CUBATURE RULES ON THE TRIANGLE (SIMPLEX)


The purpose of this homepage is to gather the nodes and weights of numerical rules on the triangle. There are several standard representations. We decided to use:
All the quadrature formulas are available as Matlab files. In particular, they are easily downloadable as zip file.

I point out that a huge amount of work have been done by Nico Schlömer, even on other domains. If You are interested in Phyton codes, take a look at his stunning homepage at https://github.com/nschloe/quadpy#triangle.


7PTS

Deg. card Err. NW OD Q
3 7 2.220e-16 0 0 PI

» SOURCE: Rule converted from Python file: https://github.com/nschloe/quadpy#triangle. Thanks to: Nico Schlömer.

» MATLAB FILES:
ALBRECHT COLLATZ

Deg. card Err. NW OD Q
3 6 3.680e-16 0 0 PI

» SOURCE: J. Albrecht, L. Collatz, "Zur numerischen Auswertung mehrdimensionaler Integrale" ZAMM, Volume 38, Issue 1-2, 1958, pp. 1-15.

» MATLAB FILES:
BERNTSEN ESPELID #I

Deg. card Err. NW OD Q
13 37 1.510e-12 0 0 PI

» SOURCE: J. Berntsen, T.O. Espelid, "Degree 13 symmetric quadrature rules for the triangle", Reports in Informatics, Dept. of Informatics, University of Bergen, (1990).

» MATLAB FILES:
BERNTSEN ESPELID #I [Corrected]

Deg. card Err. NW OD Q
13 37 5.970e-16 0 0 PI

» SOURCE: J. Berntsen, T.O. Espelid, "Degree 13 symmetric quadrature rules for the triangle", Reports in Informatics, Dept. of Informatics, University of Bergen, (1990).

» MATLAB FILES:
BERNTSEN ESPELID #II

Deg. card Err. NW OD Q
13 40 3.960e-16 0 0 PI

» SOURCE: J. Berntsen, T.O. Espelid, "Degree 13 symmetric quadrature rules for the triangle", Reports in Informatics, Dept. of Informatics, University of Bergen, (1990).

» MATLAB FILES:
BERNTSEN ESPELID #III

Deg. card Err. NW OD Q
13 36 4.930e-16 3 6 NO

» SOURCE: J. Berntsen, T.O. Espelid, "Degree 13 symmetric quadrature rules for the triangle", Reports in Informatics, Dept. of Informatics, University of Bergen, (1990).

» MATLAB FILES:
BERNTSEN ESPELID #IV

Deg. card Err. NW OD Q
13 40 3.960e-16 0 0 PI

» SOURCE: J. Berntsen, T.O. Espelid, "Degree 13 symmetric quadrature rules for the triangle", Reports in Informatics, Dept. of Informatics, University of Bergen, (1990).

» MATLAB FILES:
COOLS HAEGEMANS

Deg. card Err. NW OD Q
8 15 6.000e-14 0 3 PO

» SOURCE: R. Cools, A. Haegemans, "Construction of minimal cubature formulae for the square and the triangle using invariant theory", Department of Computer Science, K.U.Leuven, TW Reports vol.TW96, Sept. 1987.

» MATLAB FILES:
COOLS HAEGEMANS # [Corrected]

Deg. card Err. NW OD Q
8 15 3.610e-16 0 3 PO

» SOURCE: R. Cools, A. Haegemans, "Construction of minimal cubature formulae for the square and the triangle using invariant theory", Department of Computer Science, K.U.Leuven, TW Reports vol.TW96, Sept. 1987.

» MATLAB FILES:
COWPER

Deg. card Err. NW OD Q
2 3 2.780e-16 0 0 PI
3 4 2.220e-16 1 0 NI
4 6 2.240e-15 0 0 PI
5 7 4.790e-16 0 0 PI
6 12 8.550e-15 0 0 PI
7 13 8.650e-15 1 0 NI

» SOURCE: G.R. Cowper, "Gaussian quadrature formulas for triangles", Numerical Methods in Engineering, Volume 7, Issue 3, (1973), pp. 405-408.

» MATLAB FILES:
DAY TAYLOR

Deg. card Err. NW OD Q
6 11 1.080e-15 0 0 PI

» SOURCE: D.M. Day and M.A. Taylor "A new 11 point degree 6 formula for the triangle",PAMM Proc. Appl. Math. Mech. 7 1022501-1022502 (2007).

» MATLAB FILES:
DEDONCKER ROBINSON

Deg. card Err. NW OD Q
9 19 8.690e-15 0 0 PI
11 28 3.510e-14 0 0 PI

» SOURCE: E. de Doncker and I. Robinson, "Algorithm 612: TRIEX: Integration Over a Triangle Using Nonlinear Extrapolation". ACM Trans. Math. Softw., March 1984.

