function [x,w]=knots_exponential_leja(n,lambda)

% [x,w] = knots_exponential_leja(n,lambda)
%
% returns the collocation points (x) and the weights (w) 
% for the weighted Leja sequence for integration 
% w.r.t to the weight function 
%
% rho(x)=exp(-lambda*abs(x)), lambda>0
%
% i.e. the density of an exponential random variable 
% with rate parameter lambda.
%
% Knots and weights have been precomputed (up to 50) 
% for the case lambda=1 following the work
% 
% A. Narayan, J. Jakeman, "Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation"
% SIAM Journal on Scientific Computing,  Vol. 36, No. 6, pp. A2952--A2983, 2014
%
% An error is raised if more than 50 points are requested.
%
% Knots are sorted increasingly before returning (weights are returned in
% the corresponding order).

%-------------------------------------------------------------
% Sparse Grid Matlab Kit
% Copyright (c) 2009-2023 L. Tamellini, F. Nobile, C. Piazzola
% See LICENSE.txt for license
%-------------------------------------------------------------


if n>50
   error('SparseGKit:OutOfTable',strcat('this number of points is not available:',num2str(n)))

else
    
    [X,W] = get_ExponentialLejaPrecomputedKnotsAndWeights50;
    x = X(1:n);
    w = W(1:n,n);
    
    % modifies points according to lambda (the weigths are unaffected)
    x = x/lambda;
    
    % sort knots increasingly and weights accordingly. Weights need to be row vectors
    [x,sorter]=sort(x);
    w=w(sorter)';

end

end

function [X,W] = get_ExponentialLejaPrecomputedKnotsAndWeights50() % first node x0 = 0

   X = [    1.0000000000000000e+00   0.0000000000000000e+00   4.5615527962140003e+00   8.5514197172080024e+00   2.4062436394242832e+00   1.4401260865619140e+01   1.9817077040003067e+01   3.3655358024700011e-01   2.6266020351040002e+01   1.0975731094319631e+01   3.3660509413176108e+01   6.1768514265233962e+00   4.1283746314100362e+01   1.6569374351022725e+00   2.2879928146020173e+01   5.0513385430402096e+01   1.6735121842210170e+01   5.9322288689999993e+01   3.4569344152817303e+00   3.7386783630300002e+01   6.9506596452699995e+01   1.1677435779999908e-01   7.8766552372199897e+01   2.9763800451810031e+01   4.6426557089499227e+01   8.9512916945472000e+01   1.2453579578372999e+01   6.4162210449968001e+01   1.0104258423996515e+02   7.2915647861999995e+00   1.1131013094120001e+02   5.4624852497804341e+01   6.4593427161009009e-01   8.4027938259999644e+01   1.2403424205225900e+02   1.8329715421311679e+01   1.3499970305040193e+02   3.1650827622165441e+01   9.5394355841000007e+01   5.2901155921479992e+00   1.4764986136019502e+02   7.3612817195736838e+01   4.3747523014000002e+01   1.5997025315505266e+02   9.7565105711545392e+00   1.1741402802999966e+02   1.7284422193158599e+02   2.4519020692643611e+01   1.0600010450013701e+02   1.3270807575508201e+00];
   
   W = [ 1.0000000000000000e+00   1.0000000000000000e+00   7.1922359228732502e-01   6.9834397192895992e-01   5.7788419962981297e-01   5.8781820437776666e-01   5.9491767313945088e-01   3.7911895510029825e-01   3.7472887091368345e-01   3.6527883836697311e-01   3.6351749376603476e-01   3.5360277494987652e-01   3.5307536410448781e-01   3.1758574525562594e-01   3.1121478355506960e-01   3.0870377847310099e-01   3.0117354762649601e-01   2.9943695226010530e-01   2.5972087297161972e-01   2.5931913568739484e-01   2.5907914741243215e-01   2.7868166422259777e-01   2.7929418532631356e-01   2.8085514463100880e-01   2.8170106372114034e-01   2.8210459552364436e-01   2.8512848339643437e-01   2.8555981139718234e-01   2.8583115871496817e-01   2.9014196157723859e-01   2.9038731056153239e-01   2.9089723523048938e-01   2.1204222481371413e-01   2.1133486198784682e-01   2.1086420178590154e-01   2.0755733463515061e-01   2.0716269729518083e-01   2.0546016567908601e-01   2.0493938909401835e-01   1.9370121096050338e-01   1.9351154275766830e-01   1.9313295674656583e-01   1.9250449229262639e-01   1.9234154917368884e-01   1.8940836252011556e-01   1.8922047296718830e-01   1.8909366983608381e-01   1.8816951894148159e-01   1.8796633043966171e-01   1.2299504304311087e-01
   0.0000000000000000e+00  -0.0000000000000000e+00   2.1922359439311540e-01   2.3361952237917982e-01   2.8215732125234594e-01   2.7843248190001119e-01   2.7590480475550883e-01   1.2444539877371102e-01   1.2148150385242448e-01   1.1568275187304472e-01   1.1463406017000748e-01   1.0968659839602354e-01   1.0942979435516248e-01   1.0257850937184747e-01   1.0140234795925777e-01   1.0094796119743105e-01   9.9666729719619068e-02   9.9376237431047357e-02   9.4654468146867773e-02   9.4607983868127007e-02   9.4580614778613209e-02   7.7672065694463069e-02   7.7150430875067399e-02   7.5865750823438785e-02   7.5184549363903092e-02   7.4863223669524734e-02   7.2648699584362597e-02   7.2337742554033913e-02   7.2144056336815490e-02   6.9489027264518971e-02   6.9339274072147711e-02   6.9033730127754850e-02   4.3134162412476480e-02   4.2904597240523874e-02   4.2753082235047327e-02   4.1746612718442246e-02   4.1627391631882449e-02   4.1129302321872611e-02   4.0978542077650670e-02   3.8340175560908504e-02   3.8295949082709578e-02   3.8208870322165481e-02   3.8067621159604555e-02   3.8031228162620084e-02   3.7443254255502392e-02   3.7405911503922890e-02   3.7380855389240283e-02   3.7205692236695675e-02   3.7167543300609118e-02   3.4161034926716766e-02
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   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   3.1071146521389492e-01   3.1687066828240462e-01   3.2930211455456215e-01   3.3157302019809787e-01   3.4290395822372233e-01   3.4349693848620921e-01   3.6334948957035940e-01   3.6680846076834323e-01   3.6815372912955707e-01   3.7202482391131070e-01   3.7290752104185171e-01   3.8880269948514690e-01   3.8896060377814978e-01   3.8905402728570665e-01   3.5838761174184275e-01   3.5743748208270543e-01   3.5507074489828733e-01   3.5380661890754322e-01   3.5320807474794730e-01   3.4896843152316148e-01   3.4836997403193015e-01   3.4799596627342660e-01   3.4262102058948929e-01   3.4231693489074078e-01   3.4169265746500682e-01   2.3121064218259221e-01   2.3022742824340892e-01   2.2957673292017994e-01   2.2517350768400363e-01   2.2465062122294874e-01   2.2244259480568346e-01   2.2177190950674777e-01   2.0923718945440978e-01   2.0902659208995414e-01   2.0861003653003651e-01   2.0792910951876373e-01   2.0775329806889109e-01   2.0481136365141650e-01   2.0462398164239790e-01   2.0449800739736151e-01   2.0360508584954776e-01   2.0340999626983627e-01   1.8281104944946869e-01
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