Numerical Evaluation of Continued Fractions
Previous article Next article Numerical Evaluation of Continued FractionsG. BlanchG. Blanchhttps://doi.org/10.1137/1006092PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Leo A. Aroian, Continued fractions for the incomplete Beta function, Ann. Math. Statistics, 12 (1941), 218–223 MR0005193 0025.31901 CrossrefGoogle Scholar[2] Friedrich L. Bauer, The quotient-difference and epsilon algorithms, Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959x+462 MR0102594 Google Scholar[3] Gertrude Blanch, On the computation of Mathieu functions, J. Math. Phys. Mass. Inst. Tech., 25 (1946), 1–20 MR0016690 0061.27702 ISIGoogle Scholar[4] Gertrude Blanch and , Ida Rhodes, Table of characteristic values of Mathieu's equation for large values of the parameter, J. Washington Acad. Sci., 45 (1955), 166–196 MR0071144 Google Scholar[5] Christoffel Jacob Bouwkamp, Theoretische en Numerieke Behandeling van de Buiging door een Ronde Opening, Dissertation, University of Groningen, 1941, 60–, Groningen, Batavia MR0017632 Google Scholar[6] British Assn. for the Adv. Sci., Mathematical Tables, Bessel Functions, Vol. X, Cambridge University Press, 1952, Part II. (Introduction by J. C. P. Miller) 0049.36204 Google Scholar[7] Evelyn Frank, Corresponding type continued fractions, Amer. J. Math., 68 (1946), 89–108 MR0015533 0060.16607 CrossrefISIGoogle Scholar[8] Evelyn Frank, On continued fraction expansions for binomial quadratic surds, Numer. Math., 4 (1962), 85–95, pp. 303–307; 5 (1963), pp. 113–117 10.1007/BF01386298 MR0140181 0107.10303 CrossrefGoogle Scholar[9] E. D. Hellinger and , H. S. Wall, Contributions to the analytic theory of continued fractions and infinite matrices, Ann. of Math. (2), 44 (1943), 103–127 MR0008102 0060.16502 CrossrefGoogle Scholar[10] R. E. Lane, Interpolation by means of continued fractions, Fraternal Actuarial Assoc. Proc., 19 (1944-46), Google Scholar[11] Walter Leighton and , W. J. Thron, On value regions of continued fractions, Bull. Amer. Math. Soc., 48 (1942), 917–920 MR0007069 0060.16405 CrossrefGoogle Scholar[12] Walter Leighton and , W. J. Thron, On the convergence of continued fractions to meromorphic functions, Ann. of Math. (2), 44 (1943), 80–89 MR0007805 0060.16404 CrossrefGoogle Scholar[13] Yudell L. Luke, The Padé table and the $\tau$-method, J. Math. Phys., 37 (1958), 110–127 MR0099114 0084.34604 CrossrefGoogle Scholar[14] Nathaniel Macon and , Margaret Baskervill, On the generation of errors in the digital evaluation of continued fractions, J. Assoc. Comput. Mach., 3 (1956), 199–202 MR0080987 CrossrefISIGoogle Scholar[15] A. A. Markoff, Masters Thesis, On certain applications of algebraic continued fractions, Thesis, St. Petersburg, 1884 Google Scholar[16] J. H. Müller, On the application of continued fractions to the evaluation of certain integrals, with special reference to the incomplete Beta function, Biometrika, 22 (1920-1), 284–297 CrossrefGoogle Scholar[17] National Bureau of Standards, Tables relating to Mathieu functions, Columbia University Press, New York, 1951 0045.40109 Google Scholar[18] Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., New York, N. Y., 1950xii+524 MR0037384 0041.18206 Google Scholar[19] A. Pringsheim, Ueber die Konvergenzkreiterien für Kettenbrüche mit komplexem Gliedern, Sitzungsber. der Math-Phys. Klasse der Kgl. Bayer. Akad. Wiss., München, 35 (1905), 359–380 Google Scholar[20] R. D. Richtmyer, , Marjorie Devaney and , N. Metropolis, Continued fraction expansions of algebraic numbers, Numer. Math., 4 (1962), 68–84 10.1007/BF01386297 MR0136574 0101.28101 CrossrefGoogle Scholar[21] Heinz Rutishauser, Der Quotienten-Differenzen-Algorithmus, Mitt. Inst. Angew. Math., ETH, 7, Birkhäuser Verlags, Basel, 1956 0077.11103 Google Scholar[22] Heinz Rutishauser, Stabile Sonderfälle des Quotienten-Differenzen-Algorithmus, Numer. Math., 5 (1963), 95–112 10.1007/BF01385882 MR0175288 0196.48404 CrossrefGoogle Scholar[23] H. Schwerdtfeger, Moebius transformations and continued fractions, Bull. Amer. Math. Soc., 52 (1946), 307–309 MR0015532 0060.16202 CrossrefISIGoogle Scholar[24] W. T. Scott and , H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc., 