» MATLAB FILES:
DEDONCKER ROBINSON # [Corrected]

Deg. card Err. NW OD Q
9 19 4.860e-16 0 0 PI
11 28 5.130e-16 0 0 PI

» SOURCE: E. de Doncker and I. Robinson, "Algorithm 612: TRIEX: Integration Over a Triangle Using Nonlinear Extrapolation". ACM Trans. Math. Softw., March 1984.

» MATLAB FILES:
DUNAVANT

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 3.680e-16 0 0 PI
3 4 2.220e-16 1 0 NI
4 6 2.240e-15 0 0 PI
5 7 1.790e-13 0 0 PI
6 12 2.750e-15 0 0 PI
7 13 4.160e-15 1 0 NI
8 16 3.870e-15 0 0 PI
9 19 3.430e-15 0 0 PI
10 25 1.100e-14 0 0 PI
11 27 1.200e-13 0 3 PO
12 33 4.540e-15 0 0 PI
13 37 9.070e-15 0 0 PI
14 42 6.710e-15 0 0 PI
15 48 3.600e-14 0 9 PO
16 52 1.980e-14 0 6 PO
17 61 1.100e-14 0 0 PI
18 70 5.080e-13 3 6 NO
19 73 1.650e-14 0 0 PI
20 79 4.740e-14 3 9 NO

» SOURCE: D.A. Dunavant, "High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle" International Journal for Numerical Methods in Engineering, 21(6), June 1985, pp. 1129-1148.

» MATLAB FILES:
DUNAVANT # [Corrected]

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 3.680e-16 0 0 PI
3 4 2.220e-16 1 0 NI
4 6 2.010e-16 0 0 PI
5 7 3.150e-16 0 0 PI
6 12 3.120e-16 0 0 PI
7 13 3.750e-16 1 0 NI
8 16 6.110e-16 0 0 PI
9 19 3.190e-16 0 0 PI
10 25 5.070e-16 0 0 PI
11 27 8.880e-16 0 3 PO
12 33 3.890e-16 0 0 PI
13 37 4.790e-16 0 0 PI
14 42 8.470e-16 0 0 PI
15 48 8.470e-16 0 9 PO
16 52 9.020e-16 0 6 PO
17 61 5.640e-16 0 0 PI
18 70 1.110e-15 3 6 NO
19 73 8.100e-16 0 0 PI
20 79 9.910e-16 3 9 NO

» SOURCE: D.A. Dunavant, "High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle" International Journal for Numerical Methods in Engineering, 21(6), June 1985, pp. 1129-1148.

» MATLAB FILES:
GATERMANN

Deg. card Err. NW OD Q
7 12 2.000e-13 0 0 PI

» SOURCE: K. Gatermann. "The Construction of Symmetric Cubature Formulas for the Square and the Triangle", Computing, 40, (1988), pp. 229-240.

» MATLAB FILES:
GATERMANN # [Corrected]

Deg. card Err. NW OD Q
7 12 5.070e-16 0 0 PI

» SOURCE: K. Gatermann. "The Construction of Symmetric Cubature Formulas for the Square and the Triangle", Computing, 40, (1988), pp. 229-240.

» MATLAB FILES:
HAMMER MARLOWE STROUD #I

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 1.180e-16 0 0 PI
3 4 2.220e-16 1 0 NI
5 7 1.600e-16 0 0 PI

» SOURCE: P.C. Hammer, O.J. Marlowe and A.H. Stroud, "Numerical Integration Over Simplexes and Cones", Vol. 10, No. 55, Jul. 1956, pp. 130-137

» MATLAB FILES:
HAMMER MARLOWE STROUD #II

Deg. card Err. NW OD Q
2 3 2.780e-16 0 0 PI

» SOURCE: P.C. Hammer, O.J. Marlowe and A.H. Stroud, "Numerical Integration Over Simplexes and Cones", Vol. 10, No. 55, Jul. 1956, pp. 130-137

» MATLAB FILES:
HAMMER STROUD

Deg. card Err. NW OD Q
2 3 3.680e-16 0 0 PI
3 4 2.220e-16 1 0 NI

» SOURCE: P. C. Hammer and A. H. Stroud, "Numerical Evaluation of Multiple Integrals II," Math. Comp. 12 (1958), pp. 272-280.

» MATLAB FILES:
HILLION #I

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 2.780e-16 0 0 PI
3 4 3.890e-16 0 0 PI

» SOURCE: P. Hillion, "Numerical Integration on a Triangle", International Journal for Numerical Methods in Engineering, Vol. 11, (1977), pp.797-815.

» MATLAB FILES:
HILLION #II

Deg. card Err. NW OD Q
2 3 3.680e-16 0 0 PI
3 4 2.430e-16 0 0 PI

» SOURCE: P. Hillion, "Numerical Integration on a Triangle", International Journal for Numerical Methods in Engineering, Vol. 11, (1977), pp.797-815.

» MATLAB FILES:
HILLION #III

Deg. card Err. NW OD Q
2 3 1.670e-16 0 0 PI
3 4 2.220e-16 1 0 NI

» SOURCE: P. Hillion, "Numerical Integration on a Triangle", International Journal for Numerical Methods in Engineering, Vol. 11, (1977), pp.797-815.

» MATLAB FILES:
HILLION #IV

Deg. card Err. NW OD Q
2 3 1.110e-16 0 1 PO
3 5 5.550e-16 0 0 PI

» SOURCE: P. Hillion, "Numerical Integration on a Triangle", International Journal for Numerical Methods in Engineering, Vol. 11, (1977), pp.797-815.

» MATLAB FILES:
HILLION #V

Deg. card Err. NW OD Q
2 4 2.500e-16 0 0 PI

» SOURCE: P. Hillion, "Numerical Integration on a Triangle", International Journal for Numerical Methods in Engineering, Vol. 11, (1977), pp.797-815.

» MATLAB FILES:
LAURIE

Deg. card Err. NW OD Q
8 19 9.580e-16 0 0 PI

» SOURCE: D.P. Laurie "Algorithm 584: CUBTRI: Automatic Cubature over a Triangle". ACM Trans. Math. Softw., June 1982.

» MATLAB FILES:
LAURSEN GELLERT #I

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 3.680e-16 0 0 PI
3 4 2.220e-16 1 0 NI
4 6 2.240e-15 0 0 PI
5 7 2.180e-15 0 0 PI
6 12 8.550e-15 0 0 PI
7 13 8.650e-15 1 0 NI
8 16 5.180e-15 0 0 PI
9 19 3.430e-15 0 0 PI
10 25 4.160e-15 0 0 PI

» SOURCE: M.E. Laursen, M. Gellert, "Some criteria for numerically integrated matrices and quadrature formulas for triangles", International Journal for Numerical Methods in Engineering, Volume 12, Issue 1, 1978, pp.67-76.

» MATLAB FILES:
LAURSEN GELLERT #II

Deg. card Err. NW OD Q
2 3 2.780e-16 0 0 PI
3 6 1.700e-15 0 0 PI
4 7 5.470e-15 0 0 PI
5 9 4.950e-15 0 0 PI
7 15 1.320e-14 0 0 PI
9 21 4.310e-15 0 0 PI
10 25 6.720e-15 0 0 PI

» SOURCE: M.E. Laursen, M. Gellert, "Some criteria for numerically integrated matrices and quadrature formulas for triangles", International Journal for Numerical Methods in Engineering, Volume 12, Issue 1, 1978, pp.67-76.

» MATLAB FILES:
LETHER

Deg. card Err. NW OD Q
1 4 4.440e-16 0 0 PI
2 4 4.440e-16 0 0 PI
3 9 3.890e-16 0 0 PI
4 9 3.890e-16 0 0 PI
5 16 5.200e-16 0 0 PI
6 16 5.200e-16 0 0 PI
7 25 5.970e-16 0 0 PI
8 25 5.970e-16 0 0 PI
9 36 7.290e-16 0 0 PI
10 36 7.290e-16 0 0 PI
11 49 1.440e-15 0 0 PI
12 49 1.440e-15 0 0 PI
13 64 9.990e-16 0 0 PI
14 64 9.990e-16 0 0 PI
15 81 1.550e-15 0 0 PI
16 81 1.550e-15 0 0 PI
17 100 9.370e-16 0 0 PI
18 100 9.370e-16 0 0 PI
19 121 1.120e-15 0 0 PI
20 121 1.120e-15 0 0 PI
21 144 1.320e-15 0 0 PI
22 144 1.320e-15 0 0 PI
23 169 1.110e-15 0 0 PI
24 169 1.110e-15 0 0 PI
25 196 1.390e-15 0 0 PI
26 196 1.390e-15 0 0 PI
27 225 1.330e-15 0 0 PI
28 225 1.330e-15 0 0 PI
29 256 1.080e-15 0 0 PI
30 256 1.660e-15 0 0 PI
31 289 1.050e-15 0 0 PI
32 289 1.050e-15 0 0 PI
33 324 1.200e-15 0 0 PI
34 324 1.220e-15 0 0 PI
35 361 1.770e-15 0 0 PI
36 361 1.770e-15 0 0 PI
37 400 1.460e-15 0 0 PI
38 400 1.460e-15 0 0 PI
39 441 1.430e-15 0 0 PI
40 441 1.430e-15 0 0 PI
41 484 1.880e-15 0 0 PI
42 484 1.880e-15 0 0 PI
43 529 1.160e-15 0 0 PI
44 529 1.160e-15 0 0 PI
45 576 1.610e-15 0 0 PI
46 576 1.610e-15 0 0 PI
47 625 1.490e-15 0 0 PI
48 625 1.490e-15 0 0 PI
49 676 1.190e-15 0 0 PI
50 676 1.190e-15 0 0 PI

» SOURCE: F.G. Lether, "Computation of double integrals over a triangle", JCAM, vol 2, no 3, (1976), pp. 219-224

» MATLAB FILES:
LYNESS JESPERSEN #I

Deg. card Err. NW OD Q
2 3 2.780e-16 0 0 PI
3 4 2.220e-16 1 0 NI
4 6 2.220e-16 0 0 PI
5 7 4.790e-16 0 0 PI
6 12 1.240e-13 0 0 PI
7 13 4.110e-14 1 0 NI
8 16 8.380e-15 0 0 PI
9 19 8.750e-15 0 0 PI
11 27 1.360e-13 0 3 PO

» SOURCE: J.N. Lyness, D. Jespersen, "Moderate Degree Symmetric Quadrature Rules for the Triangle", J. Inst. Maths Applies, 15 (1975), pp.19-32.

» MATLAB FILES:
LYNESS JESPERSEN #I [Corrected]

Deg. card Err. NW OD Q
2 3 2.780e-16 0 0 PI
3 4 2.220e-16 1 0 NI
4 6 2.220e-16 0 0 PI
5 7 4.790e-16 0 0 PI
6 12 2.310e-16 0 0 PI
7 13 4.160e-16 1 0 NI
8 16 5.830e-16 0 0 PI
9 19 3.050e-16 0 0 PI
11 27 4.580e-16 0 3 PO

» SOURCE: J.N. Lyness, D. Jespersen, "Moderate Degree Symmetric Quadrature Rules for the Triangle", J. Inst. Maths Applies, 15 (1975), pp.19-32.

» MATLAB FILES:
LYNESS JESPERSEN #II

Deg. card Err. NW OD Q
2 4 1.390e-16 0 0 PI
3 7 2.220e-16 0 0 PI
4 10 6.660e-16 3 0 NI
5 10 2.500e-16 0 0 PI
6 16 9.440e-16 4 4 NO
7 16 2.800e-13 0 0 PI
8 21 7.280e-12 3 0 NI
9 22 3.570e-15 0 0 PI
11 28 3.540e-14 0 0 PI

» SOURCE: J.N. Lyness, D. Jespersen, "Moderate Degree Symmetric Quadrature Rules for the Triangle", J. Inst. Maths Applies, 15 (1975), pp.19-32.

» MATLAB FILES:
LYNESS JESPERSEN #II [Corrected]

Deg. card Err. NW OD Q
2 4 2.780e-17 0 0 PI
3 7 2.080e-17 0 0 PI
4 10 3.330e-16 3 0 NI
5 10 1.820e-17 0 0 PI
6 16 3.330e-16 4 4 NO
7 16 1.110e-16 0 0 PI
8 21 4.090e-16 3 6 NO
9 22 1.110e-16 0 0 PI
11 28 2.220e-16 0 0 PI

» SOURCE: J.N. Lyness, D. Jespersen, "Moderate Degree Symmetric Quadrature Rules for the Triangle", J. Inst. Maths Applies, 15 (1975), pp.19-32.

» MATLAB FILES:
LYNESS JESPERSEN #III

Deg. card Err. NW OD Q
4 9 3.890e-16 0 0 PI
6 13 2.440e-15 0 0 PI
8 16 2.910e-15 1 0 NI

» SOURCE: J.N. Lyness, D. Jespersen, "Moderate Degree Symmetric Quadrature Rules for the Triangle", J. Inst. Maths Applies, 15 (1975), pp.19-32.

» MATLAB FILES:
PAPANICOLOPULOS A #I

Deg. card Err. NW OD Q
15 48 5.410e-16 3 0 NI
17 58 7.770e-16 0 6 PO
21 87 5.240e-16 0 0 PI
22 94 1.630e-15 0 6 PO
23 102 9.720e-16 0 0 PI
24 112 1.370e-15 0 0 PI
25 118 6.750e-16 0 6 PO

» SOURCE:

» MATLAB FILES:
PAPANICOLOPULOS A #II

Deg. card Err. NW OD Q
22 96 6.730e-16 0 0 PI
25 120 7.550e-15 6 0 NI

» SOURCE:

» MATLAB FILES:
PAPANICOLOPULOS A #III

Deg. card Err. NW OD Q
16 52 2.310e-06 0 0 PI
18 66 6.110e-16 0 0 PI

» SOURCE:

» MATLAB FILES:
PAPANICOLOPULOS A #III [Corrected]

