It is described how the Hermite-Pad\'e polynomials corresponding to an
algebraic approximant for a power series may be used to predict coefficients of
the power series that have not been used to compute the Hermite-Pad\'e
polynomials. A recursive algorithm is derived and some numerical examples are
given.Comment: 9 pages, ISRN Applied Mathematics, in pres

Homeier, Herbert H. H.

English

arXiv

It is described how the Hermite-Padé polynomials corresponding to an algebraic approximant for a power series may be used to predict coefficients of the power series that have not been used to compute the Hermite-Padé polynomials. A recursive algorithm is derived, and some numerical examples are given

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International Scholarly Research NetworkISRN Applied MathematicsVolume 2011, Article ID 958968, 7 pagesdoi:10.5402/2011/958968Research ArticleSeries Prediction Based on Algebraic ApproximantsHerbert H. H. Homeier1, 21 Institut fu¨r Physikalische und Theoretische Chemie, Universita¨t Regensburg, 93040 Regenburg, Germany2 IT Service, science + computing ag, Ingolsta¨dter Str. 22, 80807 Mu¨nchen, GermanyCorrespondence should be addressed to Herbert H. H. Homeier, herbert.homeier@na-net.ornl.govReceived 28 March 2011; Accepted 9 May 2011Academic Editors: H. Huo, E. Kita, and Y. SongCopyright q 2011 Herbert H. H. Homeier. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.It is described how the Hermite-Pade´ polynomials corresponding to an algebraic approximant fora power series may be used to predict coefficients of the power series that have not been used tocompute the Hermite-Pade´ polynomials. A recursive algorithm is derived, and some numericalexamples are given.1. IntroductionUsing sequence transformation and extrapolation algorithms for the prediction of furthersequence elements from a finite number of known sequence elements is a topic of growingimportance in applied mathematics. For a short introduction, see the book of Brezinski andRedivo Zaglia 1, Section 6.8. We mention theoretical work on prediction properties of Pade´approximants and related algorithms like the epsilon algorithm, and the iterated Aitken andTheta algorithms 2–5, Levin-type sequence transformations 6, 7, the E algorithm 4, 8,and applications on perturbation series of physical problems 7, 9.Here, we will concentrate on a different class of approximants, namely, the algebraicapproximants. For a general introduction to these approximants and the related Hermite-Pade´ polynomials see 10. Programs for these approximants are available 11. We summa-rize those properties that are important for the following.Consider a function f of complex variable zwith a known formal power seriesfz ∞∑j0fjzj . 1.12 ISRN Applied MathematicsThe Hermite-Pade´ polynomials HPPs corresponding to a certain algebraic approximant areN 	 1 polynomials Pnz with degree pn  degPn, n  0 · · ·N such that the order conditionN∑n0Pnzfz n  O(zM)1.2holds for small z. Since one of the coefficients of the polynomials can be normalized to unity,the order condition 1.2 gives rise to a system ofM linear equations forN	∑Nn0 pn unknownpolynomial coefficients. Thus, the coefficient of zm of the Taylor expansion at z  0 of theleft hand side of 1.2 must be zero for m  0, . . . ,M − 1. In order to have exactly as manyequations as unknowns, we chooseM  N 	N∑n0pn 1.3and assume that the linear system 1.2 has a solution. Then, the HPPs Pnz are uniquelydefined upon specifying the normalization. The algebraic approximant under considerationthen is that pointwise solution az of the algebraic equationP0z 	N∑n1Pnzazn  0 1.4for which the Taylor series of az coincides with the given power series at least up to orderzM−1.We note that for N  1, the algebraic approximants are nothing but the well-knownPade´ approximants.Although we assumed that the power series of f is known, quite often in practice,only a finite number of coefficients is really known. These coefficients then may be used tocompute the Hermite-Pade´ polynomials and the algebraic approximant under consideration.We note that the higher coefficients of the Taylor series of az may be considered aspredictions for the higher