References

This window will jump to the reference of your interest.



Alf86
P. Alfeld, Trivariate adaptive cubature, Approximation Theory V (Boston) (C.K. Chui, L.L. Schumaker, and J.D. Ward, eds.), Academic Press, 1986, pp. 231--234.

BE85
J. Berntsen and T.O. Espelid, On the construction of higher degree three dimensional embedded integration rules, Reports in Informatics 16, Dept. of Informatics, University of Bergen, 1985.

BE88
J. Berntsen and T.O. Espelid, On the construction of higher degree three-dimensional embedded integration rules, SIAM J. Numer. Anal. 25 (1988), 222--234.

BE90
J. Berntsen and T.O. Espelid, Degree 13 symmetric quadrature rules for the triangle, Reports in Informatics 44, Dept. of Informatics, University of Bergen, 1990.

Bec87
T. Becker, Konstruktion von interpolatorischen Kubaturformeln mit Anwendungen in der Finit-Element-Methode, Ph.D. thesis, Technische Hochschule Darmstadt, 1987.

Bec92
M. Beckers, Numerical integration in high dimensions, Ph.D. thesis, Katholieke Universiteit Leuven, 1992.

BH86
M. Beckers and A. Haegemans, Construction of three-dimensional invariant cubature formulae, Report TW 85, Dept. of Computer Science, K.U. Leuven, 1986.

BH88
H. Brass and G. Hämmerlin (eds.), Numerical integration III, Internat. Ser. Appl. Math., vol. 85, Basel, Birkhäuser, 1988.

BH90
M. Beckers and A. Haegemans, The construction of cubature formulae for the tetrahedron, Report TW 128, Dept. of Computer Science, K.U. Leuven, 1990.

CG86
A.M. Cohen and D.A. Gismalla, Some integration formulae for symmetric functions ot two variables, Int. J. Comput. Math. 19 (1986), no. 1, 57--68.

CG97
R. Cools and A. Genz, An adaptive numerical cubature algorithm for simplices , Unpublished, 1997.

CH85
R. Cools and A. Haegemans, Construction of fully symmetric cubature formulae of degree 4k - 3 for fully symmetric planar regions, Report TW 71, Dept. of Computer Science, K.U. Leuven, 1985.

CH86a
R. Cools and A. Haegemans, Automatic computation of knots and weights of cubature formulae for circular symmetric planar regions, Report TW 77, Dept. of Computer Science, K.U. Leuven, 1986.

CH86b
R. Cools and A. Haegemans, Tables of degree 2k - 1/2k + 1 pairs of cubature formulae for symmetric planar regions, obtained by optimal addition of knots, Report TW 76, Dept. of Computer Science, K.U. Leuven, 1986.

CH86c
R. Cools and A. Haegemans, Tables of sequences of cubature formulae for circular symmetric planar regions, Report TW 83, Dept. of Computer Science, K.U. Leuven, 1986.

CH87a
R. Cools and A. Haegemans, Construction of fully symmetric cubature formulae of degree 4k - 3 for fully symmetric planar regions, J. Comput. Appl. Math. 17 (1987), 173--180.

CH87b
R. Cools and A. Haegemans, Construction of minimal cubature formulae for the square and the triangle using invariant theory, Report TW 96, Dept. of Computer Science, K.U. Leuven, 1987.

CH87c
R. Cools and A. Haegemans, Construction of sequences of embedded cubature formulae for circular symmetric planar regions, Numerical Integration (Dordrecht) (P. Keast and G. Fairweather, eds.), Reidel Publ. Comp., 1987, pp. 165--172.

CH88a
R. Cools and A. Haegemans, Another step forward in searching for cubature formulae with a minimal number of knots for the square, Computing 40 (1988), 139--146.

CH88b
R. Cools and A. Haegemans, Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller's lower bound, Numerical Integration III (Basel) (H. Brass and G. Hämmerlin, eds.), Birkhäuser Verlag, 1988, pp. 25--36.

CH88c
R. Cools and A. Haegemans, An embedded pair of cubature formulae of degree 5 and 7 for the triangle, BIT 28 (1988), 357--359.

CH89
R. Cools and A. Haegemans, On the construction of multi-dimensional embedded cubature formulae, Numer. Math. 55 (1989), 735--745.