Deg. card Err. NW OD Q
16 52 5.670e-16 0 0 PI
18 66 6.110e-16 0 0 PI

» SOURCE:

» MATLAB FILES:
PAPANICOLOPULOS A #IV

Deg. card Err. NW OD Q
10 24 3.160e-08 0 0 PI
11 27 3.160e-08 0 0 PI
12 31 3.310e-07 0 3 PO
13 36 3.310e-07 0 0 PI
14 40 4.970e-07 0 3 PO
15 46 4.970e-07 0 0 PI
16 51 2.310e-06 0 6 PO
17 57 5.830e-16 0 0 PI
18 64 4.860e-16 1 0 NI
19 70 4.160e-16 0 0 PI
20 78 4.890e-16 0 0 PI
21 85 4.640e-16 0 0 PI
22 93 7.560e-16 0 0 PI
23 100 6.660e-16 0 0 PI
24 109 7.810e-16 0 0 PI
25 117 5.410e-16 0 0 PI

» SOURCE:

» MATLAB FILES:
PAPANICOLOPULOS A #IV [Corrected]

Deg. card Err. NW OD Q
10 24 2.080e-15 0 0 PI
11 27 9.920e-16 0 0 PI
12 31 2.210e-15 0 3 PO
13 36 3.050e-16 0 0 PI
14 40 9.510e-16 0 3 PO
15 46 5.000e-16 0 0 PI
16 51 1.780e-15 0 6 PO
17 57 5.830e-16 0 0 PI
18 64 4.860e-16 1 0 NI
19 70 4.160e-16 0 0 PI
20 78 4.890e-16 0 0 PI
21 85 4.640e-16 0 0 PI
22 93 7.560e-16 0 0 PI
23 100 6.660e-16 0 0 PI
24 109 7.810e-16 0 0 PI
25 117 5.410e-16 0 0 PI

» SOURCE:

» MATLAB FILES:
PAPANICOLOPULOS B

Deg. card Err. NW OD Q
11 26 4.720e-16 0 1 PO
20 77 1.450e-15 0 8 PO
22 92 1.720e-15 0 11 PO

» SOURCE: S-A. Papanicolopulos, "Efficient computation of cubature rules with application to new, asymmetric rules on the triangle",JCAM 204 (2016), pp. 77-83.

» MATLAB FILES:
PAPANICOLOPULOS C #I

Deg. card Err. NW OD Q
7 15 2.500e-16 0 0 PI
10 25 3.890e-16 0 0 PI
11 28 2.780e-16 1 0 NI
12 33 3.960e-16 0 0 PI
13 37 5.550e-16 0 0 PI

» SOURCE: S-A. Papanicolopulos, "Computation of moderate-degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving", Computer and Mathematics with Applications, 69 (2015), pp. 650-666.

» MATLAB FILES:
PAPANICOLOPULOS C #II

Deg. card Err. NW OD Q
11 28 1.600e-14 3 0 NI
12 33 1.170e-15 3 0 NI
13 37 7.490e-16 0 0 PI

» SOURCE: S-A. Papanicolopulos, "Computation of moderate-degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving", Computer and Mathematics with Applications, 69 (2015), pp. 650-666.

» MATLAB FILES:
PAPANICOLOPULOS C #III

Deg. card Err. NW OD Q
11 30 4.580e-16 0 0 PI
13 37 7.080e-16 1 0 NI

» SOURCE: S-A. Papanicolopulos, "Computation of moderate-degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving", Computer and Mathematics with Applications, 69 (2015), pp. 650-666.

» MATLAB FILES:
PAPANICOLOPULOS C #III [Corrected]

Deg. card Err. NW OD Q
11 30 4.580e-16 0 0 PI
13 37 7.080e-16 1 0 NI

» SOURCE: S-A. Papanicolopulos, "Computation of moderate-degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving", Computer and Mathematics with Applications, 69 (2015), pp. 650-666.

» MATLAB FILES:
PAPANICOLOPULOS C #IV

Deg. card Err. NW OD Q
11 30 7.250e-16 0 0 PI
13 37 1.940e-15 1 0 NI

» SOURCE: S-A. Papanicolopulos, "Computation of moderate-degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving", Computer and Mathematics with Applications, 69 (2015), pp. 650-666.

» MATLAB FILES:
PAPANICOLOPULOS C #V

Deg. card Err. NW OD Q
11 30 4.160e-16 0 0 PI
13 37 1.440e-15 1 0 NI

» SOURCE: S-A. Papanicolopulos, "Computation of moderate-degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving", Computer and Mathematics with Applications, 69 (2015), pp. 650-666.

» MATLAB FILES:
RADON

Deg. card Err. NW OD Q
5 7 2.780e-16 0 0 PI

» SOURCE: J. RADON, "Zur mechanischen Kubatur", Monatshefte für Mathematik, 52 (1948), pp.286-300.

