# Bibliography

References

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Wynn, P. (1961b), "L'e-algoritmo e la tavola di Pade," Rendu. Mat. Roma, 20, 403-408.

Wynn, P. (1962a), "The numerical efficiency of certain continued fraction expansions," Proc. Kon. Ned. Akad. Wetensch. A, 65, 127-154.

Wynn, P. (1962b), "Acceleration techniques for iterated vector and matrix problems," Math. Comp., 16, 301-322.

Wynn, P. (1963), "Continued fractions whose coefficients obey a non-commutative law of multiplication," Arch. Rat. Mech. Anal., 12, 273-312.

Wynn, P. (1964), "General purpose vector epsilon algorithm procedures," Num. Math., 6, 22-36.

Wynn, P. (1966a), "Upon systems of recursions which obtain among quotients of the Pade table," Num. Math., 8, 264-269.

Wynn, P. (1966b), "On the convergence and stability of the e-algorithm," SI AM J. Num. Anal., 3, 91-122.

Wynn, P. (1967), "A general system of orthogonal polynomials," Quart. J. Math., 18, 81-96.

Wynn, P. (1968), "Upon the Pade table derived from a Stieltjes series," SI A M J. Num. A nal., 5, 805-834.

Wynn, P. (1973), "On the zeros of certain confluent hypergeometric functions," Proc. Am. Math. Soc., 40, 173-182.

Wynn, P. (1974), "Some recent developments in the theories of continued fractions and the Pade table," Rocky Mm. J. Math., 4, 297-323.

Wynn, P. (1977), "The transformation of series by the use of Pade quotients and more general approximants." in E. B. Saff and R. S. Varga (eds.), Pade and Rational Approximation, Academic Press, New York, pp. 121-146.

Young, R. C., Biedenharn, L. C., and Feenberg, E. (1957), "Continued fraction approximants to the Brillouin-Wigner Perturbation series," Phys. Rev., 106, 1151-1155.

Zinn-Justin, J. (1970), "Strong interaction dynamics with Pade approximants," Phys. Repts., 1C, 56-102.

Zinn-Justin, J. (1971), "Convergence of Pade approximants in the general case," in A. Visconti (ed.), Colloquium on Advanced Computing Methods in Theoretical Physics, CNRS, Marseille.

Zinn-Justin, J. (1973), "Recent developments in the theory of Pade approximants," in A. Visconti (ed.), International Colloquium on Advanced Computing Methods in Theoretical Physics, Vol. 2, CNRS, Marseille, p. C-XIII-1.

Zohar, S. (1974), "The solution of a Toeplitz set of linear equations," J. Assoc. Comp. Mech., 21, 272-276.

Bibliography of General Pade-Approximant Reviews

Baker, G. A., Jr. (1965), "The theory and application of the Pade approximant method," Advances in Theoretical Physics, 1, 1-58.

Baker, G. A., Jr. (1975), The Essentials of Padé Approximants, Academic Press, New York.

Baker, G. A., Jr. and Gammel, J. L., eds. (1970), The Padé Approximant in Theoretical Physics, Academic Press, London.

Basdevant, J.-L. (1972), "The Padé approximant and its physical applications," Fortschr. Phys., 20, 282-331.

Bausset, M., ed. (1973), Accélération des Convergences, Centre Universitaire de Toulon.

Bessis, D., ed. (1972), Cargise Lectures in Physics, Vol. 5, Gordon and Breach, New York.

Bessis, D., Gilewicz, J., and Mery, P., eds. (1975), Proceedings of a Workshop on Padé Approximants, C.N.R.S., Marseille.

Brezinski, C. (1977), Accélération de la Convergence en Analyse Numérique, Springer Lecture Notes in Mathematics, No. 584.

Brezinski, C. (1980), "Padé type approximation and general orthogonal polynomials," Birkhäuser Verlag.

Cabannes, H., ed. (1976), Padé Approximant Method and its Application to Mechanics (Proceedings of the Euromech Colloquium, Toulon, 1975), Springer Lecture Notes in Physics, No. 47.

Chui, C. K. (1976), "Recent results on Padé approximants and related problems," in G. G. Lorentz, C. K. Chui and L. L. Schumaker (eds.), Approximation Theory, Academic Press, New York, pp. 79-116.

