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History of Newton–Cotes weights

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2026-09-17 13:10 zeta3 table-build@2.0+395f185d use existing Newton-Cotes tag current reviewed
2026-09-17 13:08 zeta3 table-build@2.0+395f185d tighten Newton-Cotes definition
2026-09-17 13:07 zeta3 table-build@2.0+395f185d clarify Newton-Cotes normalisations
2026-09-17 13:06 zeta3 table-build@2.0+395f185d fix Newton-Cotes prose and snippet
2026-09-17 13:05 zeta3 with Codex CLI, table-build@0 table-build@2.0+395f185d Newton-Cotes weights
2026-09-17 13:00 zeta3 table-build@2.0+395f185d claim Newton-Cotes weights draft

What changed between 2026-09-17 13:00 and 2026-09-17 13:05

from line 105 (915 lines, 912 more than before) @@ -105,3 +105,915 @@ 
 Display properties:   number-header: $w_{n,j}$ or $C_{n,j}$-Numbers: []+Numbers:+- params:+    n: '1'+    j: '0'+    normalisation: step+  number: 1/2+  comment: 'Trapezoidal rule: $\tfrac{1}{2}(1,1)$.'+- params:+    n: '1'+    j: '0'+    normalisation: unit-interval+  number: 1/2+- params:+    n: '1'+    j: '1'+    normalisation: step+  number: 1/2+- params:+    n: '1'+    j: '1'+    normalisation: unit-interval+  number: 1/2+- params:+    n: '2'+    j: '0'+    normalisation: step+  number: 1/3+  comment: 'Simpson''s rule: $\tfrac{1}{3}(1,4,1)$.'+- params:+    n: '2'+    j: '0'+    normalisation: unit-interval+  number: 1/6+- params:+    n: '2'+    j: '1'+    normalisation: step+  number: 4/3+- params:+    n: '2'+    j: '1'+    normalisation: unit-interval+  number: 2/3+- params:+    n: '2'+    j: '2'+    normalisation: step+  number: 1/3+- params:+    n: '2'+    j: '2'+    normalisation: unit-interval+  number: 1/6+- params:+    n: '3'+    j: '0'+    normalisation: step+  number: 3/8+  comment: 'Simpson''s 3/8 rule: $\tfrac{3}{8}(1,3,3,1)$.'+- params:+    n: '3'+    j: '0'+    normalisation: unit-interval+  number: 1/8+- params:+    n: '3'+    j: '1'+    normalisation: step+  number: 9/8+- params:+    n: '3'+    j: '1'+    normalisation: unit-interval+  number: 3/8+- params:+    n: '3'+    j: '2'+    normalisation: step+  number: 9/8+- params:+    n: '3'+    j: '2'+    normalisation: unit-interval+  number: 3/8+- params:+    n: '3'+    j: '3'+    normalisation: step+  number: 3/8+- params:+    n: '3'+    j: '3'+    normalisation: unit-interval+  number: 1/8+- params:+    n: '4'+    j: '0'+    normalisation: step+  number: 14/45+  comment: 'Boole''s rule: $\tfrac{2}{45}(7,32,12,32,7)$.'+- params:+    n: '4'+    j: '0'+    normalisation: unit-interval+  number: 7/90+- params:+    n: '4'+    j: '1'+    normalisation: step+  number: 64/45+- params:+    n: '4'+    j: '1'+    normalisation: unit-interval+  number: 16/45+- params:+    n: '4'+    j: '2'+    normalisation: step+  number: 8/15+- params:+    n: '4'+    j: '2'+    normalisation: unit-interval+  number: 2/15+- params:+    n: '4'+    j: '3'+    normalisation: step+  number: 64/45+- params:+    n: '4'+    j: '3'+    normalisation: unit-interval+  number: 16/45+- params:+    n: '4'+    j: '4'+    normalisation: step+  number: 14/45+- params:+    n: '4'+    j: '4'+    normalisation: unit-interval+  number: 7/90+- params:+    n: '5'+    j: '0'+    normalisation: step+  number: 95/288+  comment: 'The closed Newton-Cotes rule with $n=5$: $\tfrac{5}{288}(19,75,50,50,75,19)$.'