Copyright (C) 2009-2012: Zydrunas Gimbutas and Hong Xiao
Contact: Zydrunas Gimbutas gimbutas@cims.nyu.edu, Hong Xiao hxiao@ucdavis.edu
Date: June 18, 2020
Version 1.10
triasymq.f - fully symmetric (D_3) quadrature for triangle
triarotq.f - rotationally symmetric quadrature for triangle
triaarbq.f - arbitrary symmetric (or asymmetric) quadrature for triangle
triaintq.f - interpolation nodes/quadrature for smooth functions on triangle
squaresymq.f - fully symmetric (D_4) quadrature for a unit square
squaresymvq.f - centro/rotationally symmetric quadrature for a unit square
squarearbq.f - arbitrary symmetric (or asymmetric) quadrature for a unit square
squareintq.f - interpolation nodes/quadrature for smooth functions on a square
tetraarbq.f - arbitrary symmetric (or asymmetric) quadrature for tetrahedron
tetraintq.f - interpolation nodes/quadrature for smooth functions on tetrahedron
cubearbq.f - arbitrary symmetric (or asymmetric) quadrature for a unit cube
makefile - run a simple test (triasymq)
triasymq.f - construct (or rather retrieve) a fully symmetric (D_3)
quadrature formula on the user-defined triangle in the plane.
triasymq_dr.f - driver routines for triasymq.f
makefile.triasymq - makefile for triasymq.f
( make -f makefile.triasymq )
c
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1 3 6 6 7 12 15 16 19 25
c
c
c n 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------
c nodes 28 33 37 42 49 55 60 67 73 79
c
c
c n 21 22 23 24 25 26 27 28 29 30
c -----------------------------------------------------------------
c nodes 87 96 103 112 120 130 141 150 159 171
c
c
c n 31 32 33 34 35 36 37 38 39 40
c -----------------------------------------------------------------
c nodes 181 193 204 214 228 243 252 267 282 295
c
c
c n 41 42 43 44 45 46 47 48 49 50
c -----------------------------------------------------------------
c nodes 309 324 339 354 370 385 399 423 435 453
triarotq.f - construct (or rather retrieve) a rotationally symmetric
quadrature formula on the user-defined triangle in the plane.
triarotq_dr.f - driver routines for triarotq.f
makefile.triarotq - makefile for triarotq.f
( make -f makefile.triarotq )
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1 3 6 6 7 12 12 16 19 24
c
c
c n 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------
c nodes 27 33 36 42 48 54 60 66 70 78
c
c
c n 21 22 23 24 25 26 27 28 29 30
c -----------------------------------------------------------------
c nodes 85 93 102 111 117 129 138 147 156 168
c
c
c n 31 32 33 34 35 36 37 38 39 40
c -----------------------------------------------------------------
c nodes 177 189 201 213 225 237 249 262 277 291
c
c
c n 41 42 43 44 45 46 47 48 49 50
c -----------------------------------------------------------------
c nodes 303 318 333 348 363 378 393 414 429 444
triaarbq.f - construct (or rather retrieve) an asymmetric
quadrature formula on the user-defined triangle in the plane.
triaarbq_dr.f - driver routines for triaarbq.f
makefile.triaarbq - makefile for triaarbq.f
( make -f makefile.triaarbq )
c
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1 3 4 6 7 11 12 16 19 24
c
c
c n 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------
c nodes 27 32 36 41 46 53 58 65 70 78
c
c
c n 21 22 23 24 25 26 27 28 29 30
c -----------------------------------------------------------------
c nodes 85 93 101 109 117 128 136 147 156 168
c
c
c n 31 32 33 34 35 36 37 38 39 40
c -----------------------------------------------------------------
c nodes 177 189 201 212 225 236 249 262 274 290
c
c
c n 41 42 43 44 45 46 47 48 49 50
c -----------------------------------------------------------------
c nodes 302 318 332 348 363 378 393 412 429 444
c
triaintq.f - construct (or rather retrieve) fully symmetric (D_3)
interpolation nodes on the user-defined triangle in the plane.
triaintq_dr.f - driver routines for triaintq.f
makefile.triaintq - makefile for triaintq.f
( make -f makefile.triaintq )
c interp 0 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------------
c quadr 1 2 4 5 7 8 10 12 14 15 17
c -----------------------------------------------------------------------
c nodes 1 3 6 10 15 21 28 36 45 55 66
c -----------------------------------------------------------------------
c cond # 1.0 1.0 1.4 1.9 2.1 3.4 4.3 4.8 4.8 6.5 8.1
c
c
c interp 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------------
c quadr 19 20 22 24 25 27 28 30 32 33
c -----------------------------------------------------------------------
c nodes 78 91 105 120 136 153 171 190 210 231
c -----------------------------------------------------------------------
c cond # 15.7 19.2 21.4 38.6 31.4 44.3 75.3 117 153 194
squaresymq.f - construct (or rather retrieve) a fully symmetric (D_4)
quadrature formula for the unit square [-1,1]^2.
squaresymq_dr.f - driver routines for squaresymq.f
makefile.squaresymq - makefile for squaresymq.f
( make -f makefile.squaresymq )
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1 4 4 8 8 12 12 20 20 28
c
c
c n 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------
c nodes 28 37 37 48 48 57 57 72 72 85
c
c
c n 21
c -----------------------------------------------------------------
c nodes 85
squaresymvq.f - construct (or rather retrieve) a variably symmetric
quadrature formula for the unit square [-1,1]^2.
squaresymvq_dr.f - driver routines for squaresymvq.f
makefile.squaresymvq - makefile for squaresymvq.f
( make -f makefile.squaresymvq )
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1f 4f 4f 7c 7c 12f 12f 17r 17r* 24r
c
c
c n 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------
c nodes 24r* 33r 33r* 43c 43c* 54c 54c* 67c 67c* 81r
c
c
c n 21 22 23 24 25 26 27 28 29
c -----------------------------------------------------------------
c nodes 81r* 96r 96r* 115c 115c* 132r 132r* 152r 152r*
c
c
c f - fully symmetric (D_4 symmetry)
c r - rotational symmetry only (rotate by 90 degrees)
c c - center symmetry only (rotate by 180 degrees)
c
c * - number of quadrature nodes is less than (n+1)*(n+2)/2/3
squarearbq.f - construct (or rather retrieve) a quadrature formula
for the unit square [-1,1]^2.
squarearbq_dr.f - driver routines for squarearbq.f
makefile.squarearbq - makefile for squarearbq.f ( make -f makefile.squarearbq )
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1f 3 4f 6 7c 10 12f 16 17r* 22
c
c
c n 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------
c nodes 24r* 31 33r* 40 43c* 52 54c* 64 67c* 78
c
c
c n 21 22 23 24 25 26 27 28 29 30
c -----------------------------------------------------------------
c nodes 81r* 93 96r* 109 115c* 127 132r* 146 152r* 167
c
c
c f - fully symmetric (D_4 symmetry)
c r - rotational symmetry only (rotate by 90 degrees)
c c - center symmetry only (rotate by 180 degrees)
c
c * - number of quadrature nodes is less than (n+1)*(n+2)/2/3
squareintq.f - construct (or rather retrieve)
interpolation nodes for the unit square [-1,1]^2.
squareintq_dr.f - driver routines for squareintq.f
makefile.squareintq - makefile for squareintq.f
( make -f makefile.squareintq )
c interp 0 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------------
c quadr 1 2 4 5 7 9 11 12 14 16 18
c -----------------------------------------------------------------------
c nodes 1 3 6 10 15 21 28 36 45 55 66
c -----------------------------------------------------------------------
c cond # 1.0 1.4 1.8 2.1 2.8 3.1 3.8 3.6 4.1 4.6 5.7
c
c
c interp 11 12 13 14 15 16 17 18 19 20
c -----------------------------------------------------------------------
c quadr 19 20 22 23 26 27 28 30 32 34
c -----------------------------------------------------------------------
c nodes 78 91 105 120 136 153 171 190 210 231
c -----------------------------------------------------------------------
c cond # 8.4 6.1 7.7 6.9 8.8 11.1 9.1 10.7 10.0 13.0
tetraarbq.f - construct (or rather retrieve) a quadrature formula for
the standard tetrahedron, defined by vertices
(-1,-1/Sqrt[3],-1/Sqrt[6]), (0,2/Sqrt[3],-1/Sqrt[6]),
(1,-1/Sqrt[3],-1/Sqrt[6]), and (0,0,3/Sqrt[6]).
Only orders 1 and 5 are fully symmetric.
tetraarbq_dr.f - driver routines for tetraarbq.f
makefile.tetraarbq - makefile for tetraarbq.f ( make -f makefile.tetraarbq )
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1f 4 6 11 14f 23 31 44 56 74
c
c n 11 12 13 14 15
c -----------------------------------------------------------------
c nodes 95 122 146 177 214
c
c
c f - fully symmetric
tetraintq.f - construct (or rather retrieve) fully symmetric
interpolation nodes for
the standard tetrahedron, defined by vertices
(-1,-1/Sqrt[3],-1/Sqrt[6]), (0,2/Sqrt[3],-1/Sqrt[6]),
(1,-1/Sqrt[3],-1/Sqrt[6]), and (0,0,3/Sqrt[6]).
tetraintq_dr.f - driver routines for tetraintq.f
makefile.tetraintq - makefile for tetraintq.f ( make -f makefile.tetraintq )
c interp 0 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------------
c quadr 1 2 3 5 6 7 9 10 11 13 15
c -----------------------------------------------------------------------
c nodes 1 4 10 20 35 56 84 120 165 220 286
c -----------------------------------------------------------------------
c cond # 1.0 1.0 1.6 1.9 3.4 3.5 4.6 10.7 12.7 16.8 37.0
cubearbq.f - construct (or rather retrieve) a quadrature formula for
the unit cube [-1,1]^3. These are non-symmetric quadrature nodes in
general, but come in pairs for orders 3, 5, 7, 9, 11, 13, and 15.
cubearbq_dr.f - driver routines for cubearbq.f
makefile.cubearbq - makefile for cubearbq.f ( make -f makefile.cubearbq )
c n 1 2 3 4 5 6 7 8 9 10
c -----------------------------------------------------------------
c nodes 1c 4 6c 10 13c* 22 26c* 42 48c* 73
c
c
c n 11 12 13 14 15
c -----------------------------------------------------------------
c nodes 82c* 115 129c* 173 188c*
c
c
c c - centro-symmetric
c * - number of quadrature nodes is less than (n+1)*(n+2)*(n+3)/6/4
lege2eva.f - orthogonal polynomials on a unit square
lege3eva.f - orthogonal polynomials on a unit cube
ortho2eva.f - orthogonal polynomials on a triangle
ortho3eva.f - orthogonal polynomials on a tetrahedron
tetragauc.f - tensor product rule for smooth functions on a tetrahedron
gaussq.f - Gaussian type quadratures (from netlib.org)
legeexps.f - Legendre expansion routines
prini.f - printing routines
gen_table - helper script to generate *_table.txt files
tables - raw data files for import to matlab, octave, and fortran
-
All tables have bee recomputed in quad precision, moderate improvements have been made for squareq and cubeq routines.
-
The library is released under a modified 3-clause BSD license.
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 (2009), pp. 663-676