» MATLAB FILES:
STROUD

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 4 2.150e-16 0 0 PI
3 4 2.150e-16 0 0 PI
4 9 4.720e-16 0 0 PI
5 9 6.660e-16 0 0 PI
6 16 4.160e-16 0 0 PI
7 16 4.160e-16 0 0 PI
8 25 1.010e-15 0 0 PI
9 25 1.010e-15 0 0 PI
10 36 9.990e-16 0 0 PI
11 36 9.990e-16 0 0 PI
12 49 7.910e-16 0 0 PI
13 49 7.910e-16 0 0 PI
14 64 1.090e-15 0 0 PI
15 64 1.090e-15 0 0 PI
16 81 1.120e-15 0 0 PI
17 81 1.120e-15 0 0 PI
18 100 1.010e-15 0 0 PI
19 100 1.010e-15 0 0 PI
20 121 1.090e-15 0 0 PI
21 121 1.360e-15 0 0 PI
22 144 7.950e-16 0 0 PI
23 144 9.600e-16 0 0 PI
24 169 1.320e-15 0 0 PI
25 169 1.320e-15 0 0 PI
26 196 1.310e-15 0 0 PI
27 196 1.310e-15 0 0 PI
28 225 1.240e-15 0 0 PI
29 225 1.240e-15 0 0 PI
30 256 1.280e-15 0 0 PI
31 256 1.600e-15 0 0 PI
32 289 8.880e-16 0 0 PI
33 289 1.020e-15 0 0 PI
34 324 1.330e-15 0 0 PI
35 324 1.330e-15 0 0 PI
36 361 1.550e-15 0 0 PI
37 361 1.550e-15 0 0 PI
38 400 1.140e-15 0 0 PI
39 400 1.140e-15 0 0 PI
40 441 1.270e-15 0 0 PI
41 441 1.270e-15 0 0 PI
42 484 1.520e-15 0 0 PI
43 484 1.520e-15 0 0 PI
44 529 1.330e-15 0 0 PI
45 529 1.330e-15 0 0 PI
46 576 1.750e-15 0 0 PI
47 576 2.050e-15 0 0 PI
48 625 1.860e-15 0 0 PI
49 625 1.860e-15 0 0 PI
50 676 1.080e-15 0 0 PI

» SOURCE: A.H. Stroud, "Approximate calculation of multiple intregrals", Prentice Hall, Englewoods Cliffs, N.J., 1971

» MATLAB FILES:
TAYLOR

Deg. card Err. NW OD Q
10 24 7.220e-16 0 0 PI
11 27 3.640e-15 0 0 PI
12 32 1.610e-15 0 0 PI

» SOURCE: M.A. Taylor, "Asymmetric cubature formulas for polynomial integration in the triangle and square", JCAM 218 (2008), pp. 184-191.

» MATLAB FILES:
TAYLOR WINGATE BOS

Deg. card Err. NW OD Q
2 3 1.730e-13 0 0 PI
4 6 1.870e-13 0 0 PI
5 10 1.770e-13 0 0 PI
7 15 4.590e-13 0 0 PI
9 21 3.330e-13 0 0 PI
11 28 5.890e-13 0 0 PI
13 36 3.940e-13 0 0 PI
14 45 4.530e-13 0 0 PI
16 55 7.820e-13 0 0 PI
18 66 5.010e-13 0 0 PI
20 78 7.680e-13 0 0 PI
21 91 8.940e-13 0 0 PI
23 105 7.020e-13 0 0 PI
25 120 5.660e-13 0 0 PI

» SOURCE: Mark A. Taylor, Beth A. Wingate, Len P. Bos, "Several new quadrature formulas for polynomial integration in the triangle", unpublished note, see arXiv.

» MATLAB FILES:
TAYLOR WINGATE BOS # [Corrected]

Deg. card Err. NW OD Q
2 3 2.120e-16 0 0 PI
4 6 2.220e-16 0 0 PI
5 10 1.770e-13 0 0 PI
7 15 4.590e-13 0 0 PI
9 21 3.450e-16 0 0 PI
11 28 4.340e-16 0 0 PI
13 36 5.460e-16 0 0 PI
14 45 4.530e-13 0 0 PI
16 55 7.820e-13 0 0 PI
18 66 4.770e-16 0 0 PI
20 78 6.040e-16 0 0 PI
21 91 5.970e-16 0 0 PI
23 105 8.740e-16 0 0 PI
25 120 6.970e-16 0 3 PO

» SOURCE: Mark A. Taylor, Beth A. Wingate, Len P. Bos, "Several new quadrature formulas for polynomial integration in the triangle", unpublished note, see arXiv.