CH90
R. Cools and A. Haegemans, An embedded family of cubature formulae for n-dimensional product regions, Report TW 132, Dept. of Computer Science, K.U. Leuven, 1990.

CH94
R. Cools and A. Haegemans, An embedded family of cubature formulae for n-dimensional product regions, J. Comput. Appl. Math. 51 (1994), 251--262.

CK2000
R. Cools and KJ. Kim, A survey of known and new cubature formulas for the unit disk, Report TW 300, Dept. of Computer Science, K.U. Leuven, 2000.

Coo89
R. Cools, The construction of cubature formulae using invariant theory and ideal theory, Ph.D. thesis, Katholieke Universiteit Leuven, 1989.

Cow73
G.R. Cowper, Gaussian quadrature formulas for triangles, Internat. J. Numer. Methods Engrg. 7 (1973), 405--408.

dD79
E. de Doncker, New Euler-Maclaurin expansions and their application to quadrature over the s-dimensional simplex, Math. Comp. 33 (1979), 1003--1018.

Dob70
L.N. Dobrodeev, Cubature formulas of the seventh order of accuracy for a hypersphere and a hypercube, Zh. vychisl. Mat. mat. Fiz. 10 (1970), 187--188, (Russian) U.S.S.R. Comput. Maths Math. Phys. 10: 252--253, 1970 (English).

Dob78
L.N. Dobrodeev, Cubature rules with equal coefficients for integrating functions with respect to symmetric domains, Zh. vychisl. Mat. mat. Fiz. 18 (1978), 846--852, (Russian) U.S.S.R. Comput. Maths Math. Phys. 18: 27--34, 1978 (English).

DR84
P.J. Davis and P. Rabinowitz, Methods of numerical integration, Academic Press, London, 1984.

Dun85a
D.A. Dunavant, Economical symmetric quadrature rules for complete polynomials over a square domain, Internat. J. Numer. Methods Engrg. 21 (1985), 1777--1784.

Dun85b
D.A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Internat. J. Numer. Methods Engrg. 21 (1985), 1129--1148.

Dun86
D.A. Dunavant, Efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube, Internat. J. Numer. Methods. Engrg. 23 (1986), 397--407.

DW91
P. Dellaportas and D. Wright, Positive embedded integration in Bayesian analysis, Statistics and Computing 1 (1991), 1--12.

EG92
T.O. Espelid and A. Genz (eds.), Numerical integration -- recent developments, software and applications, NATO ASI Series C: Math. and Phys. Sciences, vol. 357, Dordrecht, Kluwer Academic Publishers, 1992.

Eng70a
H. Engels, Über gleichgewichtete Kubaturformeln für ein Dreiecksgebiet, Elek. Daten. 12 (1970), 535--539.

Eng70b
H. Engels, Über gleichgewichtete Kubaturformeln für Kreis- und Sechseck-Gebiet, Elek. Daten. 5 (1970), 216--223.

Eng80
H. Engels, Numerical quadrature and cubature, Academic Press, London, 1980.

Eri86
S.S. Eriksen, On the development of embedded, fully symmetric quadrature rules for the square for adaptive numerical integration over two-dimensional, rectangular regions, Master's thesis, University of Bergen, 1986.

Esp87
T.O. Espelid, On the construction of good fully symmetric integration rules, SIAM J. Numer. Anal. 24 (1987), 855--881.

Fra71a
R. Franke, Obtaining cubatures for rectangles and other planar regions by using orthogonal polynomials, Math. Comp. 25 (1971), 803--813.

Fra71b
R. Franke, Orthogonal polynomials and approximate multiple integration, SIAM J. Numer. Anal. 8 (1971), 757--765.

Gat88
K. Gatermann, The construction of symmetric cubature formulas for the square and the triangle, Computing 40 (1988), 229--240.

Gat92
K. Gatermann, Linear representations of finite groups and the ideal theoretical construction of G-invariant cubature formulas, Numerical Integration -- Recent Developments, Software and Applications (Dordrecht) (T.O. Espelid and A. Genz, eds.), NATO ASI Series C: Math. and Phys. Sciences, vol. 357, Kluwer Academic Publishers, 1992, pp. 25--35.

GK96
A. Genz and B.D. Keister, Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight., J. Comput. Appl. Math. 71(2) (July 1996), 299--309.

GM78
A. Grundmann and H.M. Möller, Invariant integration formulas for the n-simplex by combinatorial methods, SIAM J. Numer. Anal. 15 (1978), 282--290.

GM80
A.C. Genz and A.A. Malik, An adaptive algorithm for numerical integration over an N-dimensional rectangular region, J. Comput. Appl. Math. 6 (1980), 295--302.

GM83
A.C. Genz and A.A. Malik, An imbedded family of fully symmetric numerical integration rules, SIAM J. Numer. Anal. 20 (1983), 580--588.

God94
G. Godzina, Dreidimensionale Kubaturformeln für zentralsymmetrische Integrale, Ph.D. thesis, Universität Erlangen-Nürnberg, 1994.

GS99
B. Griener and H.J. Schmid, An interactive tool to visualize common zeros of twodimensional polynomials , J. Comput. Appl. Math. 112 (1999), 83--94.

Gün75
C. Günther, Zweidimensionale Quadraturformeln vom Grad 7 mit 14 Punkten, Numer. Math. 24 (1975), 309--316.

Hae75
A. Haegemans, Tables of circularly symmetrical integration formulas of degree 2d-1 for two-dimensional circularly symmetrical regions, Report TW 27, K.U. Leuven Applied Mathematics and Programming Division, 1975.

Hae76
A. Haegemans, Circularly symmetrical integration formulas for two-dimensional circularly symmetrical regions, BIT 16 (1976), 52--59.

Hae82
A. Haegemans, Construction of known and new cubature formulas of degree five for three-dimensional symmetric regions, using orthogonal polynomials, Numerical Integration (Basel), Birkhäuser Verlag, 1982, pp. 119--127.

HC87
A. Haegemans and R. Cools, Construction of three-dimensional cubature formulae with points on regular and semi-regular polytopes, Numerical Integration (Dordrecht) (P. Keast and G. Fairweather, eds.), Reidel Publ. Comp., 1987, pp. 153--163.

Hem73
P.W. Hemker, A sequence of nested cubature rules, NW 3/73, Stichting Mathematisch Centrum, 1973.

Hil77
P. Hillion, Numerical integration on a triangle, Internat. J. Numer. Methods Engrg. 11 (1977), 797--815.

Hil81
P. Hillion, Numerical integration on a tetrahedron, Calcolo 18 (1981), 117--130.

HP76
A. Haegemans and R. Piessens, Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials, Numer. Math. 25 (1976), 139--148.

HP77
A. Haegemans and R. Piessens, Construction of cubature formulas of degree seven and nine symmetric planar regions, using orthogonal polynomials, SIAM J. Numer. Anal. 14 (1977), 492--508.

HX99
S. Heo and Y. Xu, Constructing symmetric cubature formulae on a triangle, In Advances in Computational Mathematics, eds. Z. Chen et al, Marcel Dekker, New York (1999), 203-221.

Hue73
C.B. HuelsmanIII, Quadrature formulas over fully symmetric planar regions, SIAM J. Numer. Anal. 10 (1973), 539--552.

Hun75
D.R. Hunkins, Cubatures of precision 2k and 2k+1 for hyperrectangles, Math. Comp. 29 (1975), 1098--1104.

Jan98
T. Jankewitz, Zweidimensionale Kubaturformeln , Master's thesis, Universität Erlangen, 1998

Jan98x
T. Jankewitz, Personal communication, August 1998.

Jin84
Y. Jinyun, Symmetric gaussian quadrature formulae for tetrahedronal regions, Comp. Meth. Appl. Mech. Eng. 43 (1984), 349--353.

Kea79
P. Keast, Some fully symmetric quadrature formulae for product spaces, J. Inst. Math. Applics 23 (1979), 251--264.

Kea86
P. Keast, Moderate-degree tetrahedral quadrature formulas, Comput. Methods Appl. Mech. Engrg. 55 (1986), 339--348.

KF87
P. Keast and G. Fairweather (eds.), Numerical integration -- recent developments, software and applications, NATO ASI Series C: Math. and Phys. Sciences, vol. 203, Dordrecht, D. Reidel Publishing Company, 1987.

Kim97
K.J. Kim, Invariant cubature formulas over a unit cube, Ph.D. thesis, Yonsei University, South Korea, 1997.

KS97
K.J. Kim and M.S. Song, Symmetric quadrature formulas over a unit disk, Korean J. Comput. Appl. Math. 4 (1997), 179--192.

KS98
K.J. Kim and M.S. Song, Invariant cubature formulas over a unit cube, Comm. Korean Math. Soc. 13 (1998), 913--931.

Kon77
S.I. Konjaev, Ninth-order quadrature formulas invariant with respect to the icosahedral group, Dokl. Akad. Nauk SSSR 233 (1977), 784--787, (Russian) Soviet Math. Dokl. 18: 497--501, 1977 (English).

KT89
S.I. Konyaev and L.A. Tolmacheva, Numerical location of nodes and weights in a sixth-order quadrature formula for a disk, Voprosy Vychisl. i Prikl. Mat. (Tashkent) 86 (1989), 59--63,151--152, (Russian) Summary Math. Rev. 91f:65051.

Lau82
D.P. Laurie, CUBTRI -- automatic cubature over a triangle, ACM Trans. Math. Software 8 (1982), 210--218.

LG78
M.E. Laursen and M. Gellert, Some criteria for numerically integrated matrices and quadrature formulas for triangles, Internat. J. Numer. Methods Engrg. 12 (1978), 67--76.

LJ75
J.N. Lyness and D. Jespersen, Moderate degree symmetric quadrature rules for the triangle, J. Inst. Math. Appl. 15 (1975), 19--32.

LKS80
J. Linden, N. Kroll, and H.J. Schmid, Minimale Kubaturformeln für Integrale über dem Einheitsquadrat, Tech. Report 373, Universität Bonn, 1980.

Lyn65a
J.N. Lyness, Integration rules of hypercubic symmetry over a certain spherically symmetric space, Math. Comp. 19 (1965), 471--476.

Lyn65b
J.N. Lyness, Symmetric integration rules for hypercubes: II. rule projection and rule extension, Math. Comp. 19 (1965), 394--407.

Mae89
J.I. Maeztu, On symmetric cubature formulae for planar regions, IMA J. Numer. Anal. 9 (1989), 167--183.

Mae91
K. Maesen, Constructie van cubatuurformules m.b.v. lineaire representaties, Master's thesis, Dept. of Computer Science, K.U. Leuven, 1991.

MdlM95
J.I. Maeztu and E. Sainz de la Maza, An invariant quadrature rule of degree 11 for the tetrahedron, C. R. Acad. Sci. Paris 321 (1995), 1263--1267.

Moa74
T. Moan, Experiences with orthogonal polynomials and ``best'' numerical integration formulas on a triangle; with particular reference to finite element approximations, Z. Angew. Math. Mech. 54 (1974), 501--508.

Möl73
H.M. Möller, Polynomideale und Kubaturformeln, Ph.D. thesis, Universität Dortmund, 1973.

Möl76
H.M. Möller, Kubaturformeln mit minimaler Knotenzahl, Numer. Math. 25 (1976), 185--200.

Möl80
H.M. Möller, Linear abhängige Punktfunktionale bei zweidimensionalen Interpolations- und Approximationsproblemen, Math. Z. 173 (1980), 35--49.

Möl82
H.M. Möller, An immediate construction of numerical integration and differentiation formulae, Multivariate approximation theory II (Basel) (W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, 1982, pp. 275--283.

MOV82
C. Majorana, S. Odorizzi, and R. Vitaliani, Shortened quadrature rules for finite elements, Adv. Eng. Software 4 (1982), 52--57.

MP78
C.R. Morrow and T.N.L. Patterson, Construction of algebraic cubature rules using polynomial ideal theory, SIAM J. Numer. Anal. 15 (1978), 953--976.

MP85
C.R. Morrow and T.N.L. Patterson, The construction of algebraic cubature formulae by the distribution of nodes along selected lines, SIAM J. Numer. Anal. 22 (1985), 1178--1190.

MP87
M. Mori and R. Piessens (eds.), Numerical quadrature, Amsterdam, North-Holland, 1987.

MR77
F. Mantel and P. Rabinowitz, The application of integer programming to the computation of fully symmetric integration formulas in two and three dimensions, SIAM J. Numer. Anal. 14 (1977), 391--425.

Mys81
I.P. Mysovskikh, Interpolatory cubature formulas, Izdat. `Nauka', Moscow-Leningrad, 1981, (Russian).

Neu82
G. Neumann, Boolesche Interpolatorische Kubatur, Ph.D. thesis, Universität Gesamthochshule Siegen, 1982.

PH75a
R. Piessens and A. Haegemans, Cubature formulas of degree eleven for symmetric planar regions, J. Comput. Appl. Math. 1 (1975), no. 2, 79--83.

PH75b
R. Piessens and A. Haegemans, Cubature formulas of degree nine for symmetric planar regions, Math. Comp. 29 (1975), 810--815.

Phi79
G.M. Phillips, Seventh degree integration rules for the cube, BIT 19 (1979), 98--103.

Phi80a
G.M. Phillips, Seventh degree integration rules for the sphere, BIT 20 (1980), 117--119.

Phi80b
G.M. Phillips, A survey of one-dimensional and multidimensional numerical integration, Computer Physics Communications 20 (1980), 17--27.

Phi81
G.M. Phillips, Seventh degree integration rules for R3 , BIT 21 (1981), 126--128.

Ras83
G.G. Rasputin, Construction of cubature formulas containing prespecified knots, Metody Vychisl. 13 (1983), 122--128, (Russian)

Ras86
G.G. Rasputin, Construction of cubature formulas containing preassigned nodes, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1986), 44--51, (Russian) Soviet Math. (Iz. VUZ) 30: 58--67, 1986 (English).

RKEB87
P. Rabinowitz, J. Kautsky, S. Elhay, and J.C. Butcher, On sequences of imbedded integration rules, Numerical Integration (Dordrecht) (P. Keast and G. Fairweather, eds.), Reidel Publ. Comp., 1987, pp. 113--139.

Sch78
H.J. Schmid, On cubature formulae with a minimal number of knots, Numer. Math. 31 (1978), 281--297.

Sch83
H.J. Schmid, Interpolatorische Kubaturformeln, Dissertationes Math., vol. CCXX, Polish Scientific Publishers, Warszawa, 1983.

SE87
T. Sørevik and T.O. Espelid, Fully symmetric integration rules for the 4-cube, Reports in Informatics 28, Dept. of Informatics, University of Bergen, 1987.

SE89
T. Sørevik and T.O. Espelid, Fully symmetric integration rules for the 4-cube, BIT 29 (1989), 148--153.

SF73
G. Strang and G.J. Fix, An analysis of the finite element method, Prentice-Hall, London, 1973.

Sko87
M.D. Skogen, Fullt symmetriske og gode integrasjonsregler for enhetskulen og kuleskall, Master's thesis, Institutt for Informatikk, University of Bergen, 1987.

Sør85
T. Sørevik, Full-symmetriske integrasjonsregler for enhets 4-kuben, Master's thesis, Institutt for Informatikk, Universitetet i Bergen, 1985.

Sto97
S.B. Stoyanova, Cubature formulae of the seventh degree of accuracy for the hypersphere, J. Comput. Appl. Math. 84 (1997), no. 1, 15--21.

Str71
A.H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, Englewood Cliffs, N.J., 1971.

VC91
P. Verlinden and R. Cools, A cubature formula of degree 19 with 68 nodes for integration over the square, Report TW 149, Dept. of Computer Science, K.U. Leuven, 1991.

VC94
P. Verlinden and R. Cools, The algebraic construction of a minimal cubature formula of degree 11 for the square, Cubature Formulas and their Applications (Russian) (Krasnoyarsk) (M.V. Noskov, ed.), 1994, pp. 13--23.

Ver93
P. Verlinden, Cubature formulas and asymptotic expansions, Ph.D. thesis, Katholieke Universiteit Leuven, 1993.

Wal95
S. Waldron, Symmetries of linear functionals, Approximation Theory VIII (C.K. Chui and L.L. Schumaker, eds.), World Scientific Publishing Co., Inc., 1995, pp. 541--550.

WX2003
S. Wandzura and H. Xiao, Symmetric quadrature rules on a triangle, Computers and Mathematics with Applications 45 (2003), 1829--1840.

WB86
J. W. Wissman and T. Becker, Partially symmetric cubature formulas for even degrees of exactness, SIAM J. Numer. Anal. 23 (1986), 676--685.

Wei91
S. Weiß, Über Kubaturformeln vom Grad 2k-2, Master's thesis, Universität Erlangen, 1991.



This page was last modified on Tuesday 2 August 2005, 04:55:44 PM CEST.

Valid HTML 4.01!