function [x,w]=knots_triangular_leja(n,a,b)

% [x,w] = knots_triangular_leja(n,a,b)
%
% 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)= 2/(b-a)^2 (b-x)
%
% i.e. a linear decreasing pdf over the interval [a,b]
%
% Knots and weights have been precomputed (up to 50) 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_TriangularLejaPrecomputedKnotsAndWeights50();
    x = X(1:n);
    w = W(1:n,n);
    
    % modifies points according to the interval (the weigths are unaffected)
    x = (b-a)*x + a;
    
    % 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_TriangularLejaPrecomputedKnotsAndWeights50()

X = [   0.0000000000000000e+00   6.6666666010600006e-01   2.9449494690417050e-01   9.2378944730515400e-01   1.1071536817303690e-01   8.1109429511699993e-01   4.6356431984492996e-01   4.0027718122997581e-02   9.8438053380500001e-01   2.0310781603679998e-01   5.7257373209127793e-01   8.7428149865727001e-01   1.4487131815120000e-02   3.8019755310903119e-01   9.6296717571300006e-01   7.3727678569437116e-01   1.5166126223610712e-01   6.9779404003265807e-02   9.9685144187204533e-01   5.1899593357941831e-01   2.5111530772373275e-01   8.4356924260400001e-01   5.0400535699499831e-03   6.2446402370342091e-01   9.4526192770699979e-01   3.3967878090241183e-01   8.8565796293075094e-02   7.7243637587414016e-01   9.9185970836608273e-01   2.6036911320449104e-02   4.2449883238268998e-01   9.0015648111999780e-01   1.7713079693699424e-01   7.0057335862214010e-01   5.3379925843479989e-02   9.7456216977093679e-01   4.9276978250962000e-01   2.2883109010311767e-01   5.9917464819556998e-01   1.7513656519999999e-03   8.5931965466499993e-01   3.1732795894999183e-01   9.3536126826607990e-01   1.2957796102681002e-01   7.9074643046500004e-01   9.6599719327200018e-03   9.9938009921233983e-01   5.4552238966198008e-01   4.0151013550713427e-01   7.1889373327300010e-01];

W = [   1.0000000000000000e+00   4.9999999507949988e-01   2.1702966995470160e-01   1.8808447116748922e-01   1.0952592853313430e-01   8.9481573612216911e-02   6.5519589343677645e-02   5.8461474264109865e-02   5.1765855509535656e-02   2.1151640467391805e-02   2.6326959085587576e-02   2.5429804389054830e-02  -1.3147408331702070e-02   7.1597820133597959e-03   1.2428057202029604e-02   1.4649361770129968e-02   1.4437336109716748e-02   7.8016994932922348e-03   8.4457000115923282e-03   8.9426491694336186e-03   8.7299093546796911e-03   8.5154500387863638e-03  -3.2106693984807950e-03   6.2026502943412099e-03   6.1660509936797463e-03   6.5386310167225743e-03   6.3252239621693013e-03   5.5390977878813754e-03   5.3241592348973509e-03  -6.1477632632434470e-05   3.2595058345277297e-03   3.5133165138662710e-03   3.7928115984348994e-03   3.3159275570224002e-03   2.6208617837006781e-03   2.7746631886619993e-03   2.9235339040432464e-03   2.9704573201755247e-03   2.9441874947560504e-03  -1.5966592158042029e-04  -2.4033735181138474e-04   1.0678689178686873e-03   1.2492729624791058e-03   2.1385423395047596e-03   2.1685325760787666e-03   5.8406809433154275e-03   3.8570428483084384e-03   1.7551899144044830e-03   1.1717764607247837e-03   1.1790092386322446e-03
   0.0000000000000000e+00   5.0000000492049990e-01   2.7608903071948249e-01   1.9379971481296351e-01   2.3827630641360764e-01   3.0200737659835464e-01   1.2811642681075361e-01   1.3138822410174139e-01   1.4100468362140617e-01   1.2173966717093400e-01   1.0192176022879496e-01   1.1638884164548523e-01   1.0257038037161222e-01   9.2916315722018236e-02   8.4776625217762927e-02   4.8941318564953477e-02   4.7934029022284733e-02   5.1619418952414148e-02   5.0539576355082036e-02   5.3468147243848400e-02   5.4225749128538975e-02   5.7867613345191866e-02   5.6350718981567009e-02   3.8332507772543246e-02   3.8570202792792073e-02   4.1083845955300728e-02   4.1304420425644361e-02   4.7238318880353881e-02   5.2186791481203414e-02   4.7147421108619782e-02   4.1700313203891899e-02   4.0095371667058628e-02   4.0734860644232297e-02   1.8190403040206242e-02   2.1050411386338608e-02   1.9047287112666362e-02   2.4541546394240835e-02   2.3636452811013797e-02   1.9138011017759481e-02   2.0537974240851110e-02   2.0700272612274560e-02   2.3091012175958030e-02   2.4245059024260919e-02   2.2880177553688551e-02   2.2586833187731092e-02   2.3114942898941369e-02   2.2258049637576625e-02   2.6346665541900079e-02   2.4628184984385744e-02   2.4921437888849891e-02
   0.0000000000000000e+00   0.0000000000000000e+00   5.0688129932581560e-01   5.8299467298767460e-01   4.5854650416772702e-01   4.0869196921548129e-01   2.4528129438986540e-01   2.5285267977430964e-01   2.6310124995958800e-01   1.5895652327051302e-01   1.9520709086283689e-01   1.8573102208790568e-01   2.0681264174713773e-01   1.5758168723823840e-01   1.3918307853569023e-01   1.2626596726551134e-01   1.2495681626506565e-01   1.3767948020531079e-01   1.3592699437095498e-01   1.3280072814166188e-01   1.2505343428157994e-01   1.1305469014990591e-01   1.2447819698946969e-01   1.0712329671770238e-01   1.0722130860636311e-01   9.9720451796587575e-02   9.7872680208642412e-02   8.6871993168877337e-02   8.2594078434259019e-02   9.2990118070463132e-02   7.2057779988345366e-02   6.9680124437819541e-02   7.3631714101081103e-02   6.1999720039648085e-02   6.5753086638579206e-02   6.4562905428312140e-02   6.1699792143594383e-02   6.4844697069917356e-02   6.1382235246624742e-02   6.3829711329570535e-02   6.3926489647428933e-02   4.2115284061225597e-02   3.7700978984797621e-02   5.4703544956138210e-02   5.5617229850904559e-02   5.1823037689239973e-02   5.4728892271827881e-02   6.1420109667941175e-02   6.8388465934201975e-02   6.8242130466228429e-02
   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   3.5121141031872818e-02   2.2141481781480052e-02   4.5977190505020887e-02   1.7276097310470521e-02   1.7659001428669489e-02   2.3560282445766266e-02   1.5955936066438775e-02   1.3860215697607540e-02   7.4445764815257798e-03   1.1839765519584312e-02   1.3457968132486680e-02   2.3776638600787790e-02   6.5783338335316721e-03   6.2558921233308602e-03   7.0804282028380204e-03   5.9886052924695374e-03   7.0688156953119738e-03   7.2414446452857663e-03   5.4114629773381346e-03   5.9603642317521705e-03   6.8796415305483619e-03   6.7222988787633744e-03   5.7908339132059661e-03   5.7342596405438300e-03   6.7978488998799384e-03   1.1035145256576454e-02   7.9558990898295438e-03   6.3415511159093888e-03   1.1040926201986002e-02   9.8132711123615976e-03   3.2390681518553034e-03   3.8267070755251221e-03   1.3308250588701333e-03   4.0928017422906431e-03   3.8061486689187477e-03   3.5099321450960622e-03   3.5764106535635535e-03   3.5533804226457384e-03   3.3579645335120611e-03   1.1676408977434425e-03   2.9194615379765031e-03   2.5683206934376400e-03   3.0226699580468315e-03  -2.2218027040312881e-04   4.7363001161749886e-03   3.6782252358611859e-03   3.6322020246685694e-03
   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   1.7150977910405113e-01   2.2218852156839863e-01   3.0178203837818174e-01   2.8850623109872847e-01   2.7431627936588349e-01   1.3168870631996615e-01   1.6157968008447321e-01   1.5564667202921342e-01   1.9405437499457454e-01   1.6552997055089100e-01   1.5716857534554304e-01   1.5302013050544022e-01   1.5448679217628306e-01   7.6243638881062711e-02   8.4786040609964877e-02   9.3165403152072224e-02   8.6749555162631053e-02   7.9304746276747135e-02   9.8719172226712185e-02   7.9775255611705176e-02   7.9858681597503187e-02   7.8598739885978847e-02   7.5713111803966879e-02   6.3304830358236683e-02   5.9485942614061316e-02   8.8908212029311692e-02   6.4363821024062698e-02   6.2224905111185939e-02   5.5943119791259065e-02   6.8673103745323302e-02   5.1398990235987235e-02   5.5711230872893210e-02   6.1094806357567887e-02   6.4382251197006135e-02   6.2124629847389319e-02   6.6411979465811830e-02   6.6539891053494823e-02   6.3354095410417324e-02   6.2853022952176665e-02   4.5979076968226062e-02   4.5317360872890518e-02   5.3062484991156865e-02   4.8357444613586931e-02   4.2102560452071579e-02   3.9705365739453984e-02   3.9740494734991146e-02
   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00  -6.8346631499472232e-02   4.0638070606231980e-02   3.8971589277411456e-02   2.9991071516997196e-02   4.3708304105330109e-02   4.9274853416273030e-02   3.5923095298304342e-02   4.6364049028449414e-02   5.1213530138856204e-02   5.9190632007624429e-02   2.5597076847766215e-02   2.4859613824064666e-02   2.7032109655376252e-02   2.5900628087941576e-02   2.7451971906941746e-02   2.7749786746627838e-02   3.5548352762053752e-02   3.2882138575894566e-02   2.5720558286615805e-02   2.5916733102366890e-02   2.7355712509139377e-02   2.7456743551858249e-02   2.0020346618485018e-02   8.8640632278152408e-03   1.8135143585726397e-02   2.4412542660337339e-02   2.9261491245677056e-02   2.7769584472799407e-02   1.1633826831624906e-02   1.3290645269821296e-02   1.1104965908671905e-02   1.4379951619475235e-02   1.3974269899099763e-02   1.3332121784917167e-02   1.3496303005029650e-02   1.3572339362457943e-02   1.4364776159617919e-02   1.5191878093360653e-02   1.4420974127083723e-02   1.5431308746682684e-02   1.3940185677579804e-02   1.8215507673118643e-02   8.9100194348124377e-03   1.1442036565936765e-02   1.1686788684540227e-02
   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   2.0138648316081931e-01   1.9578031450025263e-01   1.8572844915514752e-01   2.2156865565006073e-01   1.8974486645081981e-01   2.0148817715471976e-01   1.8519838738378211e-01   1.4609171775488211e-01   1.2652896764648924e-01   1.0431093772583523e-01   1.0327974249411655e-01   1.0899852060301526e-01   1.0796104901272721e-01   1.0046541474262963e-01   9.6672582121436812e-02   8.8184856518059351e-02   9.3286070672004340e-02   7.7392765251908954e-02   7.7514026458276555e-02   8.0898697505295616e-02   8.1356567460626306e-02   8.5574598099907848e-02   8.7739834020084700e-02   8.4511249431103122e-02   6.2877835112767608e-02   5.9468966013567201e-02   6.1790338536599036e-02   5.0082622953996153e-02   5.2303293389017451e-02   5.1366143655012388e-02   3.6060981453199997e-02   4.0763805590851528e-02   2.9130933173982999e-02   3.4343380728403095e-02   3.4637543845831613e-02   4.4988909648127476e-02   4.7834632824623843e-02   4.2422338137257973e-02   4.1981198908195315e-02   4.3130748018920438e-02   4.1972544318794727e-02   3.3803980786330186e-02   4.8474491982111803e-02   4.7962412013877957e-02
   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   1.6380485554776877e-02   3.2578339943736240e-02   1.2481754993350638e-01   1.0805253980618780e-01   1.1109824011586951e-01   3.6813052027260429e-02   8.0518415790308079e-02   9.2348603136747989e-02   9.7623013272043230e-02   9.6939047820000748e-02   4.6734389981879954e-02   5.1810676655230231e-02   5.6055202113790621e-02   5.3893596559981022e-02   5.1605970691215461e-02   6.9624303094822454e-02   5.4169149488707311e-02   5.4231896712087377e-02   5.3507804482967720e-02   5.4264579084607377e-02   5.7204659986622046e-02   5.8042326774944371e-02   1.8981561485238503e-02   4.5575531688076480e-02   4.7702597354921682e-02   5.0728758492255596e-02   4.5252531369732049e-02   1.3343071572180430e-02   2.0706296785948937e-02   2.8463591264808401e-02   3.1427029692674267e-02   2.9649196716910933e-02   3.9260477315076772e-02   3.9522486113186650e-02   3.4660312791368948e-02   3.3955951145543180e-02   2.8959674879358979e-02   2.8782193417234172e-02   3.5695035399486916e-02   3.1805019385575908e-02   2.7356789002638941e-02   2.5985368235377997e-02   2.6003372692934401e-02
   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00   0.0000000000000000e+00  -2.0462115180608018e-03   3.8603121053343379e-04   9.5771882328034882e-04   1.7446863277034721e-03   1.2392357265652255e-03   1.0718036971597228e-03   3.0251707487674081e-03   5.6777703928601497e-04   5.2505705269933188e-04   6.2706189769759599e-04  -1.6426656318805060e-04   5.1671362132713918e-04   6.1654875193992620e-04   1.3624454940728864e-05   1.8328217608474666e-04   4.1958438325475592e-04   4.4178507729814969e-04   5.6086070979600566e-04   5.6760381495038817e-04   4.7707544834784749e-04  -2.8054139172808477e-03  -5.7084228514001924e-04   4.7388792606814902e-04  -3.7946603648074059e-04  -1.7327031152136296e-04   6.9518708562220171e-04   6.2261165285321964e-04  -9.7141086570579180e-04   5.7514681114582749e-04   4.2750865680818624e-04   2.9894249251842975e-04   3.2601683261023118e-04   3.2118168539410326e-04   2.8388120332799711e-04   3.8257679773857618e-04   3.0923535322950493e-04   3.1933605166912485e-04   3.0707895897751184e-04   7.4822431484498084e-04   1.6717898572239735e-04   2.7827759998634463e-04   2.8200717284701819e-04
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