SC2011 International Conference Scientific Computing on S. - - PDF document

EXTENSIONS OF PAD´ E-TYPE APPROXIMANTS Ernst Joachim Weniger Theoretical Chemistry University of Regensburg, Germany joachim.weniger@chemie.uni-regensburg.de

SC2011

International Conference Scientific Computing on

• S. Margherita di Pula, Sardinia, Italy

October 10-14, 2011 (with corrections)

1

Glory and Misery of Power Series ⊲ Power series are the most important tools of calculus. ⊲ They possess highly advantageous analytical features. ⇒ They are used abundantly in all applications

• f mathematics.

⊲ From a purely numerical point of view a power series expansion is a mixed blessing. ⊲ Power series converge in their circles of con- vergence, which may shrink to a single point, and they diverge outside of their circles of convergence. ⇒ With the exception of a few fortunate cases like exp(z) =

∞

• n=0

zn/n! power series are not suited to evaluate a func- tion effectively and reliably for all z ∈ C.

2

Rational Functions ⊲ Rational functions are ratios of two polyno- mials P and Q of degrees m, n ∈ N0: Pm(z) Qn(z) =

m

µ=0 pµzµ

n

ν=0 qνzν

= (z − α1) · · · (z − αm) (z − β1) · · · (z − βn) . ⊲ Power series and rational functions have largely complementary properties:

• Power series expansions for numerous func-

tions are explicitly known.

• For few functions only, rational approxi-

mants are explicitly known.

• The construction of a finite set of ratio-

nal approximants to a given function is usually a nontrivial numerical problem.

3

Advantages of Rational Functions ⊲ Rational approximants normally have (much) better numerical properties than the power series from which they are derived. ⊲ A function, whose power series has a nonzero, but finite radius of convergence must have at least one singularity on the boundary of the circle of convergence. ⇒ Rational approximants can approximate func- tions outside the circles of convergence of their power series. ⊲ Numerical problems can only occur in the im- mediate vicinity of the poles of the rational function. ⊲ But even the poles of a rational approximant – the zeros of its denominator polynomial – can provide useful information about the function. ⊲ Rational functions can simulate the cut of a function.

4

Pad´ e Approximants ⊲ Formal power series for f : C → C: f(z) =

∞

• n=0

γn zn . ⊲ Pad´ e approximant to f: [m/n]f(z) = P [m/n](z) Q[m/n](z) , m, n ∈ N0 , P [m/n](z) = p0 + p1z + · · · + pmzm , Q[m/n](z) = q0 + q1z + · · · + qnzn . ⊲ Analyticity of [m/n]f(z) at z = 0 is guaran- teed by the Baker condition q0 = 1. ⇒ The remaining m + n + 1 polynomial coeffi- cients p0, p1, . . . , pm and q1, q2, . . . , qn are de- termined by requiring that the modified accu- racy-through-order relationship Q[m/n](z) f(z) − P [m/n](z) = O

• zm+n+1

holds as z → 0.

5

⇒ The accuracy-through-order relationship leads to a system of m+n+1 coupled linear equa- tions. ⊲ The denominator coefficients q1, q2, . . . , qn are determined via the n equations

min(m+ν,n)

• κ=0

qκ γm+ν−κ = 0 , 1 ≤ ν ≤ n , ⊲ The numerator coefficients p0, p1, . . . , pm are determined via the m + 1 equations

min(µ,n)

• κ=0

qκ γµ−κ = pµ , 0 ≤ µ ≤ m , which correspond to the m + 1 leading terms

• f a Cauchy product of two power series.

⇒ The difficult part is the computation of the denominator coefficients q1, q2, . . . , qn. ⊲ These equations show that the coefficients γ0, γ1, . . . , γm+n of the power series for f are needed for the construction of [m/n]f(z).

