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Copy pathCDF2.m
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CDF2.m
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function cd = CDF2(k)
% central difference coefficients for the 2nd derivative
% Input:
% k: integer, such that 2k+1 points about the center are used, order of
% accuracy is 2k
% Output:
% cd: central difference coefficients at nodes 0:k
% (coefficients at nodes 0:-1:-k are the same)
if k <= 21
cd = precom_coeffs(k);
else
x = sym(0:k); % half of the -n:n grid
A=2*x.^((0:2:2*k)'); A(1)=1;% even moments Vandermonde, times 2 bc symmetry
b = zeros(k+1,1); b(2) = 2; % rhs: 2nd deriv of monomials eval at 0
cd = double(A\b); % sym central diff coeffs at 0, +/-1, +/-2, ...
end
cd = cd'; % row vector
function c = precom_coeffs(k)
% precomputed coeffs
switch k
case 0
c = 0;
case 1
c = [-2;1];
case 2
c = [-30;16;-1]/12;
case 3
c = [-490;270;-27;2]/180;
case 4
c = [-2.847222222222222e+00
1.600000000000000e+00
-2.000000000000000e-01
2.539682539682540e-02
-1.785714285714286e-03];
case 5
c = [-2.927222222222222e+00
1.666666666666667e+00
-2.380952380952381e-01
3.968253968253968e-02
-4.960317460317460e-03
3.174603174603175e-04];
case 6
c = [-2.982777777777778e+00
1.714285714285714e+00
-2.678571428571428e-01
5.291005291005291e-02
-8.928571428571428e-03
1.038961038961039e-03
-6.012506012506013e-05];
case 7
c = [-3.023594104308390e+00
1.750000000000000e+00
-2.916666666666667e-01
6.481481481481481e-02
-1.325757575757576e-02
2.121212121212121e-03
-2.266252266252266e-04
1.189286903572618e-05];
case 8
c = [-3.054844104308390e+00
1.777777777777778e+00
-3.111111111111111e-01
7.542087542087542e-02
-1.767676767676768e-02
3.480963480963481e-03
-5.180005180005180e-04
5.074290788576503e-05
-2.428127428127428e-06];
case 9
c = [-3.079535462333082e+00
1.800000000000000e+00
-3.272727272727273e-01
8.484848484848485e-02
-2.202797202797203e-02
5.034965034965035e-03
-9.324009324009324e-04
1.284429855858427e-04
-1.156931303990128e-05
5.078436450985471e-07];
case 10
c = [-3.099535462333082e+00
1.818181818181818e+00
-3.409090909090909e-01
9.324009324009325e-02
-2.622377622377622e-02
6.713286713286713e-03
-1.456876456876457e-03
2.518489913447897e-04
-3.213698066639243e-05
2.672861289992353e-06
-1.082508822446903e-07];
case 11
c = [-3.116064387952916e+00
1.833333333333333e+00
-3.525641025641026e-01
1.007326007326007e-01
-3.021978021978022e-02
8.461538461538461e-03
-2.073906485671192e-03
4.232462215655493e-04
-6.822060808128920e-05
8.085405402226867e-06
-6.237312738860727e-07
2.343092689279011e-08];
case 12
c = [-3.129953276841805e+00
1.846153846153846e+00
-3.626373626373626e-01
1.074481074481075e-01
-3.399725274725274e-02
1.023917259211377e-02
-2.765208647561589e-03
6.415521674256748e-04
-1.227970945463205e-04
1.848092663366141e-05
-2.041302350899874e-06
1.466979770679033e-07
-5.136127090629716e-09];
case 13
c = [-3.141787596368432e+00
1.857142857142857e+00
-3.714285714285714e-01
1.134920634920635e-01
-3.755252100840336e-02
1.201680672268907e-02
-3.513686176224876e-03
9.035193024578253e-04
-1.976448474126493e-04
3.549177955782703e-05
-4.999711555102590e-06
5.164991275932428e-07
-3.472021913265688e-08
1.137849693924122e-09];
case 14
c = [-3.151991678001085e+00
1.866666666666667e+00
-3.791666666666667e-01
1.189542483660131e-01
-4.089052287581699e-02