47 (1940), 155–172 MR0001320 0022.32603 CrossrefGoogle Scholar[25] Kurt Spielberg, Representation of power series in terms of polynomials, rational approximations and continued fractions, J. Assoc. Comput. Mach., 8 (1961), 613–627 MR0130102 0105.11204 CrossrefGoogle Scholar[26] Kurt Spielberg, Polynomial and continued-fraction approximations for logarithmic functions, Math. Comp., 16 (1962), 205–217 MR0145645 0105.11205 CrossrefGoogle Scholar[27] Irene Stegun and , Milton Abramowitz, Generation of Bessel functions on high speed computers, Math. Tables Aids Comput, 11 (1957), 255–257 MR0093939 0084.12101 CrossrefGoogle Scholar[28] T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), J1–J122 MR1508159 Google Scholar[29] Jerome D. Swalen and , Louis Pierce, Remarks on the continued fraction calculations of eigenvalues and eigenvectors, J. Mathematical Phys., 2 (1961), 736–739 10.1063/1.1703766 MR0130109 0108.12503 CrossrefISIGoogle Scholar[30] O. Szász, Ueber die Erhaltung bei independenter Veränderlichkeit aller ihrer Elemente, J. für Math., 147 (1916), 132–160 Google Scholar[31] P. L. Tschebycheff, Sur les fractions continues, Jour. de Math., 8 (1858), 289–323 Google Scholar[32] D. Teichroew, Use of continued fractions in high speed computing, Math. Tables and Other Aids to Computation, 6 (1952), 127–133 MR0049650 0047.12002 CrossrefGoogle Scholar[33] W. J. Thron, Two families of twin convergence regions for continued fractions, Duke Math. J., 10 (1943), 677–685 10.1215/S0012-7094-43-01063-4 MR0009214 0060.16407 CrossrefGoogle Scholar[34] W. J. Thron, Zwillingskonvergenzgebiete für Kettenbrüche $1+K(a\sb{n}/1)$, deren eines die Kreisscheibe $\vert a\sb{2n-1}\vert \leq \rho \sp{2}$ ist, Math. Z., 70 (1958/1959), 310–344 10.1007/BF01558596 MR0105488 0085.05002 CrossrefGoogle Scholar[35] W. J. Thron, Convergence regions for continued fractions and other infinite processes, Amer. Math. Monthly, 68 (1961), 734–750 MR0133444 0103.28304 CrossrefISIGoogle Scholar[36] Edward B. Van Vleck, On the convergence of the continued fraction of Gauss and other continued fractions, Ann. of Math. (2), 3 (1901/02), 1–18 MR1502271 CrossrefGoogle Scholar[37] H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948xiii+433 MR0025596 0035.03601 Google Scholar[38] H. S. Wall, Continued fractions and totally monotone sequences, Trans. Amer. Math. Soc., 48 (1940), 165–184 MR0002642 0024.21602 CrossrefGoogle Scholar[39] H. S. Wall, A theorem on arbitrary J-fractions, Bull. Amer. Math. Soc., 52 (1946), 671–679 MR0017395 0060.16604 CrossrefISIGoogle Scholar[40] H. S. Wall and , Marion Wetzel, Contributions to the analytic theory of J-fractions, Tran. Amer. Math. Soc., 55 (1944), 373–392 MR0011339 0060.16503 Google Scholar[41] Marion Wetzel, Applications of continued fractions, U.S. Army Tech. Report, 5, Office of Ordnance Research, 1954, Contract DA-04-200-ORD-177, August Google Scholar[42] Arthur Wouk, Difference equations and J-matrices, Duke Math. J., 20 (1953), 141–159 10.1215/S0012-7094-53-02014-6 MR0058111 0051.07201 CrossrefISIGoogle Scholar[43] P. Wynn, Converging factors for continued fractions. I, II, Numer. Math., 1 (1959), 272–320 10.1007/BF01386391 MR0116158 0092.05101 CrossrefGoogle Scholar[44] P. Wynn, The rational approximation of functions which are formally defined by a power series expansion, Math. Comput., 14 (1960), 147–186 MR0116457 0173.18803 CrossrefGoogle Scholar[45] P. Wynn, A comparison technique for the numerical transformation of slowly convergent series based on the use of rational functions, Numer. Math., 4 (1962), 8–14 10.1007/BF01386291 MR0136500 0138.09901 CrossrefGoogle Scholar[46] P. Wynn, The numerical efficiency of certain continued fraction expansions. IB, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math., 24 (1962), 138–148 l MR0139255 0105.10003 Google Scholar[47A] Alexey Nikolaevitch Khovanskii, The application of continued fractions and their generalizations to problems in approximation theory, Translated by Peter Wynn, P. Noordhoff N. V., Groningen, 1963xii + 212 MR0156126 0106.27204 Google Scholar[47B] A. Ya. Khintchine, Continued fractions, Translated by Peter Wynn, P. Noordhoff