In [1]:

    

# Standard values are for transformation x=(y-L)/(y+L), y in 0, inf
# *lin are for x=2/L*y-1 (linear transformation)
clear all
format long e;
N=1000;
perc=0.21;
Nfrag=round(perc*N);
n=0:N;
L=22;
t=pi+n'*pi/N;
x=cos(t);
y=L*(cot(t/2)).^2;
ysub=y(2:N);
ylintrans = L/2*(x+1);
# Chebyshev polys of the first kind
T         = cos(t*n);
# Now for arrays that do not include the endpoints
xsub      = x(2:N);
Tsub      = T(2:N,:);
Usub      = diag(1./sqrt(1-xsub.^2))*sin(acos(xsub)*n);
dTsub     = Usub*diag(n);
dT        = [-(n.^2).*(-1).^(n); dTsub ; (n).^2];
dTL       = diag(2*L./((y+L).^2))*dT;
dTlin     = 2/L*dT;
D1        = dTL/T;
D1lin     = dTlin/T;
D2        = D1^2;
D2lin     = D1lin^2;
H         = -D2+diag((y.^2.));
H         = H(2:N,2:N);
[Y, Lam]  = eig(H);
Lam       = diag(Lam);
[Lam, IX] = sort(Lam, 'ascend');
Y         = Y(:,IX);
yexact    = sqrt(ysub/(2*pi)).*besselk(1/4,1/4*ysub.^2);
eigex=4*(n'+1)-1; err=Lam(1:Nfrag)-eigex(1:Nfrag); errrms=sqrt(err'*err/(Nfrag))
figure(1)
plot(ysub(1:3*N/5), Y(1:3*N/5,1));
figure(2)
plot(ysub(1:3*N/5), Y(1:3*N/5,30));


    

    




errrms =    7.356911406826205e-11






    













    

In [14]:

    

Lam(3)


    

    




ans =    1.099999999998837e+01

In [16]:

    

Lam(3) #L=15


    

    




ans =    1.100000000000614e+01

In [17]:

    

Lam(10)


    

    




ans =    3.900000000001418e+01

In [18]:

    

Lam(1)


    

    




ans =    3.000000000004093e+00

In [19]:

    

eigex=4*(n'+1)-1; err=Lam(1:Nfrag)-eigex(1:Nfrag); errrms=sqrt(err'*err/(Nfrag));


    

In [20]:

    

errrms


    

    




errrms =    3.113279365871696e+00

In [21]:

    

err


    

    