1.377364981080151e-02
-4.304265565875473e-03
1.204692403277100e-03
-2.934726522187823e-04
6.049033733333998e-05
-1.020774442500112e-05
1.349784386777008e-06
-1.308685182692451e-07
8.259945926263994e-09
-2.543605797264240e-10];
case 15
c = [-3.1608805668899742E+00
1.8750000000000000E+00
-3.8602941176470590E-01
1.2391067538126362E-01
-4.4020897832817339E-02
1.5495356037151704E-02
-5.1241256736612771E-03
1.5400897200985658E-03
-4.1013258850450935E-04
9.4516152083343722E-05
-1.8373939965002018E-05
2.9202066060079494E-06
-3.6352366185901429E-07
3.3187282739453544E-08
-1.9734872564981174E-09
5.7304222559056440E-11];
case 16
c = [-3.1686930668899742E+00
1.8823529411764706E+00
-3.9215686274509803E-01
1.2842563926155257E-01
-4.6955624355005159E-02
1.7172342621259030E-02
-5.9626189657149406E-03
1.9046520209914629E-03
-5.4684345133934576E-04
1.3826362819049139E-04
-3.0152106609234084E-05
5.5375769713928528E-06
-8.3091122710631844E-07
9.7654533118392042E-08
-8.4202122943919670E-09
4.7322196693930480E-10
-1.2997429511882761E-11];
case 17
c = [-3.175613482114888e+00
1.888888888888889e+00
-3.976608187134503e-01
1.325536062378168e-01
-4.970760233918128e-02
1.879851143009038e-02
-6.811054865974774e-03
2.293518475277220e-03
-7.023900330536486e-04
1.921066757069808e-04
-4.610560216967540e-05
9.525950861503180e-06
-1.656092031956731e-06
2.351846672601275e-07
-2.616603605461590e-08
2.136892944460298e-09
-1.138259736040642e-10
2.965545414475924e-12];
case 18
c = [-3.181786321621061e+00
1.894736842105263e+00
-4.026315789473684e-01
1.363408521303258e-01
-5.228981544771018e-02
2.037029332223840e-02
-7.662436724221621e-03
2.702181767235707e-03
-8.752860411899313e-04
2.561422342759744e-04
-6.668846028113762e-05
1.520398068535483e-05
-2.980965657522116e-06
4.916118205953633e-07
-6.623277876324650e-08
6.993467818233704e-09
-5.423472859958352e-10
2.745247755114856e-11
-6.801925593520176e-13];
case 19
c = [-3.187326487826047e+00
1.900000000000000e+00
-4.071428571428571e-01
1.398268398268398e-01
-5.471485036702428e-02
2.188594014680971e-02
-8.511198945981555e-03
3.126562878115673e-03
-1.063899868247694e-03
3.302405234772384e-04
-9.223959448847006e-05
2.286932094755456e-05
-4.959117983251078e-06
9.243326418485737e-07
-1.449092917183756e-07
1.856354325281152e-08
-1.864641621376157e-09
1.376436721661754e-10
-6.636473349353468e-12
1.567442313158276e-13];
case 20
c = [-3.192326487826047e+00
1.904761904761905e+00
-4.112554112554113e-01
1.430453604366648e-01
-5.699463579898362e-02
2.334500282326369e-02
-9.352965874705006e-03
3.563034618935240e-03
-1.266547462199636e-03
4.140947002849385e-04
-1.229861259846267e-04
3.278755691405672e-05
-7.748621848829809e-06
1.600576003200993e-06
-2.841358661144621e-07
4.243095600642634e-08
-5.179560059378215e-09
4.960132330312626e-10
-3.492880710186036e-11
1.607633141700796e-12
-3.627222275962422e-14];
case 21
c = [-3.196861635218337e+00
1.909090909090909e+00
-4.150197628458498e-01
1.460254721124286e-01
-5.914031620553360e-02
2.474794770446944e-02
-1.018434061912323e-02
4.008413946302145e-03
-1.481558171962969e-03
5.072660078490497e-04
-1.590524385900891e-04
4.518535187218441e-05
-1.150552941189881e-05
2.595051534601611e-06
-5.114445590060318e-07
8.662986851312044e-08
-1.234695127667996e-08
1.439091024781492e-09
-1.316547344608583e-10
8.862077693625641e-12
-3.901475667559581e-13
8.425603428484139e-15];
end