» MATLAB FILES:
TAYLOR WINGATE BOS SIAM

Deg. card Err. NW OD Q
13 36 3.940e-13 0 0 PI
16 55 7.820e-13 0 0 PI
18 66 5.010e-13 0 0 PI
20 78 7.680e-13 0 0 PI

» SOURCE: Mark A. Taylor, Beth A. Wingate, Len P. Bos, "Several new quadrature formulas for polynomial integration in the triangle", unpublished note, see arXiv.

» MATLAB FILES:
TAYLOR WINGATE BOS SIAM # [Corrected]

Deg. card Err. NW OD Q
13 36 5.690e-16 0 0 PI
16 55 7.820e-13 0 0 PI
18 66 6.280e-16 0 0 PI
20 78 6.590e-16 0 0 PI

» SOURCE: Mark A. Taylor, Beth A. Wingate, Len P. Bos, "Several new quadrature formulas for polynomial integration in the triangle", unpublished note, see arXiv.

» MATLAB FILES:
VIOREANU ROKHLIN

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 4.200e-16 0 0 PI
4 6 1.530e-16 0 0 PI
5 10 4.430e-16 0 0 PI
7 15 6.110e-16 0 0 PI
8 21 5.270e-16 0 0 PI
10 28 7.420e-16 0 0 PI
12 36 7.490e-16 0 0 PI
14 45 1.030e-15 0 0 PI
15 55 1.620e-15 0 0 PI
17 66 1.550e-15 0 0 PI
19 78 1.400e-15 0 0 PI
20 91 5.970e-16 0 0 PI
22 105 3.030e-14 0 0 PI
24 120 2.020e-14 0 0 PI
25 136 2.120e-14 0 0 PI
27 153 2.520e-13 0 0 PI
28 171 2.860e-15 0 0 PI
30 190 2.900e-14 0 0 PI
32 210 4.080e-12 0 0 PI

» SOURCE: B. Vioreanu and V. Rokhlin, "Spectra of Multiplication Operators as a Numerical Tool", Methods and Algorithms for Scientific Computing, SIAM J. Sci. Comput., 36,1 (2014), pp.267-288.

» MATLAB FILES:
VIOREANU ROKHLIN # [Corrected]

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 1.420e-16 0 0 PI
4 6 1.530e-16 0 0 PI
5 10 3.050e-16 0 0 PI
7 15 3.610e-16 0 0 PI
8 21 5.620e-16 0 0 PI
10 28 4.720e-16 0 0 PI
12 36 4.440e-16 0 0 PI
14 45 5.000e-16 0 0 PI
15 55 4.440e-16 0 0 PI
17 66 5.340e-16 0 0 PI
19 78 8.530e-16 0 0 PI
20 91 4.300e-16 0 0 PI
22 105 5.690e-16 0 0 PI
24 120 6.180e-16 0 0 PI
25 136 6.380e-16 0 0 PI
27 153 5.000e-16 0 0 PI
28 171 5.270e-16 0 0 PI
30 190 6.420e-16 0 0 PI
32 210 6.830e-16 0 0 PI

» SOURCE: B. Vioreanu and V. Rokhlin, "Spectra of Multiplication Operators as a Numerical Tool", Methods and Algorithms for Scientific Computing, SIAM J. Sci. Comput., 36,1 (2014), pp.267-288.

» MATLAB FILES:
WALKINGTON

Deg. card Err. NW OD Q
5 7 5.000e-16 0 0 PI

» SOURCE: Walkington. Rule converted from Python file: https://github.com/nschloe/quadpy#triangle Thanks to: Nico Schlömer.

» MATLAB FILES:
WANDZURA XIAO

Deg. card Err. NW OD Q
5 7 2.500e-16 0 0 PI
10 25 8.850e-16 0 0 PI
15 54 1.390e-15 0 0 PI
20 85 7.670e-16 0 0 PI
25 126 9.160e-16 0 0 PI
30 175 1.750e-15 0 0 PI

» SOURCE: S. Wandzura, H. Xiao, "Symmetric Quadrature Rules on a Triangle", Computers and Mathematics with Applications 45 (2003), pp. 1829-1840.

» MATLAB FILES:
WILLIAMS SHUNN JAMESON

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 3.680e-16 0 0 PI
4 6 7.830e-14 0 0 PI
5 10 3.940e-15 0 0 PI
7 15 5.670e-14 0 0 PI
8 21 1.140e-12 0 0 PI
10 28 4.040e-13 0 0 PI
12 36 1.320e-12 0 0 PI

» SOURCE: D.M. Williams, L. Shunn, A. Jameson, "Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements", Journal of Computational and Applied Mathematics, 266 (2014), pp.18-38,