err =

   4.092726157978177e-12
   1.633360113828530e-12
   6.139089236967266e-12
   1.383781977892795e-11
   1.322675302617426e-11
   1.606181854185706e-11
   1.472599819862808e-11
   1.272226768378459e-11
   1.409716787748039e-11
   1.417532757841400e-11
   1.798383664208814e-11
   1.721645048746723e-11
   1.656275117056794e-11
   1.576694330651662e-11
   2.084021843984374e-11
   1.898570189950988e-11
   2.282263267261442e-11
   2.485478489688830e-11
   2.813749233609997e-11
   2.634692464198451e-11
   2.836486601154320e-11
   2.944489096989855e-11
   2.894751105486648e-11
   3.119282609986840e-11
   2.923172814917052e-11
   2.913225216616411e-11
   2.886224592657527e-11
   3.051070507353870e-11
   3.385025593161117e-11
   3.350919541844632e-11
   3.414868388063041e-11
   3.285549610154703e-11
   3.404920789762400e-11
   3.311129148642067e-11
   3.379341251275036e-11
   3.393552105990238e-11
   3.140598892059643e-11
   3.558398020686582e-11
   3.419131644477602e-11
   3.726086106325965e-11
   3.620925781433471e-11
   3.799982550845016e-11
   3.856825969705824e-11
   3.836930773104541e-11
   3.927880243281834e-11
   3.825562089332379e-11
   4.123990038351621e-11
   4.376943252282217e-11
   4.385469765111338e-11
   4.442313183972146e-11
   4.101252670807298e-11
   4.490630090003833e-11
   4.362732397567015e-11
   4.337152859079652e-11
   4.342837200965732e-11
   4.544631337921601e-11
   4.561684363579843e-11
   4.828848432225641e-11
   4.621369953383692e-11
   4.777689355250914e-11
   4.951061782776378e-11
   4.740741132991388e-11
   5.005063030694146e-11
   5.132960723130964e-11
   4.746425474877469e-11
   4.877165338257328e-11
   4.934008757118136e-11
   4.860112312599085e-11
   5.081801646156237e-11
   5.172751116333529e-11
   5.292122295941226e-11
   5.104539013700560e-11
   5.184119800105691e-11
   5.201172825763933e-11
   5.564970706473105e-11
   5.348965714802034e-11
   5.530864655156620e-11
   5.394440449890681e-11
   5.593392415903509e-11
   5.621814125333913e-11
   5.496758603840135e-11
   5.599076757789589e-11
   5.570655048359185e-11
   5.428546501207165e-11
   5.513811629498377e-11
   5.553602022700943e-11
   5.388756108004600e-11
   5.650235834764317e-11
   5.633182809106074e-11
   5.661604518536478e-11
   5.888978193979710e-11
   5.741185304941610e-11
   5.917399903410114e-11
   6.036771083017811e-11
   6.116351869422942e-11
   6.053824108676054e-11
   6.036771083017811e-11
   6.070877134334296e-11
   6.309619493549690e-11
   6.389200279954821e-11
   6.627942639170215e-11
   6.508571459562518e-11
   6.190248313941993e-11
   6.315303835435770e-11
   6.025402399245650e-11
   5.877609510207549e-11
   6.224354365258478e-11
   6.508571459562518e-11
   6.838263288955204e-11
   6.610889613511972e-11
   6.411937647499144e-11
   6.923528417246416e-11
   6.480149750132114e-11
   6.480149750132114e-11
   6.696154741803184e-11
   6.707523425575346e-11
   6.912159733474255e-11
   6.889422365929931e-11
   6.889422365929931e-11
   7.338485374930315e-11
   7.480593922082335e-11
   7.378275768132880e-11
   6.912159733474255e-11
   6.838263288955204e-11
   7.190692485892214e-11
   7.258904588525183e-11
   7.304379323613830e-11
   7.315748007385992e-11
   7.696598913753405e-11
   7.071321306284517e-11
   7.185008144006133e-11
   7.150902092689648e-11
   7.469225238310173e-11
   7.480593922082335e-11
   7.446487870765850e-11
   6.900791049702093e-11
   6.798472895752639e-11
   7.798917067702860e-11
   7.469225238310173e-11
   7.446487870765850e-11
   7.980816008057445e-11
   8.003553375601768e-11
   7.480593922082335e-11
   7.423750503221527e-11
   7.582912076031789e-11
   7.480593922082335e-11
   7.594280759803951e-11
   8.083134162006900e-11
   7.810285751475021e-11
   7.889866537880152e-11
   7.855760486563668e-11
   7.946709956740960e-11
   8.174083632184193e-11
   7.844391802791506e-11
   7.798917067702860e-11
   7.787548383930698e-11
   8.117240213323385e-11
   8.071765478234738e-11
   7.742073648842052e-11
   8.230927051045001e-11
   8.469669410260394e-11
   8.265033102361485e-11
   8.367351256310940e-11
   8.287770469905809e-11
   8.276401786133647e-11
   8.265033102361485e-11
   8.219558367272839e-11
   8.253664418589324e-11
   8.037659426918253e-11
   8.844835974741727e-11
   8.299139153677970e-11
   8.299139153677970e-11
   8.037659426918253e-11
   8.287770469905809e-11
   7.912603905424476e-11
   8.117240213323385e-11
   8.674305718159303e-11
   8.753886504564434e-11
   8.583356247982010e-11
   8.071765478234738e-11
   8.469669410260394e-11
   9.038103598868474e-11
   8.708411769475788e-11
   8.412825991399586e-11
   8.526512829121202e-11
   8.628830983070657e-11
   9.367795428261161e-11
   9.038103598868474e-11
   8.969891496235505e-11
   9.208633855450898e-11
   9.072209650184959e-11
   8.560618880437687e-11
   9.015366231324151e-11
   8.981260180007666e-11
   8.719780453247949e-11
   8.947154128691182e-11
   8.981260180007666e-11
   9.890754881780595e-11
   8.913048077374697e-11
   9.083578333957121e-11
   9.072209650184959e-11
   9.038103598868474e-11
   1.006128513836302e-10
   9.106315701501444e-11
   8.753886504564434e-11
   8.571987564209849e-11
   9.333689376944676e-11
   9.242739906767383e-11
   8.230927051045001e-11
   8.515144145349041e-11
   1.169837560155429e-10
   1.127773430198431e-10
   5.445599526865408e-11
   1.946318661794066e-10
   4.879439075011760e-10
   1.473381416872144e-09
   3.420836947043426e-10
   1.250748482561903e-08
   4.380808604764752e-09
   1.024131961457897e-07
  -8.132019502227195e-09
   7.526274430347257e-07
  -1.063162926584482e-07
   4.895441520602617e-06
  -9.825103006733116e-07
   2.846123049948801e-05
  -4.843207875637745e-06
   1.467228752289884e-04
  -1.506245803284401e-05
   6.710119500894507e-04
   6.907760962349130e-06
   2.704881985550855e-03
   4.284213815708426e-04
   9.563483644569715e-03
   3.267674493486084e-03
   2.943436453074355e-02
   1.636404672933622e-02
   7.846747890414463e-02
   6.246179003164798e-02
   1.813822473710616e-01
   1.904620161114963e-01
   3.686240872806366e-01
   4.769731670165811e-01
   6.747462691935198e-01
   1.005867435921914e+00
   1.139315189748913e+00
   1.836372958651737e+00
   1.801469989307066e+00
   2.987918795067912e+00
   2.691558542232656e+00
   4.449283369268301e+00
   3.829314433841887e+00
   6.192709752176825e+00
   5.230441706399574e+00
   8.180380044483627e+00
   6.914433869431377e+00
   1.036531937626683e+01
   8.910879932654098e+00
   1.269853792506728e+01
   1.125388372305633e+01
   1.515062610995574e+01
   1.396133001784347e+01
   1.772508461469670e+01
   1.702053357059049e+01
   2.044373482139235e+01
   2.040302761294288e+01

In [ ]: