
% Coefficients for the propagators for the Schoedinger equation 
% with a time-dependent Hamiltonian
function [a, b, nex]=coefsSplit(m)
% a: matrix coefficients with the coefficients a_{ij};
% b: matrix coefficients with the coefficients b_{ij};
% nex: number of stages per step
if ( m == 1)
% 8-stage 4th-order optimised (SM_{8}^{[4]})
    nex=8;

a=[ 0.0565436438082349,   0.013657706809987716, -0.03436754777974814;
    0.1518765115304851,  -0.06621736226664678,   0.07546973535118673;
    0.07444694250251396,  0.20831893021621906,  -0.04276374038685533;
   -0.011712456096658058,-0.0021714894642745654, 0.008284688848618608;
    0., 0.13726887385387365, 0.;
    0.008284688848618608,-0.0021714894642745654, -0.011712456096658058;
   -0.04276374038685533,  0.20831893021621906,    0.07444694250251396;
    0.07546973535118673, -0.06621736226664678,    0.1518765115304851;
   -0.03436754777974814,  0.013657706809987716,   0.0565436438082349];


b=[ 0.10916518501485574,  0.013444365009573012, -0.019256622788629017;
    0.1881946990723202,  -0.03970769739425204,   0.05839565588582722;
    0.14115441625629258,  0.5706426558268328,   -0.043047573981921036;
   -0.10006019670089686, -0.3221571012199316,   -0.056767784980071075;
   -0.056767784980071075, -0.3221571012199316,   -0.10006019670089686;
   -0.043047573981921036, 0.5706426558268328,    0.14115441625629258;
    0.05839565588582722, -0.03970769739425204,   0.1881946990723202;
   -0.019256622788629017, 0.013444365009573012,  0.10916518501485574;
 0., 0., 0.];
  
elseif ( m == 2)
    nex=11;

% 11-stage 6th-order (autonomous): (SM_{11}^{[6]})
    nex=11;

a=[ 0.06422945556041365, -0.025341566517244504, 0.007599565747693973;
 -0.044424869662260426, -0.024510802572977775, 0.008244001069595266;
 0.2054755288210808, 0.02143961776111986, -0.00844862011879387;
 0.06628319090337748, 0.12295086312304439, -0.021180474543329136;
 -0.003551371843914318, 0.3143923641703535, 0.0035513718439143397;
 0.0034887127303764045, -0.1867082537420732, -0.0034887127303764175;
 -0.0034887127303764175, -0.1867082537420732, 0.0034887127303764045;
 0.0035513718439143397, 0.3143923641703535, -0.003551371843914318;
 -0.021180474543329136, 0.12295086312304439, 0.06628319090337748;
 -0.00844862011879387, 0.02143961776111986, 0.2054755288210808;
 0.008244001069595266, -0.024510802572977775, -0.044424869662260426;
 0.007599565747693973, -0.025341566517244504, 0.06422945556041365];

b= [0.1989318844867104, -0.016617016619520697, 0.0020156156354758656;
 -0.030836241539055953, -0.011909451581405113, 0.0016887898227496028;
 0.07965098544654711, 0.04499424637998612, 0.009110447840217101;
 0.08286433933376311, 0.1865482510472211, -0.06564804324862945;
 0.012909944487358058, -0.011760166914960044, -0.01290994448735806;
 0, 0.06193271982180175, 0;
 -0.01290994448735806, -0.011760166914960044, 0.012909944487358058;
 -0.06564804324862945, 0.1865482510472211, 0.08286433933376311;
 0.009110447840217101, 0.04499424637998612, 0.07965098544654711;
 0.0016887898227496028, -0.011909451581405113, -0.030836241539055953;
 0.0020156156354758656, -0.016617016619520697, 0.1989318844867104;
 0., 0., 0.];


elseif ( m == 3)
    nex=11;

% 11-stage 8th-order (autonomous): (SM_{11}^{[8]})
     
a= [0.07853904850841768, -0.055358596018743884, 0.016550797395649186;
 -0.10035959831600916, -0.1593139689007569, 0.045184103414205014;
 0.21748557677884262, 0.2096281405714991, -0.058916052519282266;
 0.08304419790770616, 0.10420247818733559, -0.019749426810985585;
 0.007699808870368553, 0.290721833658121, 0.09773915898281106;
 0.03837073411525743, -0.1676576652752326, -0.12781057054920295;
 -0.12781057054920295, -0.1676576652752326, 0.03837073411525743;
 0.09773915898281106, 0.290721833658121, 0.007699808870368553;
 -0.019749426810985585, 0.10420247818733559, 0.08304419790770616;
 -0.058916052519282266, 0.2096281405714991, 0.21748557677884262;
 0.045184103414205014, -0.1593139689007569, -0.10035959831600916;
 0.016550797395649186, -0.055358596018743884, 0.07853904850841768];

 
b= [0.15417184209270385, -0.01842607545473761, 0.007131251771948284;
 -0.006347779038079291, -0.0038906540539865287, 0.001097338895845309;
 0.11361626716381867, 0.024959240321255193, -0.013594387634411464;
 0.03428700927840067, 0.20128383652670429, -0.008359528724829266;
 -0.00011980505503017717, -0.008521705597032279, -0.00011980505503017717;
 -0.003984625917558664, 0.05363516096003841, -0.003984625917558664;
 -0.00011980505503017717, -0.008521705597032279, -0.00011980505503017717;
 -0.008359528724829266, 0.20128383652670429, 0.03428700927840067;
 -0.013594387634411464, 0.024959240321255193, 0.11361626716381867;
 0.001097338895845309, -0.0038906540539865287, -0.006347779038079291;
 0.007131251771948284, -0.01842607545473761, 0.15417184209270385;
 0., 0., 0.];
 
end
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