6

Pad´ e-Type Approximants: Heuristic Mo- tivation ⊲ The polynomial coefficients of a Pad´ e ap- proximant [m/n]f(z) can be computed if the numerical values of the power series coeffi- cients γ0, γ1, . . . γm+n are known. ⊲ No further information about the function f(z), which is to be approximated, is needed. ⊲ This feature is also shared by other compu- tational schemes for Pad´ e approximants such as Wynn’s celebrated epsilon algorithm. ⊲ This feature of Pad´ e approximants is highly advantageous, and it contributed significantly to their usefulness and popularity. ⊲ But this advantage can also become a dis-

• advantage. We often have some knowledge

about the function, which we want to ap- proximate. ⊲ Unfortunately, there is no obvious way of uti- lizing such an information in the case of Pad´ e approximants.

7

Pad´ e-Type Approximants: Theory ⊲ In 1979, Brezinski introduced his so-called Pad´ e-type approximant (m/n)f(z) = U(m/n)(z) V(m/n)(z) , m, n ∈ N0 , U(m/n)(z) = u0 + u1z + · · · + umzm , V(m/n)(z) = v0 + v1z + · · · + vnzn , which look like Pad´ e approximants [m/n]f(z). Their theory was fully developed in a mono- graph by Brezinski (1980). ⊲ However, it is now assumed that the denomi- nator polynomial V(m/n)(z) is explicitly known. ⇒ The coefficients of the numerator polyno- mial are determined via the via the modified asymptotic condition (z → 0) V(m/n)(z) f(z) − U(m/n)(z) = O(zm+1) . ⇒ Explicit expression as a Cauchy product: U(m/n)(z) =

m

• µ=0

zµ

min(µ,n)

• ν=0

vν γµ−ν .

8

Cauchy Products of Power Series ⊲ Assume Φ(z) =

∞

• n=0

ϕnzn , Φn(z) =

n

• ν=0

ϕνzν , Ψ(z) =

∞

• n=0

ψnzn . Ψn(z) =

n

• ν=0

ψνzν . ⊲ Standard form of the Cauchy product with truncation: Φ(z) Ψ(z) =

∞

• n=0

zn

n

• ν=0

ϕn−ν ψν =

N

• n=0

zn

n

• ν=0

ϕν ψn−ν + O(zN+1) , z → 0 . ⊲ Alternative Cauchy product involving partial sums Ψn(z): Φ(z) Ψ(z) =

N

• ν=0

ϕνzν ΨN−ν(z) + O(zN+1) , z → 0 .

9

Alternative Expression for Pad´ e-Type Ap- proximants ⊲ In the theory of sequence transformations, it is frequently more convenient to use the partial sums fn(z) =

n

• ν=0

γν zν n ∈ N0 .

• f a power series as input data and not their

coefficients γν. ⇒ Alternative explicit expression for the numer- ator polynomial U(m/n)(z) =

min(m,n)

• µ=0

vµ zµ fm−µ(z) and the Pad´ e-type approximant (m/n)f(z) =

min(m,n)

µ=0

vµ zµ fm−µ(z)

n

ν=0 vν zν

. ⊲ The alternative expression for U(m/n)(z) can also be used for the numerator polynomial P [m/n](z) of a Pad´ e approximant [m/n]f(z).

10

Pad´ e-Type Approximants: General Con- siderations ⊲ The computation of the numerator polyno- mials U(m/n)(z) of a Pad´ e-type approximant (m/n)f(z) is fairly simple once the denomina- tor polynomial V(m/n)(z) is explicitly known. ⇒ The real challenge is the choice of a suitable class of denominator polynomials if we want to employ Pad´ e-type approximants in conver- gence acceleration and summation processes. ⊲ The poles of a function are only rarely known. A fortunate counter-example is the digamma function: ψ(z) = −γ − 1 z +

∞

• k=1

z k(k + z) z = 0, −1, −2, · · · . A Pad´ e-Type approximant to ψ(z) was con- structed by Weniger (2003). ⇒ In general, alternative approaches for the de- termination of the denominators are neces- sary.

11

Levin’s Transformations ⊲ In 1973, Levin introduced a sequence trans- formation which utilizes the information con- tained in explicit estimates of the truncation error of the input sequence ⇒ The explicit utilization of additional informa- tion makes Levin’s transformation potentially very powerful. ⊲ Levin’s idea was later extended in articles by Levin and Sidi (1981), Sidi (1979), Sidi and Levin (1982), Weniger (1989, 2004), and Homeier (2000). ⇒ Many new, more or less closely related se- quence transformation were derived which also utilize the information contained in explicit truncation error estimates. ⊲ It can be shown that certain variants of Levin- type transformations are actually Pad´ e-type approximants [Weniger 2004].

12

Levin-Type Transformations ⊲ The Levin-type transformations considered in Weniger (2004) all possess the general structure T (n)

k

(sn, ωn) = ∆k[Pk−1(n) sn/ωn] ∆kPk−1(n)/ωn] =

k

• j=0

(−1)j k j

Pk−1(n + j) sn+j

ωn+j

k

• j=0

(−1)j k j

Pk−1(n + j)

ωn+j . Pk−1(n) is a polynomial of degree k − 1 in n, and the {ωn}∞

n=0 are remainder estimates.

⊲ Obviously, ∆k{Pk−1(n)/ωn} = 0 must hold for all finite k, n ∈ N0. ⊲ The {ωn}∞

n=0 should reproduce the leading

• rder asymptotics of the actual remainders

{rn}∞

n=0:

rn = sn − s = ωn

• c + O
• 1/n
• ,

n → ∞ .

13

Different Variants ⊲ The choice Pk−1(n) = (β + n)k−1 with β > 0 yields Levin’s transformation: L(n)

k

(β, sn, ωn) = ∆k[(β + n)k−1sn/ωn] ∆k[(β + n)k−1/ωn] =

k

• j=0

(−1)j k j

(β + n + j)k−1

(β + n + k)k−1 sn+j ωn+j

k

• j=0

(−1)j k j

(β + n + j)k−1

(β + n + k)k−1 1 ωn+j . ⊲ The choice Pk−1(n) = (β + n)k−1 with β > 0 yields Weniger transformation: S(n)

k

(β, sn, ωn) = ∆k[(β + n)k−1sn/ωn] ∆k[(β + n)k−1/ωn] =

k

• j=0

(−1)j k j

(β + n + j)k−1

(β + n + k)k−1 sn+j ωn+j

k

• j=0

(−1)j k j

(β + n + j)k−1

(β + n + k)k−1 1 ωn+j . ⊲ There are also some other transformations.

14

Rational Levin-Type Transformations. I. ⊲ The input data sn are now the partial sums fn(z) = n

ν=0 γνzν of a power series.

⊲ u variant [ωn = (β + n)γnzn (Levin 1973)]:

uT (n) k

(fn(z)), (β + n)γnzn) =

k

• j=0

(−1)j k j

Pk−1(n + j) zk−j fn+j(z)

(β + n + j)γn+j

k

• j=0

(−1)j k j

Pk−1(n + j)zk−j

(β + n + j)γn+j . ⊲ t variant [ωn = γnzn (Levin 1973)]:

tT (n) k

(fn(z)), (β + n)γnzn) =

k

• j=0

(−1)j k j

Pk−1(n + j) zk−j fn+j(z)

γn+j

k

• j=0

(−1)j k j

Pk−1(n + j)zk−j

γn+j .

15

Rational Levin-Type Transformations. II. ⊲ d variant [ωn = γn+1zn (Smith and Ford 1979)]:

dT (n) k

(fn(z)), (β + n)γnzn) =

k

• j=0

(−1)j k j

Pk−1(n + j) zk−j fn+j(z)

γn+j+1

k

• j=0

(−1)j k j

Pk−1(n + j)zk−j

γn+j+1 . ⊲ v variant [ωn = (γn+j−zγn+j+1)/(γn+jγn+j+1) (Levin 1973)]:

dT (n) k

(fn(z)), (β + n)γnzn) =

k

• j=0

(−1)j k j

• Ω(k)

n+j zk−j fn+j(z) k

• j=0

(−1)j k j

• Ω(k)

n+j zk−j

, Ω(k)

n

= Pk−1(n) γn − zγn+1 γnγn+1

16

Rational Levin-Type Transformations. III. ⊲ The u, t, and d variants are ratios of polyno- mials of degrees k+n and k in z, respectively, having the following structure:

T(n)

k

(z) =

k

• j=0

λ(k,n)

j

zk−j fn+j(z)

k

• j=0

λ(k,n)

j

zk−j =

k

• j=0

λ(k,n)

k−j zj fn+k−j(z) k

• j=0

λ(k,n)

k−j zj

. ⊲ It can be shown that the u, t, and d variants are (k + n/k)f(z) Pad´ e-Type approximants (Weniger 2004). ⊲ Taylor expansion of these rational approxi- mants about z = 0 reproduces all power se- ries coefficients used for their construction (Weniger 2004).

17

Rational Levin-Type Transformations. IV. ⊲ The v variants are ratios of polynomials of degrees k + n + 1 and k + 1 in z, respectively, having the following structure:

V(n)

k

(z) =

k

• j=0

λ(k,n)

j

zk−j fn+j(z)

k

• j=0

λ(k,n)

j

zk−j + z

k

• j=0

µ(k,n)

j

zk−j + z

k

• j=0

µ(k,n)

j

zk−j fn+j(z)

k

• j=0

λ(k,n)

j

zk−j + z

k

• j=0

µ(k,n)

j

zk−j ⊲ Contrary to Sect. VI of Weniger (2004), the v variants are (k+n+1/k+1)f(z) Pad´ e-Type approximants.

18

Open Questions ⊲ The numerators of rational Levin-type trans- formation (k+n/k)f(z) are weighted sums of the input data fn(z), fn+1(z), · · · , fn+k(z). ⊲ Their structure as well as that of other Pad´ e- type approximants is completely determined by their denominator polynomials. ⊲ Levin-type transformations are very often re- markably powerful. ⇒ Can we learn something about how to choose the denominators of other Pad´ e-type approx- imants? ⊲ Poles of rational approximants can be spuri-

• us, i.e., they can be artifacts of the approx-

imation scheme. ⇒ It makes sense to assume that the zeros of “good” denominator polynomials somehow represent the function f(z) to be approxi- mated. ⊲ Often, the cut of a function f(z) is known.

19

Euler Integral and Euler Series ⊲ Euler integral: E(z) =

∞

e−tdt 1 + zt . Prototype of a Stieltjes function. ⊲ Euler series: E(z) ∼

∞

• ν=0

(−1)ν ν! zν = 2F0(1, 1; −z) , z → 0 . Prototype of a factorially divergent, but summable Stieltjes series. ⊲ ⊲ The Euler integral has a cut along the neg- ative real axis. ⇒ The zeros of “good” denominator polynomi- als should simulate the cut.

20

Application of the Delta Transformation ⊲ Delta transformation (variant of the Weniger transformation with ωn = γn+1zn+1): δ(n)

k

• β, fn(z)
• =

k

• j=0

(−1)j k j

(β + n + j)k−1

(β + n + k)k−1 zk−jfn+j(z) γn+j+1

k

• j=0

(−1)j k j

(β + n + j)k−1

(β + n + k)k−1 zk−j γn+j+1 . Very powerful (k + n/k)f(z) Pad´ e-type ap- proximant. ⊲ ⇒ Numerator of the delta approximation to the Euler series with k = 10, n = 0, and β = 1:

10

• j=0

k

j

(n + j + 1)9

(j + 1)! z10−j = 362880 z10 + 18144000 z9 + 149688000 z8 + 399168000 z7 + 454053600 z6 + 254270016 z5 + 75675600 z4 + 12355200 z3 + 1093950 z2 + 48620 z + 4199 5

21

Denominator of the [10/10] Pad´ e Approx- imant and the L10(−z) Laguerre polyno- mial ⊲ The [10/10] Pad´ e approximant to the Euler series requires the input of the partial sums f0(z), f1(z), · · · , f20(z). Q[10/10](z) = 1 + 110 z + 4950 z2 + 118800 z3 + 1663200 z4 + 13970880 z5 + 69854400 z6 + 199584000 z7 + 299376000 z8 + 199584000 z9 + 39916800 z10 . ⊲ The Laguerre polynomials are orthogonal with respect to an integration over the positive real semiaxis. ⇒ The Laguerre polynomials with negative ar- guments have zeros on the cut of the Stieltjes integral. ⊲ It should at least in principle be possible to use Laguerre polynomials with negative ar- guments as denominators of Pad´ e-type ap- proximants to the Euler series.

22

Comparison of the Zeros Zeros of the Denominators and of L10(z) n [10/10] L10(−z) delta 1

• 2.98928
• 29.921
• 40.4575

2

• 0.88633
• 21.997
• 5.8159

3

• 0.41738
• 16.279
• 1.8475

4

• 0.23999
• 11.844
• 0.8125

5

• 0.15415
• 8.3302
• 0.4274

6

• 0.10606
• 5.5525
• 0.2515

7

• 0.07632
• 3.4014
• 0.1594

8

• 0.05651
• 1.8083
• 0.1061

9

• 0.04241
• 0.7295
• 0.0726

10

• 0.03156
• 0.1378
• 0.0496

⊲ There are enormous qualitative differences. The zeros L10(−z) are much more evenly dis- tributed than the zeros of Q([10/10])(z) or of the denominator of the delta transformation, which cluster in the immediate vicinity of the

• rigin.

⊲ A simpleminded use of Laguerre polynomials with negative arguments does not work.

23

• Bibliography. I.

Brezinski C. (1979), Rational approximation to formal power series, J. Approx. Theory 25, 295 – 317. Brezinski C. (1980), Pad´ e-Type Approxima- tion and General Orthogonal Polynomials (Birk- h¨ auser, Basel). Homeier H.H.H. (2000), Scalar Levin-type sequence transformations, J. Comput. Appl.

• Math. 122, 81 – 147. Reprinted in Brezinski
• C. (Editor), Numerical Analysis 2000, Vol.

2: Interpolation and Extrapolation, pp. 81 – 147 (Elsevier, Amsterdam). Levin D. (1973), Development of non-linear transformations for improving convergence of sequences, Int. J. Comput. Math. B 3, 371 – 388. Levin D. and Sidi A. (1981), Two new classes

• f nonlinear transformations for accelerating

the convergence of infinite integrals and se- ries, Appl. Math. Comput. 9, 175 – 215.

24

• Bibliography. II.

Sidi A. (1979), Some properties of a general- ization of the Richardson extrapolation pro- cess, J. Inst. Math. Applic. 24, 327 – 346. Sidi A. and Levin D. (1982), Rational approx- imations from the d-transformation, IMA J.

• Numer. Anal. 2, 153 – 167.

Smith D.A. and Ford W.F. (1979), Acceler- ation of linear and logarithmic convergence, SIAM J. Numer. Anal. 16, 223 – 240. Weniger E.J. (1989), Nonlinear sequence trans- formations for the acceleration of convergence and the summation of divergent series, Com-

• put. Phys. Rep. 10, 189 – 371. Los Alamos

Preprint math-ph/0306302 (http://arXiv.org). Weniger E.J. (2003), A rational approximant for the digamma function, Numer. Algor. 33, 499 – 507.

25

• Bibliography. III.

Weniger E.J. (2004), Mathematical proper- ties of a new Levin-type sequence transfor- mation introduced by ˇ C ´ ıˇ zek, Zamastil, and Sk´ ala.

• I. Algebraic theory, J. Math.

Phys. 45, 1209 – 1246.

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