coefficients of the power series. The latter are also of interest inapplications.The question then arises how to compute the Taylor series of az. If it is possible tosolve 1.4 explicitly, that is for N ≤ 4, a computer algebra system may be used to do thejob. But even then, a recursive algorithm for the computation of the coefficients of the Taylorseries would be preferable in order to reduce computational efforts.In the following section, such a recursive algorithm is obtained. In a further section,we will present numerical examples.2. The Recursive AlgorithmWe consider the HPPsPnz pn∑j0pn,jzj 2.1ISRN Applied Mathematics 3as known. Puttingaz ∞∑k0akzk, 2.2we obtain from 1.4p0∑j0p0,jzj 	N∑n1pn∑j0pn,jzj∞∑k10. . .∞∑kn0zk1	···	knn∏m1akm  0, 2.3whence, by equating the coefficient of zJ to zero, we obtain an infinite set of equations. Dueto 1.2, all the equations for J < M are satisfied exactly for aj  fj , j  0, . . . ,M − 1.As a first step, we compute aM. We note thatM > p0. Hence, the coefficient of zM doesnot involve any terms with p0,j . For this coefficient RM, we only need to consider terms in2.3 such thatM  j 	 k1 	 · · · 	 kn, and we obtain RM  0 forRM N∑n1∑j	k1	···	knMpn,jn∏m1akm. 2.4The only terms on the RHS involving aM are obtained if exactly one of the km is equal toM,that is, we have km  M, j  0, and kj  0 for j /m. Thus, we may rewrite all these terms asaMC, whereC N∑n1n pn,0fn−10 2.5and note that the rest DM  RM − aMC is independent of aM. Recalling RM  0, we obtainaM −DMC. 2.6Proceeding analogously for J > M, only terms with J  j 	k1 	 · · ·	kn need to be considered.Hence, RJ  0 forRJ N∑n1∑j	k1	···	knJpn,jn∏m1akm. 2.7Now, the only terms on the RHS involving aJ are obtained if exactly one of the km is equal toJ , that is, we have km  J , j  0, and kj  0 for j /m. Thus, we may rewrite all these terms asaJC, where C is defined above. Proceeding as before, we put DJ  RJ − aJC and obtainaJ −DJC. 2.8An equivalent form of the recursive algorithm is obtained in the following way.4 ISRN Applied MathematicsConsider for known Pn and a0, . . . , aJ−1 the expressionUJ dJJ!dzJ∣∣∣∣∣z0N∑n1Pnz⎛⎝J∑j0ajzj⎞⎠n. 2.9It is easy to see, that this expression is exactly equal to RJ , and hence, is linear in the unknownaJ . Thus, we may compute the quantities DJ by substituting aJ  0 intoUJ , which entailsDJ dJJ!dzJ∣∣∣∣∣z0N∑n1Pnz⎛⎝J−1∑j0ajzj⎞⎠n. 2.10Thus, starting from J  M, one may compute all the aJ consecutively by repeated use of 2.5,2.10, and 2.8.This concludes the derivation of the recursive algorithm.3. Modes of ApplicationBasically, there are two modes of application:a one computes a sequence of HPPs and for the resulting algebraic approximants,one predicts a fixed number of so far unused coefficients, for example, only onenew coefficient. This mode is mainly for tests,b one computes from all available coefficients certain HPPs. For the best HPPs onecomputes a larger number of predictions for so far unused coefficients.In the following examples, we concentrate on mode b. Here, it is to be expected thatthe computed values have the larger errors the higher coefficients are predicted.4. ExamplesThe examples serve to introduce to the approach. All numerical calculations in this sectionwere done using Maple Digits  16.Example 4.1. As a first example, we considerN  2, p0  p1  p2  1, and, hence,M  5. SinceN  2, we are dealing with a quadratic algebraic approximant. Then, the recursive algorithmis started by aj  fj , j  0, . . . , 4. For a5, we obtaina5  −p1,1f4 	 p2,1(2f0f4 	 2f1f3 	 f22)	 p2,0(2f1f4 	 2f2f3)p1,0 	 2p2,0f04.1and for J > 5, we obtainaJ  −p1,1aJ−1 	 p2,1∑J−1k0 aJ−k−1ak 	 p2,0∑J−1k1 akaJ−kp1,0 	 2p2,0f0. 4.2ISRN Applied Mathematics 5Table 1: The case ofN  2, p0  p1  p2  1 for 4.3. Displayed are the coefficients of the Taylor series, thepredicted coefficients, and absolute and relative errors of the predicted coefficients.j fj aj |fj − aj | rel. error %5 −.294 −.294 .001 .186 −.330 −.332 .001 .387 −.389 −.392 .002 .588 −.475 −.478 .004 .769 −.593 −.599 .006 .9310 −.756 −.765 .008 1.10Forfz  2 − 3z1/2 	 15 − z , 4.3the HPPs are determined to beP0z  1. − 1.544503593423590 z,P1z  .1947992842134984 	 .06783822675080703 z,P2z  −.5044536972622500 − .01090573365920830 z.4.4The results for the predicted coefficients given in Table 1.Example 4.2. As a second example, we consider againN  2, p0  p1  p2  1, andM  5, butnow the functionfz  171 − 2z−1/3 	 z2 − z 4.5with the HPPsP0z  −49.52369318166839 − 6.946105600281359 z,P1z  1. 	 