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using LinearAlgebra | ||
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# order 2 | ||
Qp = [ | ||
-1//4 5//4 -1//2 0 0 0 | ||
-1//4 -5//4 2 -1//2 0 0 | ||
0 0 -3//2 2 -1//2 0 | ||
0 0 0 -3//2 2 -1//2 | ||
0 0 0 0 -5//4 5//4 | ||
0 0 0 0 -1//4 -1//4 | ||
] | ||
Qm = Matrix(-Qp') | ||
B = zero(Qp); B[1,1] = -1; B[end,end] = 1 | ||
M = Diagonal([1//4, 5//4, 1, 1, 5//4, 1//4]) | ||
Dp = M \ (Qp + B / 2) | ||
Dm = M \ (Qm + B / 2) | ||
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# order 3 | ||
Qp = [ | ||
-1//12 3//4 -1//6 0 0 0 | ||
-5//12 -5//12 1 -1//6 0 0 | ||
0 -1//3 -1//2 1 -1//6 0 | ||
0 0 -1//3 -1//2 1 -1//6 | ||
0 0 0 -1//3 -5//12 3//4 | ||
0 0 0 0 -5//12 -1//12 | ||
] | ||
Qm = Matrix(-Qp') | ||
B = zero(Qp); B[1,1] = -1; B[end,end] = 1 | ||
M = Diagonal([5//12, 13//12, 1, 1, 13//12, 5//12]) | ||
Dp = M \ (Qp + B / 2) | ||
Dm = M \ (Qm + B / 2) | ||
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# order 4 | ||
Qp = [ | ||
-1//48 205//288 -29//144 1//96 0 0 0 0 0 0 0 | ||
-169//288 -11//48 33//32 -43//144 1//12 0 0 0 0 0 0 | ||
11//144 -13//32 -29//48 389//288 -1//2 1//12 0 0 0 0 0 | ||
1//32 -11//144 -65//288 -13//16 3//2 -1//2 1//12 0 0 0 0 | ||
0 0 0 -1//4 -5//6 3//2 -1//2 1//12 0 0 0 | ||
0 0 0 0 -1//4 -5//6 3//2 -1//2 1//12 0 0 | ||
0 0 0 0 0 -1//4 -5//6 3//2 -1//2 1//12 0 | ||
0 0 0 0 0 0 -1//4 -13//16 389//288 -43//144 1//96 | ||
0 0 0 0 0 0 0 -65//288 -29//48 33//32 -29//144 | ||
0 0 0 0 0 0 0 -11//144 -13//32 -11//48 205//288 | ||
0 0 0 0 0 0 0 1//32 11//144 -169//288 -1//48 | ||
] | ||
Qm = Matrix(-Qp') | ||
B = zero(Qp); B[1,1] = -1; B[end,end] = 1 | ||
m = [49//144, 61//48, 41//48, 149//144] | ||
M = Diagonal(vcat(m, ones(Int, size(Qp,1)-2*length(m)), m[end:-1:1])) | ||
Dp = M \ (Qp + B / 2) | ||
Dm = M \ (Qm + B / 2) | ||
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# order 5 | ||
Qp = [ | ||
-1//120 941//1440 -47//360 -7//480 0 0 0 0 0 0 0 | ||
-869//1440 -11//120 25//32 -43//360 1//30 0 0 0 0 0 0 | ||
29//360 -17//32 -29//120 1309//1440 -1//4 1//30 0 0 0 0 0 | ||
1//32 -11//360 -661//1440 -13//40 1 -1//4 1//30 0 0 0 0 | ||
0 0 1//20 -1//2 -1//3 1 -1//4 1//30 0 0 0 | ||
0 0 0 1//20 -1//2 -1//3 1 -1//4 1//30 0 0 | ||
0 0 0 0 1//20 -1//2 -1//3 1 -1//4 1//30 0 | ||
0 0 0 0 0 1//20 -1//2 -13//40 1309//1440 -43//360 -7//480 | ||
0 0 0 0 0 0 1//20 -661//1440 -29//120 25//32 -47//360 | ||
0 0 0 0 0 0 0 -11//360 -17//32 -11//120 941//1440 | ||
0 0 0 0 0 0 0 1//32 29//360 -869//1440 -1//120 | ||
] | ||
Qm = Matrix(-Qp') | ||
B = zero(Qp); B[1,1] = -1; B[end,end] = 1 | ||
# m = [251//720, 299//240, 41//48, 149//144] | ||
m = [251//720, 299//240, 211//240, 739//720] | ||
M = Diagonal(vcat(m, ones(Int, size(Qp,1)-2*length(m)), m[end:-1:1])) | ||
Dp = M \ (Qp + B / 2) | ||
Dm = M \ (Qm + B / 2) | ||
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# order 6 | ||
Qptmp = [ | ||
-265//128688 1146190567//1737288000 -1596619//18384000 -55265831//579096000 26269819//3474576000 2464501//144774000 0 0 0 0 0 0 | ||
-1116490567//1737288000 -8839//214480 190538869//347457600 102705469//694915200 413741//9651600 -191689861//3474576000 0 0 0 0 0 0 | ||
1096619//18384000 -135385429//347457600 -61067//321720 45137333//57909600 -253641811//694915200 70665929//579096000 -1//60 0 0 0 0 0 | ||
66965831//579096000 -208765789//694915200 -17623253//57909600 -18269//45960 410905829//347457600 -477953317//1158192000 2//15 -1//60 0 0 0 0 | ||
-49219819//3474576000 293299//9651600 26422771//694915200 -141938309//347457600 -346583//643440 2217185207//1737288000 -1//2 2//15 -1//60 0 0 0 | ||
-2374501//144774000 142906261//3474576000 -3137129//579096000 -29884283//1158192000 -630168407//1737288000 -3559//6128 4//3 -1//2 2//15 -1//60 0 0 | ||
0 0 0 0 1//30 -2//5 -7//12 4//3 -1//2 2//15 -1//60 0 | ||
0 0 0 0 0 1//30 -2//5 -7//12 4//3 -1//2 2//15 -1//60 | ||
] | ||
bnd = size(Qptmp, 1)-2 | ||
Qp = zeros(eltype(Qptmp), 2*bnd+2, 2*bnd+2) | ||
Qp[1:size(Qptmp,1), 1:size(Qptmp,2)] = Qptmp | ||
for i in 1:bnd | ||
Qp[end+1-i, end+1-size(Qptmp,1):end] = Qptmp[end:-1:1, i] | ||
end | ||
Qm = Matrix(-Qp') | ||
B = zero(Qp); B[1,1] = -1; B[end,end] = 1 | ||
m = [13613//43200, 12049//8640, 535//864, 1079//864, 7841//8640, 43837//43200] | ||
M = Diagonal(vcat(m, ones(Int, size(Qp,1)-2*length(m)), m[end:-1:1])) | ||
Dp = M \ (Qp + B / 2) | ||
Dm = M \ (Qm + B / 2) | ||
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# order 7 | ||
Qptmp = Rational{Int128}[ | ||
-265//300272 1587945773//2432203200 -1926361//25737600 -84398989//810734400 48781961//4864406400 3429119//202683600 0 0 0 0 0 0 0 | ||
-1570125773//2432203200 -26517//1501360 240029831//486440640 202934303//972881280 118207//13512240 -231357719//4864406400 0 0 0 0 0 0 0 | ||
1626361//25737600 -206937767//486440640 -61067//750680 49602727//81073440 -43783933//194576256 51815011//810734400 -1//140 0 0 0 0 0 0 | ||
91418989//810734400 -53314099//194576256 -33094279//81073440 -18269//107240 440626231//486440640 -365711063//1621468800 1//15 -1//140 0 0 0 0 0 | ||
-62551961//4864406400 799//35280 82588241//972881280 -279245719//486440640 -346583//1501360 2312302333//2432203200 -3//10 1//15 -1//140 0 0 0 0 | ||
-3375119//202683600 202087559//4864406400 -11297731//810734400 61008503//1621468800 -1360092253//2432203200 -10677//42896 1 -3//10 1//15 -1//140 0 0 0 | ||
0 0 0 -1//105 1//10 -3//5 -1//4 1 -3//10 1//15 -1//140 0 0 | ||
0 0 0 0 -1//105 1//10 -3//5 -1//4 1 -3//10 1//15 -1//140 0 | ||
0 0 0 0 0 -1//105 1//10 -3//5 -1//4 1 -3//10 1//15 -1//140 | ||
] | ||
bnd = size(Qptmp, 1)-3 | ||
Qp = zeros(eltype(Qptmp), 2*bnd+3, 2*bnd+3) | ||
Qp[1:size(Qptmp,1), 1:size(Qptmp,2)] = Qptmp | ||
for i in 1:bnd | ||
Qp[end+1-i, end+1-size(Qptmp,1):end] = Qptmp[end:-1:1, i] | ||
end | ||
Qm = Matrix(-Qp') | ||
B = zero(Qp); B[1,1] = -1; B[end,end] = 1 | ||
m = [19087//60480, 84199//60480, 18869//30240, 37621//30240, 55031//60480, 61343//60480] | ||
M = Diagonal(vcat(m, ones(Int, size(Qp,1)-2*length(m)), m[end:-1:1])) | ||
Dp = M \ (Qp + B / 2) | ||
Dm = M \ (Qm + B / 2) |
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