Ltd., Groningen, 1963iii+101 MR0161834 0117.28503 Google Scholar[48] E. G. Kogbetliantz, Computation of ${\rm Sin}\ N$${\rm Cos}\ N$ and $\root m\of N$ using an electronic computer, IBM J. Res. Develop., 3 (1959), 147–152 MR0102170 CrossrefISIGoogle Scholar[49] E. G. Kogbetliantz, Anthony Ralston and , Herbert S. Wilf, Generation of elementary functionsMathematical methods for digital computers, Wiley, New York, 1960, 7–35 MR0117907 Google Scholar[50] H. J. Maehly, Rational approximations for transcendental functionsInformation processing, UNESCO, Paris, 1960, 57–62 MR0148206 0112.35303 Google Scholar[51A] Hans J. Maehly, Methods for fitting rational approximations. I. Telescoping procedures for continued fractions, J. Assoc. Comput. Mach., 7 (1960), 150–162 MR0116455 0113.32502 CrossrefISIGoogle Scholar[51B] Hans J. Maehly, Methods for fitting rational approximations. II, III, J. Assoc. Comput. Mach., 10 (1963), 257–277 MR0157474 0113.32502 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Emergence of new category of continued fractions from the Sturm–Liouville problem and the Schrödinger equation30 July 2021 | São Paulo Journal of Mathematical Sciences, Vol. 15, No. 2 Cross Ref Computation and Applications of Mathieu Functions: A Historical PerspectiveChris Brimacombe, Robert M. Corless, and Mair Zamir4 November 2021 | SIAM Review, Vol. 63, No. 4AbstractPDF (29320 KB)Computational methods for birth‐death processes2 January 2018 | WIREs Computational Statistics, Vol. 10, No. 2 Cross Ref Some Sets of Relative Stability Under Perturbations of Branched Continued Fractions with Complex Elements and a Variable Number of Branches5 April 2016 | Journal of Mathematical Sciences, Vol. 215, No. 1 Cross Ref Mathematical Functions, Approximations to29 September 2014 Cross Ref Estimation for General Birth-Death Processes13 June 2014 | Journal of the American Statistical Association, Vol. 109, No. 506 Cross Ref Predicting the molecular complexity of sequencing libraries24 February 2013 | Nature Methods, Vol. 10, No. 4 Cross Ref Transition probabilities for general birth–death processes with applications in ecology, genetics, and evolution9 October 2011 | Journal of Mathematical Biology, Vol. 65, No. 3 Cross Ref An efficient numerical parameter estimation scheme for the space-dependent dispersion coefficient of a solute transport equation in porous mediaMathematical and Computer Modelling, Vol. 51, No. 3-4 Cross Ref A Laplace transform finite difference method for the Black–Scholes equationMathematical and Computer Modelling, Vol. 51, No. 3-4 Cross Ref Analytical solution for pulsatile viscous flow in a straight elliptic annulus and application to the motion of the cerebrospinal fluidPhysics of Fluids, Vol. 20, No. 9 Cross Ref Algorithm 855ACM Transactions on Mathematical Software, Vol. 32, No. 3 Cross Ref Mathematical Functions, Approximations to15 August 2006 Cross Ref Adaptive sampling applied to multivariate, multiple output rational interpolation models with application to microwave circuits19 June 2002 | International Journal of RF and Microwave Computer-Aided Engineering, Vol. 12, No. 4 Cross Ref Creating accurate multivariate rational interpolation models of microwave circuits by using efficient adaptive sampling to minimize the number of computational electromagnetic analysesIEEE Transactions on Microwave Theory and Techniques, Vol. 49, No. 8 Cross Ref Evaluating Padé Approximants of the Matrix LogarithmNicholas J. Higham31 July 2006 | SIAM Journal on Matrix Analysis and Applications, Vol. 22, No. 4AbstractPDF (144 KB)Algorithms for the computation of all Mathieu functions of integer ordersACM Transactions on Mathematical Software, Vol. 26, No. 3 Cross Ref An implementation of a Fourier series method for the numerical inversion of the Laplace transformACM Transactions on Mathematical Software, Vol. 25, No. 3 Cross Ref A Complete Method for the Computations of Mathieu Characteristic Numbers of Integer OrdersFayez A. Alhargan12 July 2006 | SIAM Review, Vol. 38, No. 2AbstractPDF (1505 KB)On the stability of branching continued fractionsJournal of Mathematical Sciences, Vol. 79, No. 6 Cross Ref A Survey of Truncation Error Analysis for