Ciaessens, G. (1976), "Some aspects of the rational Hermite interpolation table and its applications," Ph. D. Thesis, University of Antwerp.

Gilewicz, J. (1978), Approximants de Padé, Springer Lecture Notes in Mathematics, No. 667.

Gragg, W. B. (1972), "The Padé table and its relation to certain algorithms of numerical analysis," SI AM Review, 14, 1-62.

Graves-Morris, P. R., ed. (1973), Padé Approximants and Their Applications, Academic Press, London.

Graves-Morris, P. R., ed. (1973), Padé Approximants, Inst, of Phys. Press, London.

Jones, W. B. and Thron, W. J. (1974), "Proceedings of the international conference on Padé approximants, continued fractions and related topics," Rocky Mtn. J. Math., 4, 135-397.

Padé, H. ( 1892), "Sur la representation approchée d'une fonction par des fractions rationelles," Annales de l'Ecole Normale, 9, suppl. 3, 93.

Saff, E. B. and Varga, R. S., eds. (1977), Padé and Rational Interpolation, Academic Press, New York.

Warner, D. D. (1974), "Hermite interpolation with rational functions," Ph. D. Thesis, University of California.

Wimp, J. (1981), "Sequence Transformations and Their Applications," Academic Press, New York.

Wuytack, L. (1976), "Applications of Padé approximation in numerical analysis," in Schaback, R. and Scherer, K. (eds.), Approximation Theory, Springer Lecture Notes in Mathematics, Vol. 556, pp. 453-461.

Wuytack, L., ed. (1979), Padé Approximation and Its Applications, Springer Lecture Notes in Mathematics, Vol. 765.

Zinn-Justin, J. (1970), "Strong interaction dynamics with Padé approximants," Phys. Reports, 1C, 56-102.

Selected bibliography on Continued Fractions and Method of Moments

Akhiezer, N. I. (1965), The Classical Moment Problem, Oliver and Boyd, London.

Jones, W. B. and Thron, W. J. (1980), "Continued fractions: Analytic theory and applications," in G.-C. Rota (ed.), Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass.

Khovanskii, A. N. (1963), Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, Noordhoff.

Perron, O. (1957), Die Lehre von der Kettenbrüchen, Vol. 2, Teubner B. G., Stuttgart.

Shohat, J. A. and Tamarkin, J. D. (1963), The Problem of Moments, American Math. Soc. Publ., Vol. 1, Providence, R.I.

Vorobyev, Yu. V. (1965), Method of Moments in Applied Mathematics, Gordon and Breach, New York.

Wall, H. S. (1929), "On the Pade approximants associated with the continued fraction and series of Stieltjes," Trans. Am. Math. Soc., 31, 91-115.

Wall, H. S. (1948), The Analytic Theory of Continued Fractions, Van Nostrand, Princeton.

Index for Part I and Part II

Acceleration of convergence, 1:16-17, 69-90 Accuracy, numerical, 1:58-65 Accuracy-through-order, 1:1-2 Aitken's A2 method, 1:69-74 rj-Algorithm, 1:80-84 e-Algorithm, 1:76-80, 84-90 generalized, II: 14-16 for vector sequences, 11:55 Algorithms for rational interpolation, 11:6-17 Anharmonic oscillator, II: 170— 171 Arzela's theorem, 1:175 Associated continued fraction, 1:127 /(-stability, 11:151 Asymptotic convergence, 1:200 Asymptotic expansion, 1:221

Backward recurrence method, 1:115 Baker algorithm, 1:66 Baker definition, 1:21-22; 11:52 Baker-Gammel approximants, 11:21 —32, 63

asymptotic, 11:30-31 Baker, Gammel and Wills, theorem of, 1:32-33 conjecture of, 1:286 Beardon's theorems, 1:237 Behte-Salpeter equation, II: 108-113 Bigradients, 1:37-42, 136-137

polynomial, 1:39 Binomial function, 1:139, 144 Biorthogonal algorithm, 11:76-79 Blocks, 1:24-31