» MATLAB FILES:
WITHERDEN VINCENT

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
4 6 1.390e-16 0 0 PI
5 7 1.980e-16 0 0 PI
6 12 2.360e-16 0 0 PI
7 15 3.750e-16 0 0 PI
8 16 8.050e-16 0 0 PI
9 19 3.260e-16 0 0 PI
10 25 4.860e-16 0 0 PI
11 28 1.490e-15 0 0 PI
12 33 6.520e-16 0 0 PI
13 37 5.830e-16 0 0 PI
14 42 4.370e-16 0 0 PI
15 49 5.620e-16 0 0 PI
16 55 6.110e-16 0 0 PI
17 60 4.090e-16 0 0 PI
18 67 9.000e-16 0 0 PI
19 73 6.380e-16 0 0 PI
20 79 7.460e-16 0 0 PI

» SOURCE: F.D: Witherden, P.E. Vincent. the identification of symmetric quadrature rules for finite element methods". Computers and Mathematics with Applications 69 (2015) 1232-1241.

» MATLAB FILES:
XIAO GIMBUTAS

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 4.200e-16 0 0 PI
3 6 1.250e-15 0 0 PI
4 6 1.250e-15 0 0 PI
5 7 7.080e-16 0 0 PI
6 12 4.300e-16 0 0 PI
7 15 7.150e-16 0 0 PI
8 16 9.990e-16 0 0 PI
9 19 1.120e-15 0 0 PI
10 25 6.660e-16 0 0 PI
11 28 5.830e-16 0 0 PI
12 33 8.470e-16 0 0 PI
13 37 1.060e-15 0 0 PI
14 42 1.140e-15 0 0 PI
15 49 8.500e-16 0 0 PI
16 55 2.030e-15 0 0 PI
17 60 9.490e-16 0 0 PI
18 67 1.500e-15 0 0 PI
19 73 1.070e-15 0 0 PI
20 79 1.250e-15 0 0 PI
21 87 1.420e-15 0 0 PI
22 96 9.990e-16 0 0 PI
23 103 2.050e-15 0 0 PI
24 112 9.850e-16 0 0 PI
25 120 1.800e-15 0 0 PI
26 130 1.960e-15 0 0 PI
27 141 1.110e-15 0 0 PI
28 150 1.390e-15 0 0 PI
29 159 1.340e-15 0 0 PI
30 171 1.640e-15 0 0 PI
31 181 1.700e-15 0 0 PI
32 193 1.710e-15 0 0 PI
33 204 1.450e-15 0 0 PI
34 214 1.870e-15 0 0 PI
35 228 1.680e-15 0 0 PI
36 243 1.180e-15 0 0 PI
37 252 1.030e-15 0 0 PI
38 267 1.850e-15 0 0 PI
39 282 1.580e-15 0 0 PI
40 295 1.420e-15 0 0 PI
41 309 1.450e-15 0 0 PI
42 324 1.440e-15 0 0 PI
43 339 1.450e-15 0 0 PI
44 354 1.380e-15 0 0 PI
45 370 1.520e-15 0 0 PI
46 385 2.440e-15 0 0 PI
47 399 1.560e-15 0 0 PI
48 423 1.490e-15 0 0 PI
49 435 1.230e-15 0 0 PI
50 453 1.390e-15 0 0 PI

» SOURCE: H. Xiao, Z. Gimbutas. "A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions". Computers and Mathematics with Applications 59 (2010) 663-676.

» MATLAB FILES:
ZHANG CUI LIU

Deg. card Err. NW OD Q
1 1 1.920e-16 0 0 PI
2 3 3.680e-16 0 0 PI
3 6 1.460e-16 0 0 PI
4 6 1.390e-16 0 0 PI
5 7 1.980e-16 0 0 PI
6 12 2.360e-16 0 0 PI
7 16 3.750e-16 0 0 PI
8 21 4.420e-13 0 0 PI
9 19 5.550e-16 0 0 PI
10 25 4.720e-16 0 0 PI
11 28 8.330e-16 0 0 PI
12 33 4.820e-16 0 0 PI
13 37 5.830e-16 0 0 PI
14 42 4.580e-16 0 0 PI
15 49 4.820e-16 0 0 PI
16 55 8.120e-16 0 0 PI
17 60 6.210e-16 0 0 PI
18 73 1.010e-15 0 0 PI
20 82 9.060e-16 0 0 PI
21 87 5.590e-16 0 0 PI
22 96 6.660e-16 0 0 PI
24 112 1.340e-15 0 0 PI
25 126 6.170e-16 0 0 PI
26 133 8.500e-16 0 0 PI
27 145 9.990e-16 0 0 PI
28 154 1.110e-15 0 0 PI
29 166 1.330e-15 0 0 PI

» SOURCE: L. Zhang, T. Cui, H. Liu, "A set of Symmetric Quadrature rules on triangles and tetrahedra", J. Comput. Math. 27, No. I, (2009), pp.89-96.

» MATLAB FILES: