# [Dune] Quadrature Rule 4 & 5 Identical

Tue Mar 6 11:05:55 CET 2012

```Hi,

I was just cross-checking quadrature rules of dune (as I had a problem) by
printing out the weights and points and it looks to me that the quadrature
rule of order 4 and order 5 are exactly the same. Is this a bug?

Please find attached my test code (integration.cc) & output (jj.txt).

Regards,
Alex
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Weights and gauss points for cubes for quadrature rule of order 1

#1	weight = 1	position: 0.5 0.5 0.5
sum of weights = 1

Weights and gauss points for cubes for quadrature rule of order 2

#1	weight = 0.125	position: 0.211325 0.211325 0.211325
#2	weight = 0.125	position: 0.788675 0.211325 0.211325
#3	weight = 0.125	position: 0.211325 0.788675 0.211325
#4	weight = 0.125	position: 0.788675 0.788675 0.211325
#5	weight = 0.125	position: 0.211325 0.211325 0.788675
#6	weight = 0.125	position: 0.788675 0.211325 0.788675
#7	weight = 0.125	position: 0.211325 0.788675 0.788675
#8	weight = 0.125	position: 0.788675 0.788675 0.788675
sum of weights = 1

Weights and gauss points for cubes for quadrature rule of order 3

#1	weight = 0.125	position: 0.211325 0.211325 0.211325
#2	weight = 0.125	position: 0.788675 0.211325 0.211325
#3	weight = 0.125	position: 0.211325 0.788675 0.211325
#4	weight = 0.125	position: 0.788675 0.788675 0.211325
#5	weight = 0.125	position: 0.211325 0.211325 0.788675
#6	weight = 0.125	position: 0.788675 0.211325 0.788675
#7	weight = 0.125	position: 0.211325 0.788675 0.788675
#8	weight = 0.125	position: 0.788675 0.788675 0.788675
sum of weights = 1

Weights and gauss points for cubes for quadrature rule of order 4

#1	weight = 0.0214335	position: 0.112702 0.112702 0.112702
#2	weight = 0.0342936	position: 0.5 0.112702 0.112702
#3	weight = 0.0214335	position: 0.887298 0.112702 0.112702
#4	weight = 0.0342936	position: 0.112702 0.5 0.112702
#5	weight = 0.0548697	position: 0.5 0.5 0.112702
#6	weight = 0.0342936	position: 0.887298 0.5 0.112702
#7	weight = 0.0214335	position: 0.112702 0.887298 0.112702
#8	weight = 0.0342936	position: 0.5 0.887298 0.112702
#9	weight = 0.0214335	position: 0.887298 0.887298 0.112702
#10	weight = 0.0342936	position: 0.112702 0.112702 0.5
#11	weight = 0.0548697	position: 0.5 0.112702 0.5
#12	weight = 0.0342936	position: 0.887298 0.112702 0.5
#13	weight = 0.0548697	position: 0.112702 0.5 0.5
#14	weight = 0.0877915	position: 0.5 0.5 0.5
#15	weight = 0.0548697	position: 0.887298 0.5 0.5
#16	weight = 0.0342936	position: 0.112702 0.887298 0.5
#17	weight = 0.0548697	position: 0.5 0.887298 0.5
#18	weight = 0.0342936	position: 0.887298 0.887298 0.5
#19	weight = 0.0214335	position: 0.112702 0.112702 0.887298
#20	weight = 0.0342936	position: 0.5 0.112702 0.887298
#21	weight = 0.0214335	position: 0.887298 0.112702 0.887298
#22	weight = 0.0342936	position: 0.112702 0.5 0.887298
#23	weight = 0.0548697	position: 0.5 0.5 0.887298
#24	weight = 0.0342936	position: 0.887298 0.5 0.887298
#25	weight = 0.0214335	position: 0.112702 0.887298 0.887298
#26	weight = 0.0342936	position: 0.5 0.887298 0.887298
#27	weight = 0.0214335	position: 0.887298 0.887298 0.887298
sum of weights = 1

Weights and gauss points for simplices for quadrature rule of order 1:

#1	weight = 0.166667	position: 0.25 0.25 0.25
sum of weights = 0.166667

Weights and gauss points for simplices for quadrature rule of order 2:

#1	weight = 0.0416667	position: 0.58541 0.138197 0.138197
#2	weight = 0.0416667	position: 0.138197 0.58541 0.138197
#3	weight = 0.0416667	position: 0.138197 0.138197 0.58541
#4	weight = 0.0416667	position: 0.138197 0.138197 0.138197
sum of weights = 0.166667

Weights and gauss points for simplices for quadrature rule of order 3:

#1	weight = 0.00416667	position: 0 0 0
#2	weight = 0.00416667	position: 1 0 0
#3	weight = 0.00416667	position: 0 1 0
#4	weight = 0.00416667	position: 0 0 1
#5	weight = 0.0375	position: 0.333333 0.333333 0
#6	weight = 0.0375	position: 0.333333 0 0.333333
#7	weight = 0.0375	position: 0 0.333333 0.333333
#8	weight = 0.0375	position: 0.333333 0.333333 0.333333
sum of weights = 0.166667

Weights and gauss points for simplices for quadrature rule of order 4:

#1	weight = 0.0197531	position: 0.25 0.25 0.25
#2	weight = 0.0119895	position: 0.0919711 0.0919711 0.0919711
#3	weight = 0.0119895	position: 0.724087 0.0919711 0.0919711
#4	weight = 0.0119895	position: 0.0919711 0.724087 0.0919711
#5	weight = 0.0119895	position: 0.0919711 0.0919711 0.724087
#6	weight = 0.0115114	position: 0.319794 0.319794 0.319794
#7	weight = 0.0115114	position: 0.0406191 0.319794 0.319794
#8	weight = 0.0115114	position: 0.319794 0.0406191 0.319794
#9	weight = 0.0115114	position: 0.319794 0.319794 0.0406191
#10	weight = 0.00881834	position: 0.443649 0.0563508 0.0563508
#11	weight = 0.00881834	position: 0.0563508 0.443649 0.0563508
#12	weight = 0.00881834	position: 0.0563508 0.0563508 0.443649
#13	weight = 0.00881834	position: 0.443649 0.443649 0.0563508
#14	weight = 0.00881834	position: 0.443649 0.0563508 0.443649
#15	weight = 0.00881834	position: 0.0563508 0.443649 0.443649
sum of weights = 0.166667

Weights and gauss points for simplices for quadrature rule of order 5:

#1	weight = 0.0197531	position: 0.25 0.25 0.25
#2	weight = 0.0119895	position: 0.0919711 0.0919711 0.0919711
#3	weight = 0.0119895	position: 0.724087 0.0919711 0.0919711
#4	weight = 0.0119895	position: 0.0919711 0.724087 0.0919711
#5	weight = 0.0119895	position: 0.0919711 0.0919711 0.724087
#6	weight = 0.0115114	position: 0.319794 0.319794 0.319794
#7	weight = 0.0115114	position: 0.0406191 0.319794 0.319794
#8	weight = 0.0115114	position: 0.319794 0.0406191 0.319794
#9	weight = 0.0115114	position: 0.319794 0.319794 0.0406191
#10	weight = 0.00881834	position: 0.443649 0.0563508 0.0563508
#11	weight = 0.00881834	position: 0.0563508 0.443649 0.0563508
#12	weight = 0.00881834	position: 0.0563508 0.0563508 0.443649
#13	weight = 0.00881834	position: 0.443649 0.443649 0.0563508
#14	weight = 0.00881834	position: 0.443649 0.0563508 0.443649
#15	weight = 0.00881834	position: 0.0563508 0.443649 0.443649
sum of weights = 0.166667

Weights and gauss points for simplices for quadrature rule of order 6:

#1	weight = 0.00302926	position: 0.0469101 0.0661748 0.0615802
#2	weight = 0.00398631	position: 0.230765 0.0534094 0.0497011
#3	weight = 0.00200181	position: 0.5 0.0347159 0.0323055
#4	weight = 0.000358752	position: 0.769235 0.0160225 0.01491
#5	weight = 7.33839e-06	position: 0.95309 0.00325705 0.00303091
#6	weight = 0.00408887	position: 0.0469101 0.314529 0.0443365
#7	weight = 0.00538068	position: 0.230765 0.253855 0.0357838
#8	weight = 0.00270202	position: 0.5 0.165005 0.0232593
#9	weight = 0.00048424	position: 0.769235 0.0761548 0.0107349
#10	weight = 9.9053e-06	position: 0.95309 0.0154808 0.00218219
#11	weight = 0.00201401	position: 0.0469101 0.638561 0.0218383
#12	weight = 0.0026503	position: 0.230765 0.51538 0.0176256
#13	weight = 0.0013309	position: 0.5 0.334995 0.0114566
#14	weight = 0.000238517	position: 0.769235 0.154611 0.00528756
#15	weight = 4.87894e-06	position: 0.95309 0.0314293 0.00107486
#16	weight = 0.00022602	position: 0.0469101 0.886915 0.00459464
#17	weight = 0.000297428	position: 0.230765 0.715825 0.00370831
#18	weight = 0.000149359	position: 0.5 0.465284 0.00241039
#19	weight = 2.67673e-05	position: 0.769235 0.214743 0.00111247
#20	weight = 5.47534e-07	position: 0.95309 0.043653 0.000226143