Calculation of Padé approximants, algebraic, 1:8-14, 43-48, 66-68 numerical, 1:61-68 Canterbury approximants, 11:41-45 Capacity, 1:274-284 Cardioid theorem, 1:151 Carleman's criterion, application of, 11:169

for Hamburger series, 1:223 for Stieltjes series, 1:200-202 Cartan's lemma, 1:274 Cauchy-Binet formula, 1:239-240 Cauchy-Jacobi problem, II: 1 C-fraction, 1:127

Characteristic function, 11:128-132 Chebychev, see Tchebyscheff Chisholm approximants, 11:41-43 Chordal metric, 1:256-263 Clenshaw-Lord algorithm, 11:58-60 C(L/M), definition of, 1:21 Coefficient problem, 1:65; 11:7 Collocation, 11:138-145 Complementary error function, 1:140, 144 Continued fractions, 1:103-157 associated, 1:127 contraction of, 1:127, 147 convergents of, see convergents, corresponding, 1:127 definition, 1:104-105, 107 divergence condition for, 1:148 elements of, 1:104, 119 equivalence transformation of, 1:105-107 periodic, 1:117, 138

recurrence relations for, 1:106, 109, 135 regular corresponding, 1:127 repeating, 1:117-138 summation formula, 1:115-116 terminating, 1:104 Convergence, in measure, 1:263-274 in capacity, 1:274-284 of row sequences, 1:238, 261 Convergents, 1:104, 111-112, 117-119 Cordellier's identity, 1:88 Crank-Nicholson methods, II: 145-153 Critical point methods, 1:55-57, 59-61:

11:32-40, 178 C-table, 1:22-23, 29, 31

Dawson integral, 1:141

generalized, 11:19 Defect, 1:53-55, 58-59 Deficiency index, 1:20 de Montessus's theorem, 1:241-252

generalization of, 1:254: 11:161 Density function, construction of, 1:180-181 Determinancy of moment problem, 1:17, 179,

193, 197-207 Determinantal formulas for Pade approxi-

mants, 1:4-8. 43-48 Determinantal identities for Stieltjes series

coefficients, 1:165-166 Determinantal inequalities, 1:227, 235 for Hamburger series coefficients, 1:227, 235

for Stieltjes series coefficients, 1:208, 209 Diagonal sequences, 1:236 Diffusion processes, II: 145-153 Divergence condition, 1:148, 156-157 Divided differences, 11:2 D-log Pade approximants, 1:55; 11:33 D(m, n), definition of, 1:162 Duality, 1:31, 236

Essential singularity, 1:49 Equicontinuity, 1:174-176 Equicontinuous sequences, 1:210, 261 Equivalence transformation, 1:105-106 Error formula for Pade approximants of Hamburger series, 11:128-129 Stieltjes, 1:185, 189-193; 11:128-132 Error formula for Pade approximation, 1:6, 250

from variational principles, 11:103-106 Error function, 1:141, 11:19-20 Euclidean algorithm, 1:66, 134-135 Euler's corresponding fraction, 1:137-139 Euler's function and series, 1:17-18, 200-202 Euler-Maclaurin sum formula, 11:130 Euler's recurrence theorem, 1:106 Euler's summation formula, 1:115-116 Existence of Pade approximants, 1:27 Exploding vacuum, 11:166-167 Exponential approximants, 11:28 Exponential function, Continued fractions for, 1:139-143 Pade approximants for, 1:8-14 Saff-Varga theorems for, 1:229-233 Exponential integral, continued fraction for, 1:140; 144, 11:164- 165

first order, 1:186 Extended Stieltjes series, see Hamburger series.

Factorization, 11:43 Fisher approximants, 11:45 Forward recurrence method, 1:114-115 Frobenius definition, 1:20, 22 Frobenius identities, 1:85, 90-93

Gamma function, Binet's formula for, 1:202— 205

Gammel - Guttman - Gaunt - Joyce approximants, 11:33-36 Gammel's counterexample, 1:285 Gaussian quadrature, 11:127-132 General C-fraction, 1:129 Gnomic theory, 11:74 Green's functions, bound-state, 11:113 partial wave, free, 11:89 Jost solution, 11:90 momentum space, 11:114 standing wave, free, 11:89 S-wave, exponential potential, 11:98 three-dimensional, free, 11:81

relativistic, free, II: 108 relativistic, standing wave, free, 11:109 standing wave, free, 11:85 Gregory's series, 1:78, 81

Hadamard's determinant theorem, 1:39-42, 243

Hamburger functions as real J-fractions, 1:225 Hamburger moment, definition of, 1:208, 222 Hamburger series, definition of, 1:208, 222 Hamburger's theorem, 1:221 Hankel determinant, 1:7, 41 Hankel matrix, condition number of, 1:64-65 Hardy's puzzle, 1:78-80 Hausdorff measure, a-dimensional, 1:283 Hausdorff moment problem, 1:193-196 Herglotz functions, 1:225-227 Hermite's formula, 1:250; 11:2 High field expansions, 11:179 Hilbert space methods, 11:46, 68-72, 103-107 Homographic invariance, 1:32-33, 11:42-43, 45, 53

Hughes Jones approximants, 11:43-45 Hyperbolic tangent, 1:139, 143-144

Hypergeometric functions, 1:141-147, 197- 198

2F{( ■), Pade approximants of, 1:43-47,

147, 156

iFi( ), Pade approximants of, 1:43, 47,

147, 154, 11:19

2 /-;>( ■), Pade approximants of, 1:43, 47, 141-147, 154-155

Identities for neighboring approximants,

1:90-96 Inclusion regions, 1:171, 206 Incomplete Gamma function, 1:140, 141, 145:

11:19 Inequalities, for density function, II: 132-138 for moments, 1:186 Integer moment problem, 1:196-197 Integral equations, 11:64-79 Interlacing, 1:168-169, 210, 227; 11:135 Invariance,'homographic, 1:32-33; 11:42-43, 45, 53

Inverse hyperbolic tangent, 1:140, 144 Inverse tangent, 1:139, 144; 11:36-37

7-fraction, 1:128 Jost method, 11:121-122

Kernels, compact, 11:67-72 completely continuous, 11:67-72 finite rank, 11:65-67, 72 ^-matrix, 11:85, 107, 110-112 Kronecker's algorithm, 1:66; 11:7

Laguerre's method, II: 165 Laguerre polynomials, 1:186 Lanczos biorthogonal method, 11:76 Lanczos r-method, 11:138-145 Laplace transform, inversion of, II: 153-155 Laurent's theorem, 1:49 Lattice-cutoff field theory, 11:176-178 Lemniscates, 1:274-284 Le Roy function, 11:32 Lippman-Schwinger equation, 11:82-83, 1 13 Logarithmic capacity, 1:283 /.-stability, 11:151

Markov problem, II: 137

Matrix Pade approximants, 11:50-56

Measure, convergence in, 1:263

Meromorphic functions, convergence of sequences of, 1:259 Mittag-Leffler star, 1:50 Moment bounds, 1:186 Moment, definition of Hamburger, 1:208, 222 definition of Hausdorff, 1:195 definition of Stieltjes, 1:158 Moment method, 11:46 Moment problems, 1:178-179 Moments,

bounds for, 1:186; 11:132-138 Multi-index, 1:239

Multipoint Pade approximation, 1:254; 11:1 — 31

Multipole, 1:49

Multivariable approximants, 11:40-50

Natural logarithm, 1:140, 144 Newton interpolating polynomials, 11:2-3 Newton-Pade approximants, II: 1-31 V-point Pade approximants, 11:1-31 Nuttall's compact form, 1:16 Nuttall's theorem, 1:269

Orthogonal polynomials, 1:82-86, 209-210, 255-256

Osculatory rational interpolation, II: 1 -31

Pade approximant, 1:1-288; 11:1-180 Pade-Borel approximation, 11:29-30 Pade denominator, definition of, 1:4 Pade equations, 1:2-3 Pade-Fourier approximants, 11:62-63 Pade-Frobenius definition, 1:20, 22 Pade-Legendre series, 11:21-29 Pade numerator, definition of, 1:6 Pade table, 1:7-8, 27-31 Pade-Tchebycheff approximants, 11:56-62 Parabola theorem for continued fractions, 1:150

Parabola theorem of Saff and Varga, 1:229— 232

Paradiagonal sequences, 1:236 Peres model, II: 167 Perron's counterexample, 1:238 P-fraction, 1:130-131 Pion-Pion scattering, II: 172— 176 Poles and zeros of Pade approximants, 1:31, 49-59, 96-102, 126, 166, 263 for Hamburger functions, 1:209, 227 for Stieltjes functions, 1:186, 193

Poloids, 11:175

Polya frequency series, 1:227-235 Pommerenke's theorem, 1:271 Potential scattering, 11:79-126 Projection techniques, 11:72-79 Prong method, 11:43-44 Prym's function, 1:140, 147

Q. D. algorithm, 1:99, 125-126 generalized, 11:13-14 for T-fractions, 11:20 Q. D. Table, 1:99, 125-126 Quadratic approximants, 11:36-37, 49 Quadrature, 11:127-132 Quantum theory, connection with, 11:79-126 Quasi-analytic functions, 1:51

Ratio method, 11:33

Rational approximation, 11:155-162

Rational interpolation, 11:1-31

Ray sequences, 1:191

Real 7-fractions, 1:128, 225

Real symmetric functions, 1:160-161, 180

Regular C-fraction, 1:127

Reliability, 1:63, 132; 11:7

Rhombus rule, 1:77, 81

Riccati equation, Pade approximants for,

11:162-165 Riemann sphere, 1:257 Root problem, 1:96-102 Rouche's theorem, 1:279 Runge's theorem, 11:158-159

Saffs theorem, 1:254

Scattering theory, quantum mechanical,

11:80-92 Schwarz's lemma, 1:189, 192 Sectorial theorem of Saff and Varga, 1:229 Seidel's theorem, 1:148-150 Sequence,

acceleration of convergence of, 1:16-17,

69-90 column, 1:30 diagonal, 1:17, 32, 35 paradiagonal, 1:30 row, 1:30

Series, acceleration of convergence of,

1:16-17, 69-90 Series analysis, 1:55-57, 59-61; 11:32-40

S-fraction, 1:127-128, 165, 206 Shafer approximants, 11:36-37 Single-sign potentials, II: 106-114 Singular potentials, 11:120-126 S-matrix, partial wave, 11:89, 91 Spherical convergence, 1:256-263 Star Identity, 1:23 Stieltjes function, 1:158, 208 definition of, 1:158, 221 numerical calculation of, 1:64 Stieltjes inversion formula, 1:221 Stieltjes series, 1:158, 208 definition of, 1:158, 221 inequalities for Pade approximants of, 1:170 example of, 1:161 ,S'-fraction for, 1:165 Sturm sequence, 1:167 Sylvester's theorem, 1:23, 167

Tangent function, 1:139, 143 Tchebycheff inequalities, density function for,

11:132-138 Tchebycheff polynomials, 1:275-279 7"-fractions, 1:138; 11:17-21 Thacher-Tukey algorithm, 11:11-13 Thiele's reciprocal difference method, 11:9-11 T-matrix, 11:81-82, 103, 172-176 partial wave, 11:89 Toeplitz matrix, 1:67 Totally monotone sequence, 1:195 Totally positive series, 1:228 Trudi's theorem, 1:42 Truncation theorem, 1:34-35

Uniform boundedness convergence criterion, 1:262

Unitarity, 1:35-36; 11:43, 54

Value problem, 1:65; 11:7 Variational Pade approximants, 11:114-120 Variational principle, 11:54-55 Rayleigh-Ritz, 11:92-96 Kohn, 11:96-98, 103 Schwinger, 11:96, 99-100, 116-119 Bessis, 11:100-103 Vandermonde determinants, 1:243-244, 249 Van Vleck's theorem, 1:152 Viskovatov's method, 1:113, 129-137

Walsh theorems, 11:161-162 Watson algorithm, 1:67 Weierstrass's (approximation) theorem, 11:68 Weierstrass's (double series) theorem, 1:176 Weierstrass's (essential singularity) theorem, 1:51

Werner's algorithm, 1:131 Wronskian, 11:87 Wynn's identity, 1:78

Zeros of functions, numerical methods for. 11:16