROSES - There can be spurious poles. Perron s Example: f entire (R = ) such that every point in C is a limit point of poles of some subsequence of [m/1] f. THM (de Montessus de Ballore, 1902) Let f be meromorphic with precisely ν poles (counting multiplicity) in the disk : z <ρ, with no poles at z =0. Then lim m [m/ν] f(z) =f(z) uniformly on compact subsets of \{ν poles of f}. Furthermore, as m, the poles of [m/ν] f tend, respectively, to the ν poles of f in. Ex: f(z) =zγ(z) haspolesatz = 1, 2,... The n-th column of Padé table will converge to zγ(z) in{ z <n+1}\{ 1,..., n}.

23 ProofofdeMontessusdeBalloreTheorem: Hermite s Formula Suppose g is analytic inside and on Γ, a simple closed contour. Let z 1,z 2,...,z µ be points interior to Γ, regarded with multiplicities n 1,n 2,...,n µ. Set N := n 1 + n n µ. Then a unique poly p P N 1 such that p (j) (z k )=g (j) (z k ), j =0, 1,...,n k 1, k =1,...,µ. Moreover, p(z) = 1 2πi Γ g(z) p(z) = 1 2πi where ω(ζ) ω(z) ω(ζ)(ζ z) Γ ω(z) := ω(z)g(ζ) ω(ζ)(ζ z) µ j=1 g(ζ) dζ z C, (z z j ) n j. dζ z inside Γ,

24 IdeaofProofofdeM.deBalloreThm f meromorphic with ν poles in z <ρ. u m,ν, v m,ν PF of type (m, ν) forf. (6) fv m,ν u m,ν = O ( z m+ν+1). Let Q ν P ν have zeros at poles of f with same multiplicity. Q ν fv m,ν Q ν u m,ν = O ( z m+ν+1) = 1 2πi ζ =ρ ɛ for z <ρ ɛ. z m+ν+1 (Q ν fv m,ν ) (ζ) ζ m+ν+1 (ζ z) dζ, For v m,ν suitably normalized, integral 0 for z <ρ ɛ. Method extends to multi-point Padé.

25 What about other sequences from Padé Table, such as rows, diagonals, ray sequences? THM (Wallin) There exists f entire such that the diagonal sequence [n/n] f (z), n =0, 1, 2,..., is unbounded at every point in C except z =0. Baker-Gammel-Wills Conjecture: If f is analytic in z < 1 except for m poles ( 0), then there exists a subsequence of diagonal PAs [n/n] f (z) that converges to f locally uniformly in { z < 1}\{m poles of f}. Conjecture is FALSE! D. S. Lubinsky, Rogers-Ramanujan and..., Annals of Math, 157 (2003),

26 Next step: Consider a weaker form of convergence, such as convergence in measure or convergence in capacity. Nuttall-Pommerenke Near-diagonal PAs will be inaccurate approximations to f only on sets of small capacity (transfinite diameter). THM f analytic at andinadomaind C with cap(c \ D) =0. LetR m,n (z) denote the PA to f at. Fix r > 1, λ>1. Then for ɛ, η > 0 there exists an m 0 such that R m,n (z) f(z) <ɛ m for all m>m 0,1/λ m/n λ, andforallz in z <r, z E m,n,cap(e m,n ) <η.

27 THM (Stahl) Let f(z) be analytic at infinity. There exists a unique compact set K 0 C such that (i) D 0 := C \K 0 is a domain in which f(z) has a single-valued analytic continuation, (ii) cap(k 0 )=inf K cap(k), where the infimum is over all compact sets K C satisfying (i), (iii) K 0 K for all compact sets K C satisfying (i) and (ii). The set K 0 is called minimal set (for singlevalued analytical continuation of f(z)) and the domain D 0 C is called extremal domain.

28 THM (Stahl) Let the function f(z) bedefined by f(z) = j=0 f j z j and have all its singularities in a compact set E C of capacity zero. Then any close to diagonal sequence of Padé approximants [m/n](z) to the function f(z) converges in capacity to f(z) in the extremal domain D 0.