Padé and Continued Fraction Approximants Cross Ref A survey of truncation error analysis for Pad� and continued fraction approximantsActa Applicandae Mathematicae, Vol. 33, No. 2-3 Cross Ref The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger orderACM Transactions on Mathematical Software, Vol. 19, No. 3 Cross Ref Chebyshev approximations for cos πχ2 and sin πχ2 with applicationsApplied Numerical Mathematics, Vol. 6, No. 3 Cross Ref Truncation error bounds for modified continued fractions with applications to special functionsNumerische Mathematik, Vol. 55, No. 3 Cross Ref Algorithms for the evaluation of Bessel functions of complex argument and integer ordersApplied Mathematics Letters, Vol. 2, No. 4 Cross Ref Continued fractions in numerical analysisApplied Numerical Mathematics, Vol. 4, No. 2-4 Cross Ref Chapter I: Continued Fractions Cross Ref Free-boundary stability of straight stellaratorsPhysics of Fluids, Vol. 27, No. 10 Cross Ref A uniformly convergent alternative to the davidson-cole distribution functionAdvances in Molecular Relaxation and Interaction Processes, Vol. 24, No. 1 Cross Ref COULFG: Coulomb and Bessel functions and their derivatives, for real arguments, by Steed's methodComputer Physics Communications, Vol. 27, No. 2 Cross Ref Continued-fraction evaluation of Coulomb functions Fλ(η, x), Gλ(η, x) and their derivativesJournal of Computational Physics, Vol. 46, No. 2 Cross Ref High-precision evaluation of the regular and irregular Coulomb wavefunctionsJournal of Computational and Applied Mathematics, Vol. 8, No. 1 Cross Ref A quotient-difference algorithm for the determination of eigenvalues of periodic tridiagonal matricesComputers & Mathematics with Applications, Vol. 8, No. 2 Cross Ref An asymmetric form of the discrete Schrodinger equation with application to the inverse tunnelling problem1 January 1999 | Journal of Physics A: Mathematical and General, Vol. 14, No. 9 Cross Ref Evaluation and Inversion of the Ratios of Modified Bessel Functions, I1 ( x ) / I0 ( x ) and I1.5 ( x )/ I0.5 ( x )ACM Transactions on Mathematical Software, Vol. 7, No. 2 Cross Ref A continued fraction algorithmNumerische Mathematik, Vol. 37, No. 1 Cross Ref An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracyComputer Physics Communications, Vol. 21, No. 3 Cross Ref On Temme's Algorithm for the Modified Bessel Function of the Third KindACM Transactions on Mathematical Software, Vol. 6, No. 4 Cross Ref Evaluation of infinite series by use of continued fraction expansions: A numerical studyJournal of Computational Physics, Vol. 37, No. 2 Cross Ref On the relevance of Fuchsian differential equations for the propagation of waves in layered plasmas7 December 2012 | Radio Science, Vol. 15, No. 1 Cross Ref Gentle diversionsIEEE Communications Magazine, Vol. 17, No. 1 Cross Ref An algorithm for calculating continued fractionsJournal of Computational and Applied Mathematics, Vol. 3, No. 4 Cross Ref On a posteriori error estimates1 January 1977 | Mathematics of Computation, Vol. 31, No. 137 Cross Ref Estimates of the speed of convergence of continued fraction expansions of functions1 January 1977 | Mathematics of Computation, Vol. 31, No. 138 Cross Ref Investigation of algorithms for numerical computation of continued fractionsUSSR Computational Mathematics and Mathematical Physics, Vol. 16, No. 4 Cross Ref Surface currents on perfectly conducting elliptic cylindersIEEE Transactions on Antennas and Propagation, Vol. 23, No. 2 Cross Ref Computational Methods in Special Functions-A Survey Cross Ref Numerical stability in evaluating continued fractions1 January 1974 | Mathematics of Computation, Vol. 28, No. 127 Cross Ref An algebraic method for simulating legal systems19 August 2016 | SIMULATION, Vol. 21, No. 5 Cross Ref Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinderApplied Scientific Research, Vol. 28, No. 1 Cross Ref On the computation of Mathieu functionsJournal of Engineering Mathematics, Vol. 7, No. 1 Cross Ref A priori estimates for truncation error of continued fractionsK(1/b n )Numerische Mathematik, Vol. 19, No. 4 Cross Ref A Posteriori Bounds for the Truncation Error of Continued FractionsWilliam B. Jones and