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:

- Barycentric coordinates, since they can easily be adapted to any triangle by a simple mapping depending on the vertices of the triangle. In general the sum of the weights makes 1.
- Compact form, describing the pointsets in terms of its orbits.
- Standard coordinates, suitable for integration on the triangle T with vertices (0,0), (1,0), (0,1). In general the sum of the weights makes 1/2, i.e. the area of T.

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 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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

BERNTSEN ESPELID #III

Deg. | card | Err. | NW | OD | Q
--- | --- | --- | --- | --- | ---
13 | 36 | 4.930e-16 | 3 | 6 | N0

» 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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

COOLS HAEGEMANS

Deg. | card | Err. | NW | OD | Q
--- | --- | --- | --- | --- | ---
8 | 15 | 6.000e-14 | 0 | 3 | P0

» 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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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 | P0
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 | P0
16 | 52 | 1.980e-14 | 0 | 6 | P0
17 | 61 | 1.100e-14 | 0 | 0 | PI
18 | 70 | 5.080e-13 | 3 | 6 | N0
19 | 73 | 1.650e-14 | 0 | 0 | PI
20 | 79 | 4.740e-14 | 3 | 9 | N0

» 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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

LAURSEN GELLERT #I

Deg. | card | Err. | NW | OD | Q
--- | --- | --- | --- | --- | ---
1 | 1 | 1.920e-16 | 0 | 0 | PI
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
6 | 12 | 8.550e-15 | 0 | 0 | PI
7 | 15 | 1.320e-14 | 0 | 0 | PI
8 | 16 | 5.180e-15 | 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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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 | P0

» 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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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 | N0
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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

PAPANICOLOPULOS A #I

Deg. | card | Err. | NW | OD | Q
--- | --- | --- | --- | --- | ---
15 | 48 | 5.410e-16 | 3 | 0 | NI
17 | 58 | 7.770e-16 | 0 | 6 | P0
21 | 87 | 5.240e-16 | 0 | 0 | PI
22 | 94 | 1.630e-15 | 0 | 6 | P0
23 | 102 | 9.720e-16 | 0 | 0 | PI
24 | 112 | 1.370e-15 | 0 | 0 | PI
25 | 118 | 6.750e-16 | 0 | 6 | P0

» SOURCE:
» MATLAB FILES:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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 | P0
13 | 36 | 3.310e-07 | 0 | 0 | PI
14 | 40 | 4.970e-07 | 0 | 3 | P0
15 | 46 | 4.970e-07 | 0 | 0 | PI
16 | 51 | 2.310e-06 | 0 | 6 | P0
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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

PAPANICOLOPULOS B

Deg. | card | Err. | NW | OD | Q
--- | --- | --- | --- | --- | ---
11 | 26 | 4.720e-16 | 0 | 1 | P0
20 | 77 | 1.450e-15 | 0 | 8 | P0
22 | 92 | 1.720e-15 | 0 | 11 | P0

» 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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]

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:

- BARYCENTRIC COORDINATES: [.m]
- COMPACT FORM: [.m]
- STANDARD COORDINATES: [.m]