+- params:+    n: '5'+    j: '0'+    normalisation: unit-interval+  number: 19/288+- params:+    n: '5'+    j: '1'+    normalisation: step+  number: 125/96+- params:+    n: '5'+    j: '1'+    normalisation: unit-interval+  number: 25/96+- params:+    n: '5'+    j: '2'+    normalisation: step+  number: 125/144+- params:+    n: '5'+    j: '2'+    normalisation: unit-interval+  number: 25/144+- params:+    n: '5'+    j: '3'+    normalisation: step+  number: 125/144+- params:+    n: '5'+    j: '3'+    normalisation: unit-interval+  number: 25/144+- params:+    n: '5'+    j: '4'+    normalisation: step+  number: 125/96+- params:+    n: '5'+    j: '4'+    normalisation: unit-interval+  number: 25/96+- params:+    n: '5'+    j: '5'+    normalisation: step+  number: 95/288+- params:+    n: '5'+    j: '5'+    normalisation: unit-interval+  number: 19/288+- params:+    n: '6'+    j: '0'+    normalisation: step+  number: 41/140+  comment: 'The closed Newton-Cotes rule with $n=6$: $\tfrac{1}{140}(41,216,27,272,27,216,41)$.'+- params:+    n: '6'+    j: '0'+    normalisation: unit-interval+  number: 41/840+- params:+    n: '6'+    j: '1'+    normalisation: step+  number: 54/35+- params:+    n: '6'+    j: '1'+    normalisation: unit-interval+  number: 9/35+- params:+    n: '6'+    j: '2'+    normalisation: step+  number: 27/140+- params:+    n: '6'+    j: '2'+    normalisation: unit-interval+  number: 9/280+- params:+    n: '6'+    j: '3'+    normalisation: step+  number: 68/35+- params:+    n: '6'+    j: '3'+    normalisation: unit-interval+  number: 34/105+- params:+    n: '6'+    j: '4'+    normalisation: step+  number: 27/140+- params:+    n: '6'+    j: '4'+    normalisation: unit-interval+  number: 9/280+- params:+    n: '6'+    j: '5'+    normalisation: step+  number: 54/35+- params:+    n: '6'+    j: '5'+    normalisation: unit-interval+  number: 9/35+- params:+    n: '6'+    j: '6'+    normalisation: step+  number: 41/140+- params:+    n: '6'+    j: '6'+    normalisation: unit-interval+  number: 41/840+- params:+    n: '7'+    j: '0'+    normalisation: step+  number: 5257/17280+  comment: 'The closed Newton-Cotes rule with $n=7$: $\tfrac{7}{17280}(751,3577,1323,2989,2989,1323,3577,751)$.'+- params:+    n: '7'+    j: '0'+    normalisation: unit-interval+  number: 751/17280+- params:+    n: '7'+    j: '1'+    normalisation: step+  number: 25039/17280+- params:+    n: '7'+    j: '1'+    normalisation: unit-interval+  number: 3577/17280+- params:+    n: '7'+    j: '2'+    normalisation: step+  number: 343/640+- params:+    n: '7'+    j: '2'+    normalisation: unit-interval+  number: 49/640+- params:+    n: '7'+    j: '3'+    normalisation: step+  number: 20923/17280+- params:+    n: '7'+    j: '3'+    normalisation: unit-interval+  number: 2989/17280+- params:+    n: '7'+    j: '4'+    normalisation: step+  number: 20923/17280+- params:+    n: '7'+    j: '4'+    normalisation: unit-interval+  number: 2989/17280+- params:+    n: '7'+    j: '5'+    normalisation: step+  number: 343/640+- params:+    n: '7'+    j: '5'+    normalisation: unit-interval+  number: 49/640+- params:+    n: '7'+    j: '6'+    normalisation: step+  number: 25039/17280+- params:+    n: '7'+    j: '6'+    normalisation: unit-interval+  number: 3577/17280+- params:+    n: '7'+    j: '7'+    normalisation: step+  number: 5257/17280+- params:+    n: '7'+    j: '7'+    normalisation: unit-interval+  number: 751/17280+- params:+    n: '8'+    j: '0'+    normalisation: step+  number: 3956/14175+  comment: 'The closed Newton-Cotes rule with $n=8$: $\tfrac{4}{14175}(989,5888,-928,10496,-4540,10496,-928,5888,989)$.'