W. J. Thron3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 8, No. 4AbstractPDF (905 KB)Using efficient multivariate adaptive sampling by minimizing the number of computational electromagnetic analysis needed to establish accurate interpolation models Cross Ref Algorithm 352: characteristic values and associated solutions of Mathieu's differential equation [S22]Communications of the ACM, Vol. 12, No. 7 Cross Ref Truncation Error Bounds for Continued FractionsWilliam B. Jones and R. I. Snell14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 6, No. 2AbstractPDF (811 KB)Bibliography Cross Ref Bibliography Cross Ref Steady-State Analysis of Linear Networks Containing a Single Sinusoidally Varying CapacitorIEEE Transactions on Circuit Theory, Vol. 14, No. 3 Cross Ref Estimates of the Speed of Convergence of Certain Continued FractionsW. B. Sweezy and W. J. Thron14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 4, No. 2AbstractPDF (867 KB)Computational Aspects of Three-Term Recurrence RelationsWalter Gautschi18 July 2006 | SIAM Review, Vol. 9, No. 1AbstractPDF (4854 KB)Truncation error estimates for Stieltjes fractionsNumerische Mathematik, Vol. 9, No. 2 Cross Ref On Truncation Errors for Continued Fraction ComputationsE. P. Merkes14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 3, No. 3AbstractPDF (797 KB)Numerical aspects of Mathieu eigenvaluesRendiconti del Circolo Matematico di Palermo, Vol. 15, No. 1 Cross Ref Volume 6, Issue 4| 1964SIAM Review History Submitted:11 April 1962Accepted:24 February 1964Published online:18 July 2006 InformationCopyright © 1964 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1006092Article page range:pp. 383-421ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics
- Research Article
656
- 10.1137/1009002
- Jan 1, 1967
- SIAM Review
Computational Aspects of Three-Term Recurrence Relations
- Research Article
66
- 10.1137/0705031
- Jun 1, 1968
- SIAM Journal on Numerical Analysis
The Rate of Convergence of Some Difference Schemes
- Research Article
154
- 10.1137/0905023
- Jun 1, 1984
- SIAM Journal on Scientific and Statistical Computing
Condition Estimates
- Research Article
129
- 10.1137/0706055
- Dec 1, 1969
- SIAM Journal on Numerical Analysis
Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials
- Research Article
3522
- 10.1137/0705041
- Sep 1, 1968
- SIAM Journal on Numerical Analysis
On the Construction and Comparison of Difference Schemes
- Research Article
108
- 10.1137/0706002
- Mar 1, 1969
- SIAM Journal on Numerical Analysis
We consider the general system of n first order linear ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b, subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).)
- Research Article
144
- 10.1137/0114060
- Jul 1, 1966
- SIAM Journal on Applied Mathematics
Bounds for Truncation Error of the Sampling Expansion
- Research Article
122
- 10.1137/0705057
- Dec 1, 1968
- SIAM Journal on Numerical Analysis
One parameter operator imbedding to modify Newton method for solution of nonlinear equations
- Research Article
58
- 10.1137/0705028
- Jun 1, 1968
- SIAM Journal on Numerical Analysis
The Coupled Equation Approach to the Numerical Solution of the Biharmonic Equation by finite Differences. I
- Research Article
506
- 10.1137/090778766
- Jan 1, 2010
- SIAM Journal on Matrix Analysis and Applications
For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.
- Research Article
57
- 10.1137/0706035
- Sep 1, 1969
- SIAM Journal on Numerical Analysis
The Numerical Solution of Integral Equations on the Half-Line
- Research Article
64
- 10.1137/0322006
- Jan 1, 1984
- SIAM Journal on Control and Optimization
Global existence is proved for the solution, of a Riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations.
- Research Article
52
- 10.1137/0706034
- Sep 1, 1969
- SIAM Journal on Numerical Analysis
Numerial Methods for Volterra Integral Equations with Singular Kernels
- Research Article
148
- 10.1098/rsta.2003.1338
- Mar 15, 2004
- Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.
- Research Article
104
- 10.1137/0519039
- May 1, 1988
- SIAM Journal on Mathematical Analysis
The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.