#21	weight = 0.00567914	position: 0.0469101 0.0661748 0.29269
#22	weight = 0.00747338	position: 0.230765 0.0534094 0.236229
#23	weight = 0.00375291	position: 0.5 0.0347159 0.153548
#24	weight = 0.000672575	position: 0.769235 0.0160225 0.0708672
#25	weight = 1.37577e-05	position: 0.95309 0.00325705 0.0144059
#26	weight = 0.00766566	position: 0.0469101 0.314529 0.210731
#27	weight = 0.0100875	position: 0.230765 0.253855 0.17008
#28	weight = 0.00506564	position: 0.5 0.165005 0.110552
#29	weight = 0.000907835	position: 0.769235 0.0761548 0.051023
#30	weight = 1.85701e-05	position: 0.95309 0.0154808 0.010372
#31	weight = 0.00377578	position: 0.0469101 0.638561 0.103797
#32	weight = 0.00496868	position: 0.230765 0.51538 0.0837745
#33	weight = 0.00249513	position: 0.5 0.334995 0.0544531
#34	weight = 0.000447162	position: 0.769235 0.154611 0.0251318
#35	weight = 9.14685e-06	position: 0.95309 0.0314293 0.0051088
#36	weight = 0.000423734	position: 0.0469101 0.886915 0.0218383
#37	weight = 0.000557606	position: 0.230765 0.715825 0.0176256
#38	weight = 0.000280013	position: 0.5 0.465284 0.0114566
#39	weight = 5.01823e-05	position: 0.769235 0.214743 0.00528756
#40	weight = 1.0265e-06	position: 0.95309 0.043653 0.00107486
#41	weight = 0.00567914	position: 0.0469101 0.0661748 0.594225
#42	weight = 0.00747338	position: 0.230765 0.0534094 0.479596
#43	weight = 0.00375291	position: 0.5 0.0347159 0.311736
#44	weight = 0.000672575	position: 0.769235 0.0160225 0.143876
#45	weight = 1.37577e-05	position: 0.95309 0.00325705 0.0292471
#46	weight = 0.00766566	position: 0.0469101 0.314529 0.42783
#47	weight = 0.0100875	position: 0.230765 0.253855 0.3453
#48	weight = 0.00506564	position: 0.5 0.165005 0.224444
#49	weight = 0.000907835	position: 0.769235 0.0761548 0.103588
#50	weight = 1.85701e-05	position: 0.95309 0.0154808 0.0210573
#51	weight = 0.00377578	position: 0.0469101 0.638561 0.210731
#52	weight = 0.00496868	position: 0.230765 0.51538 0.17008
#53	weight = 0.00249513	position: 0.5 0.334995 0.110552
#54	weight = 0.000447162	position: 0.769235 0.154611 0.051023
#55	weight = 9.14685e-06	position: 0.95309 0.0314293 0.010372
#56	weight = 0.000423734	position: 0.0469101 0.886915 0.0443365
#57	weight = 0.000557606	position: 0.230765 0.715825 0.0357838
#58	weight = 0.000280013	position: 0.5 0.465284 0.0232593
#59	weight = 5.01823e-05	position: 0.769235 0.214743 0.0107349
#60	weight = 1.0265e-06	position: 0.95309 0.043653 0.00218219
#61	weight = 0.00302926	position: 0.0469101 0.0661748 0.825335
#62	weight = 0.00398631	position: 0.230765 0.0534094 0.666124
#63	weight = 0.00200181	position: 0.5 0.0347159 0.432979
#64	weight = 0.000358752	position: 0.769235 0.0160225 0.199833
#65	weight = 7.33839e-06	position: 0.95309 0.00325705 0.0406221
#66	weight = 0.00408887	position: 0.0469101 0.314529 0.594225
#67	weight = 0.00538068	position: 0.230765 0.253855 0.479596
#68	weight = 0.00270202	position: 0.5 0.165005 0.311736
#69	weight = 0.00048424	position: 0.769235 0.0761548 0.143876
#70	weight = 9.9053e-06	position: 0.95309 0.0154808 0.0292471
#71	weight = 0.00201401	position: 0.0469101 0.638561 0.29269
#72	weight = 0.0026503	position: 0.230765 0.51538 0.236229
#73	weight = 0.0013309	position: 0.5 0.334995 0.153548
#74	weight = 0.000238517	position: 0.769235 0.154611 0.0708672
#75	weight = 4.87894e-06	position: 0.95309 0.0314293 0.0144059
#76	weight = 0.00022602	position: 0.0469101 0.886915 0.0615802
#77	weight = 0.000297428	position: 0.230765 0.715825 0.0497011
#78	weight = 0.000149359	position: 0.5 0.465284 0.0323055
#79	weight = 2.67673e-05	position: 0.769235 0.214743 0.01491
#80	weight = 5.47534e-07	position: 0.95309 0.043653 0.00303091
sum of weights = 0.166667
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