+- params:+    n: '8'+    j: '0'+    normalisation: unit-interval+  number: 989/28350+- params:+    n: '8'+    j: '1'+    normalisation: step+  number: 23552/14175+- params:+    n: '8'+    j: '1'+    normalisation: unit-interval+  number: 2944/14175+- params:+    n: '8'+    j: '2'+    normalisation: step+  number: -3712/14175+- params:+    n: '8'+    j: '2'+    normalisation: unit-interval+  number: -464/14175+- params:+    n: '8'+    j: '3'+    normalisation: step+  number: 41984/14175+- params:+    n: '8'+    j: '3'+    normalisation: unit-interval+  number: 5248/14175+- params:+    n: '8'+    j: '4'+    normalisation: step+  number: -3632/2835+- params:+    n: '8'+    j: '4'+    normalisation: unit-interval+  number: -454/2835+- params:+    n: '8'+    j: '5'+    normalisation: step+  number: 41984/14175+- params:+    n: '8'+    j: '5'+    normalisation: unit-interval+  number: 5248/14175+- params:+    n: '8'+    j: '6'+    normalisation: step+  number: -3712/14175+- params:+    n: '8'+    j: '6'+    normalisation: unit-interval+  number: -464/14175+- params:+    n: '8'+    j: '7'+    normalisation: step+  number: 23552/14175+- params:+    n: '8'+    j: '7'+    normalisation: unit-interval+  number: 2944/14175+- params:+    n: '8'+    j: '8'+    normalisation: step+  number: 3956/14175+- params:+    n: '8'+    j: '8'+    normalisation: unit-interval+  number: 989/28350+- params:+    n: '9'+    j: '0'+    normalisation: step+  number: 25713/89600+  comment: 'The closed Newton-Cotes rule with $n=9$: $\tfrac{9}{89600}(2857,15741,1080,19344,5778,5778,19344,1080,15741,2857)$.'+- params:+    n: '9'+    j: '0'+    normalisation: unit-interval+  number: 2857/89600+- params:+    n: '9'+    j: '1'+    normalisation: step+  number: 141669/89600+- params:+    n: '9'+    j: '1'+    normalisation: unit-interval+  number: 15741/89600+- params:+    n: '9'+    j: '2'+    normalisation: step+  number: 243/2240+- params:+    n: '9'+    j: '2'+    normalisation: unit-interval+  number: 27/2240+- params:+    n: '9'+    j: '3'+    normalisation: step+  number: 10881/5600+- params:+    n: '9'+    j: '3'+    normalisation: unit-interval+  number: 1209/5600+- params:+    n: '9'+    j: '4'+    normalisation: step+  number: 26001/44800+- params:+    n: '9'+    j: '4'+    normalisation: unit-interval+  number: 2889/44800+- params:+    n: '9'+    j: '5'+    normalisation: step+  number: 26001/44800+- params:+    n: '9'+    j: '5'+    normalisation: unit-interval+  number: 2889/44800+- params:+    n: '9'+    j: '6'+    normalisation: step+  number: 10881/5600+- params:+    n: '9'+    j: '6'+    normalisation: unit-interval+  number: 1209/5600+- params:+    n: '9'+    j: '7'+    normalisation: step+  number: 243/2240+- params:+    n: '9'+    j: '7'+    normalisation: unit-interval+  number: 27/2240+- params:+    n: '9'+    j: '8'+    normalisation: step+  number: 141669/89600+- params:+    n: '9'+    j: '8'+    normalisation: unit-interval+  number: 15741/89600+- params:+    n: '9'+    j: '9'+    normalisation: step+  number: 25713/89600+- params:+    n: '9'+    j: '9'+    normalisation: unit-interval+  number: 2857/89600+- params:+    n: '10'+    j: '0'+    normalisation: step+  number: 80335/299376+  comment: 'The closed Newton-Cotes rule with $n=10$: $\tfrac{5}{299376}(16067,106300,-48525,272400,-260550,427368,-260550,272400,-48525,106300,16067)$.'+- params:+    n: '10'+    j: '0'+    normalisation: unit-interval+  number: 16067/598752+- params:+    n: '10'+    j: '1'+    normalisation: step+  number: 132875/74844+- params:+    n: '10'+    j: '1'+    normalisation: unit-interval+  number: 26575/149688+- params:+    n: '10'+    j: '2'+    normalisation: step+  number: -80875/99792+- params:+    n: '10'+    j: '2'+    normalisation: unit-interval+  number: -16175/199584+- params:+    n: '10'+    j: '3'+    normalisation: step+  number: 28375/6237+- params:+    n: '10'+    j: '3'+    normalisation: unit-interval+  number: 5675/12474+- params:+    n: '10'+    j: '4'+    normalisation: step+  number: -24125/5544+- params:+    n: '10'+    j: '4'+    normalisation: unit-interval+  number: -4825/11088+- params:+    n: '10'+    j: '5'+    normalisation: step+  number: 89035/12474+- params:+    n: '10'+    j: '5'+    normalisation: unit-interval+  number: 17807/24948+- params:+    n: '10'+    j: '6'+    normalisation: step+  number: -24125/5544+- params:+    n: '10'+    j: '6'+    normalisation: unit-interval+  number: -4825/11088+- params:+    n: '10'+    j: '7'+    normalisation: step+  number: 28375/6237+- params:+    n: '10'+    j: '7'+    normalisation: unit-interval+  number: 5675/12474+- params:+    n: '10'+    j: '8'+    normalisation: step+  number: -80875/99792+- params:+    n: '10'+    j: '8'+    normalisation: unit-interval+  number: -16175/199584+- params:+    n: '10'+    j: '9'+    normalisation: step+  number: 132875/74844+- params:+    n: '10'+    j: '9'+    normalisation: unit-interval+  number: 26575/149688+- params:+    n: '10'+    j: '10'+    normalisation: step+  number: 80335/299376+- params:+    n: '10'+    j: '10'+    normalisation: unit-interval+  number: 16067/598752+- params:+    n: '11'+    j: '0'+    normalisation: step+  number: 4777223/17418240+  comment: 'The closed Newton-Cotes rule with $n=11$: $\tfrac{11}{87091200}(2171465,13486539,-3237113,25226685,-9595542,15493566,15493566,-9595542,25226685,-3237113,13486539,2171465)$.'+- params:+    n: '11'+    j: '0'+    normalisation: unit-interval+  number: 434293/17418240+- params:+    n: '11'+    j: '1'+    normalisation: step+  number: 49450643/29030400+- params:+    n: '11'+    j: '1'+    normalisation: unit-interval+  number: 4495513/29030400+- params:+    n: '11'+    j: '2'+    normalisation: step+  number: -35608243/87091200+- params:+    n: '11'+    j: '2'+    normalisation: unit-interval+  number: -3237113/87091200+- params:+    n: '11'+    j: '3'+    normalisation: step+  number: 6166523/1935360+- params:+    n: '11'+    j: '3'+    normalisation: unit-interval+  number: 560593/1935360+- params:+    n: '11'+    j: '4'+    normalisation: step+  number: -17591827/14515200+- params:+    n: '11'+    j: '4'+    normalisation: unit-interval+  number: -1599257/14515200+- params:+    n: '11'+    j: '5'+    normalisation: step+  number: 28404871/14515200+- params:+    n: '11'+    j: '5'+    normalisation: unit-interval+  number: 2582261/14515200+- params:+    n: '11'+    j: '6'+    normalisation: step+  number: 28404871/14515200+- params:+    n: '11'+    j: '6'+    normalisation: unit-interval+  number: 2582261/14515200+- params:+    n: '11'+    j: '7'+    normalisation: step+  number: -17591827/14515200+- params:+    n: '11'+    j: '7'+    normalisation: unit-interval+  number: -1599257/14515200+- params:+    n: '11'+    j: '8'+    normalisation: step+  number: 6166523/1935360+- params:+    n: '11'+    j: '8'+    normalisation: unit-interval+  number: 560593/1935360+- params:+    n: '11'+    j: '9'+    normalisation: step+  number: -35608243/87091200+- params:+    n: '11'+    j: '9'+    normalisation: unit-interval+  number: -3237113/87091200+- params:+    n: '11'+    j: '10'+    normalisation: step+  number: 49450643/29030400+- params:+    n: '11'+    j: '10'+    normalisation: unit-interval+  number: 4495513/29030400+- params:+    n: '11'+    j: '11'+    normalisation: step+  number: 4777223/17418240+- params:+    n: '11'+    j: '11'+    normalisation: unit-interval+  number: 434293/17418240+- params:+    n: '12'+    j: '0'+    normalisation: step+  number: 1364651/5255250+  comment: 'The closed Newton-Cotes rule with $n=12$: $\tfrac{1}{5255250}(1364651,9903168,-7587864,35725120,-51491295,87516288,-87797136,87516288,-51491295,35725120,-7587864,9903168,1364651)$.'+- params:+    n: '12'+    j: '0'+    normalisation: unit-interval+  number: 1364651/63063000+- params:+    n: '12'+    j: '1'+    normalisation: step+  number: 150048/79625+- params:+    n: '12'+    j: '1'+    normalisation: unit-interval+  number: 12504/79625+- params:+    n: '12'+    j: '2'+    normalisation: step+  number: -1264644/875875+- params:+    n: '12'+    j: '2'+    normalisation: unit-interval+  number: -105387/875875+- params:+    n: '12'+    j: '3'+    normalisation: step+  number: 3572512/525525+- params:+    n: '12'+    j: '3'+    normalisation: unit-interval+  number: 893128/1576575+- params:+    n: '12'+    j: '4'+    normalisation: step+  number: -3432753/350350+- params:+    n: '12'+    j: '4'+    normalisation: unit-interval+  number: -1144251/1401400+- params:+    n: '12'+    j: '5'+    normalisation: step+  number: 14586048/875875+- params:+    n: '12'+    j: '5'+    normalisation: unit-interval+  number: 1215504/875875+- params:+    n: '12'+    j: '6'+    normalisation: step+  number: -2090408/125125+- params:+    n: '12'+    j: '6'+    normalisation: unit-interval+  number: -522602/375375+- params:+    n: '12'+    j: '7'+    normalisation: step+  number: 14586048/875875+- params:+    n: '12'+    j: '7'+    normalisation: unit-interval+  number: 1215504/875875+- params:+    n: '12'+    j: '8'+    normalisation: step+  number: -3432753/350350+- params:+    n: '12'+    j: '8'+    normalisation: unit-interval+  number: -1144251/1401400+- params:+    n: '12'+    j: '9'+    normalisation: step+  number: 3572512/525525+- params:+    n: '12'+    j: '9'+    normalisation: unit-interval+  number: 893128/1576575+- params:+    n: '12'+    j: '10'+    normalisation: step+  number: -1264644/875875+- params:+    n: '12'+    j: '10'+    normalisation: unit-interval+  number: -105387/875875+- params:+    n: '12'+    j: '11'+    normalisation: step+  number: 150048/79625+- params:+    n: '12'+    j: '11'+    normalisation: unit-interval+  number: 12504/79625+- params:+    n: '12'+    j: '12'+    normalisation: step+  number: 1364651/5255250+- params:+    n: '12'+    j: '12'+    normalisation: unit-interval+  number: 1364651/63063000

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