1.695055482965655 z,P2z  .1125387307324166 − .2732915349762758 z.4.6The results for the predicted coefficients given in Table 2.Example 4.3. As a final example, we consider the case N  p0  p1  p2  2, whence M  8,and the functionfz  expz 2 − 3 z−1/3 	 15 − z . 4.76 ISRN Applied MathematicsTable 2: The case ofN  2, p0  p1  p2  1 for 4.5. Displayed are the coefficients of the Taylor series, thepredicted coefficients, and absolute and relative errors of the predicted coefficients.j fj aj |fj − aj | rel. error %5 67.938 68.212 .274 .406 120.739 122.291 1.552 1.297 218.459 224.194 5.735 2.628 400.498 418.053 17.555 4.389 741.657 790.063 48.406 6.5310 1384.425 1509.437 125.012 9.03Table 3: The case ofN  2, p0  p1  p2  2 for 4.7. Displayed are the coefficients of the Taylor series, thepredicted coefficients, and absolute and relative errors of the predicted coefficients.j fj aj |fj − aj | rel. error %8 3.888956 3.878509 .010447 .279 5.356681 5.301047 .055634 1.0410 7.451679 7.275227 .176452 2.3711 10.447061 10.006950 .440111 4.2112 14.739132 13.781978 .957155 6.4913 20.903268 18.995972 1.907297 9.12The corresponding HPPs areP0z  1. − 1.027576803009053 z 	 .02070967420422950 z2,P1z  2.617867885747464 − .6563757889994458 z − 3.118191126500581 z2,P2z  −3.647182626894738 	 7.471780741166546 z − 3.356878399103086 z2.4.8The results for the predicted coefficients are displayed in Table 3.5. ConclusionsIt is seen that even rather low-order algebraic approximants, or HPPs, respectively, can leadto quite accurate predictions of the unknown coefficients of the power series, especially forfM, and the next few coefficients.References1 C. Brezinski and M. Redivo Zaglia, Extrapolation Methods. Theory and Practice, vol. 2 of Studies inComputational Mathematics, North-Holland Publishing Co., Amsterdam, The Netherlands, 1991.2 J. Gilewicz, “Numerical detection of the best Pade´ approximant and determination of the Fouriercoefficients of insufficiently sampled functions,” in Pade´ Approximants and Their Applications, P. R.Graves-Morris, Ed., pp. 99–103, Academic Press, New York, NY, USA, 1973.3 M. Pre´vost and D. Vekemans, “Partial Pade´ Prediction,”Numerical Algorithms, vol. 20, no. 1, pp. 23–50,1999.4 C. Brezinski, “Prediction properties of some extrapolation methods,” Applied Numerical Mathematics,vol. 1, no. 6, pp. 457–462, 1985.5 E. J. Weniger, “Prediction properties of Aitken’s iteratedΔ2 process, of Wynn’s epsilon algorithm, andof Brezinski’s iterated theta algorithm,” in Numerical Analysis, C. Brezinski, Ed., vol. 2 of Interpolationand Extrapolation, pp. 329–356, Elsevier, Amsterdam, The Netherlands, 2000.ISRN Applied Mathematics 76 A. Sidi and D. Levin, “Prediction properties of the t-transformation,” SIAM Journal on NumericalAnalysis, vol. 20, no. 3, pp. 589–598, 1983.7 D. Roy and R. Bhattacharya, “Prediction of unknown terms of a sequence and its application to somephysical problems,” Annals of Physics, vol. 321, no. 6, pp. 1483–1523, 2006.8 D. Vekemans, “Algorithm for the E-prediction,” Journal of Computational and Applied Mathematics, vol.85, no. 2, pp. 181–202, 1997.9 U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, “Resummation of QED perturbation series bysequence transformations and the prediction of perturbative coefficients,” Physical Review Letters, vol.85, no. 12, pp. 2446–2449, 2000.10 G. A. Baker, Jr. and P. Graves-Morris, Pade´ Approximants, vol. 59 of Encyclopedia of Mathematics and ItsApplications, Cambridge University Press, Cambridge, UK, 2nd edition, 1996.11 T. M. Feil and H. H. H. Homeier, “Programs for the approximation of real and imaginary single- andmulti-valued functions by means of Hermite-Pade´-approximants,” Computer Physics Communications,vol. 158, no. 2, pp. 124–135, 2004, Computer Physics Communications Program Library, Cataloguenumber: ADSO.

Series Prediction based on Algebraic Approximants

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