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butcher.m
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function B = butcher(method_name,symbolic)
% Usage: B = butcher(method_name,symbolic)
%
% Returns the butcher table associated with the method given by the input
% argument 'method_name'. We compute the internal stage times (root nodes)
% c, the Butcher matrix A, the method coefficients b, and the order
% of accuracy for the method q. The output table has the block structure
% B = [c, A; q, b]
% for a standard Runge-Kutta method.
%
% If the method has an embedded error indicator, we also compute the embedded
% method's coefficients b2 and order of accuracy p, and provide the output
% B = [c, A; q, b; p, b2]
%
% If the 'symbolic' input flag is both present and is 'true', then
% the resulting Butcher table is returned in symbolic format (to
% full precision).
%
% Method types are specified by the abbreviations:
% ERK - explicit Runge Kutta (strictly lower-triangular A)
% IRK - implicit Runge Kutta (dense A)
% DIRK - diagonally-implicit Runge Kutta (lower-triangular A)
% SDIRK - singly-diagonally implicit Runge Kutta (DIRK with fixed diagonal)
% ESDIRK - SDIRK method with an initial explicit stage
%
% Allowed methods are listed below. They are grouped by category
% (ERK, DIRK, IRK) and subcategory (non-embedded vs embedded).
% Each method lists:
% - the number of stages (s)
% - the order of accuracy of the stages (qs) (implicit methods
% only, since explicit method stage accuracy is limited to 1)
% - the order of accuracy of the method (q)
% - the linear order of the method (lq) -- i.e. the order of
% accuracy when the IVP RHS f(t,y) depends linearly on y
% - A,B,L-stability of the method (implicit methods only)
%
% Additionally, if the method contains an embedding, the table
% below also lists:
% - the embedding order of accuracy (p)
% - the embedding linear order of accuracy (lp)
% - A,B,L-stability of the embedding (implicit methods only)
%
% For each method, the reported values of {q,p,qs,lq,lp} correspond
% to satisfaction of the order conditions for a loose tolerance of
% 1e-8. For each method, we additionally report the tightest
% tolerance where these order conditions are still satisfied (to a
% minimum of 1e-40).
%
% Note: all method properties listed are automatically computed
% using the accompanying check_rk() function using symbolic
% variables to eliminate floating-point calculation error. This
% routine tests the full analytical order conditions up to 6;
% thereafter it merely checks Butcher's simplifying assumptions to
% assess higher orders of accuracy.
%
% Since Butcher's simplifying assumptions correspond to sufficient
% (but not necessary) conditions for order of accuracy, we may
% report an accuracy of 6 when in fact the method has higher
% order. Methods where we report 6 but have higher published order
% are marked with "+".
%
% However, since we analytically assess the *necessary* order
% conditions up to 6, any method with published accuracy <= 6 that
% does NOT attain this published order is marked with "-".
%
% Explicit, non-embedded methods:
%
% Name | s | q lq | tol
% --------------------------------------------------
% ERK-1-1 | 1 | 1 1 | 1e-40
% ERK-2-2 | 2 | 2 2 | 1e-40
% SSP2(2,2,2)-ERK | 2 | 2 2 | 1e-40
% Ascher(2,3,2)-ERK | 3 | 2 3 | 1e-40
% ARK(2,3,2)-ERK | 3 | 2 3 | 1e-40
% Ascher(2,2,2)-ERK | 3 | 2 2 | 1e-40
% SSP2(3,3,2)-lpm1-ERK | 3 | 2 2 | 1e-40
% SSP2(3,3,2)-lpm2-ERK | 3 | 2 2 | 1e-40
% SSP2(3,3,2)-lpum-ERK | 3 | 2 2 | 1e-40
% SSP2(3,3,2)-lspum-ERK | 3 | 2 2 | 1e-40
% SSP2(3,3,2)-a-ERK | 3 | 2 2 | 1e-40
% SSP2(3,3,2)-b-ERK | 3 | 2 2 | 1e-40
% Ascher(2,3,3)-ERK | 3 | 3 3 | 1e-40
% Knoth-Wolke-ERK | 3 | 3 3 | 1e-40
% SSP3(3,3,2)-ERK | 3 | 3 3 | 1e-40
% SSP3(3,3,3)-ERK | 3 | 3 3 | 1e-40
% SSPRK(3,3)-Shu-Osher-ERK | 3 | 3 3 | 1e-40
% Ascher(3,4,3)-ERK | 4 | 3 4 | 1e-17
% Cooper4-ERK | 4 | 3 3 | 1e-40
% SSP3(4,3,3)-ERK | 4 | 3 3 | 1e-40
% Ascher(4,4,3)-ERK | 5 | 3 3 | 1e-40
% DBM-5-3-ERK | 5 | 3 3 | 1e-15
% 3/8-Rule-ERK | 4 | 4 4 | 1e-40
% ERK-4-4 | 4 | 4 4 | 1e-40
% Cooper6-ERK | 6 | 4- 4 | 1e-40
% Butcher-7-6-ERK | 7 | 6 6 | 1e-40
% Butcher-7-6b-ERK | 7 | 6 6 | 1e-40
% Butcher-9-7-ERK | 9 | 6+ 7 | 1e-40
% CooperVerner-11-8-ERK | 11 | 6+ 8 | 1e-14
%
%
% Explicit, embedded methods:
%
% | | Method | Embedding |
% Name | s | q lq | p lp | tol
% -----------------------------------------------------------------
% Heun-Euler-ERK | 2 | 2 2 | 1 1 | 1e-40
% ERK-3-3 | 3 | 3 3 | 2 2 | 1e-40
% ARK3(2)4L[2]SA-ERK | 4 | 3 3 | 2 2 | 1e-25
% Bogacki-Shampine-ERK | 4 | 3 3 | 2 2 | 1e-40
% Merson-4-3-ERK | 5 | 4 4 | 3 5 | 1e-40
% Zonneveld-4-3-ERK | 5 | 4 4 | 3 3 | 1e-40
% ARK4(3)6L[2]SA-ERK | 6 | 4 4 | 3 3 | 1e-25
% Sayfy-Aburub-4-3-ERK | 6 | 4 4 | 3 3 | 1e-40
% ARK4(3)7L[2]SA-ERK | 7 | 4 4 | 3 3 | 1e-25
% Fehlberg-ERK | 6 | 5 5 | 4 4 | 1e-40
% Cash-Karp-ERK | 6 | 5 5 | 4 4 | 1e-40
% Dormand-Prince-ERK | 7 | 5 5 | 4 4 | 1e-40
% ARK5(4)8L[2]SA-ERK | 8 | 5 5 | 4 4 | 1e-25
% ARK5(4)8L[2]SAb-ERK | 8 | 5 5 | 4 4 | 1e-25
% Verner-6-5-ERK | 8 | 6 6 | 5 5 | 1e-40
% Fehlberg-8-7-ERK | 13 | 6+ 8 | 6+ 7 | 1e-40
%
%
% Diagonally-implicit, non-embedded methods:
%
% Name | s | q lq A B L | qs tol
% ------------------------------------------------------------------
% SSP2(2,2,2)-SDIRK | 2 | 2 2 Y Y Y | 1 1e-40
% Ascher(2,3,2)-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% Ascher(2,2,2)-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% ARK(2,3,2)-SDIRK | 3 | 2 2 Y Y | 2 1e-40
% SSP2(3,3,2)-lpm1-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% SSP2(3,3,2)-lpm2-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% SSP2(3,3,2)-lpum-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% SSP2(3,3,2)-lspum-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% SSP2(3,3,2)-a-DIRK | 3 | 2 2 Y Y | 1 1e-40
% SSP2(3,3,2)-b-DIRK | 3 | 2 2 Y Y | 1 1e-40
% SSP3(3,3,2)-SDIRK | 3 | 2 2 Y Y | 1 1e-40
% Ascher(2,3,3)-SDIRK | 3 | 3 3 Y Y | 1 1e-40
% SSP3(3,3,3)-ESDIRK | 3 | 3 3 | 1 1e-40
% EDIRK-3-3 | 3 | 3 3 Y | 1 1e-40
% ESDIRK-3-3 | 3 | 3 3 Y | 1 1e-40
% SSP3(4,3,3)-SDIRK | 4 | 3 3 Y | 1 1e-14
% Ascher(3,4,3)-SDIRK | 4 | 3 3 Y Y | 1 1e-17
% Ascher(4,4,3)-SDIRK | 4 | 3 3 Y Y | 1 1e-40
% Cooper4-ESDIRK | 4 | 3 3 Y | 1 1e-40
% DBM-5-3-ESDIRK | 5 | 3 3 Y Y | 1 1e-16
% SDIRK4()5L[1]SA | 5 | 4 4 Y Y | 1 1e-40
% SDIRK5()5L[1] | 5 | 5 5 Y Y | 1 1e-25
% Cooper6-ESDIRK | 6 | 4- 4 Y | 1 1e-40
%
%
% Diagonally-implicit, embedded methods:
%
% | | Method | Embedding |
% Name | s | q lq A B L | p lp A B L | qs tol
% ----------------------------------------------------------------------------------------
% SDIRK-2-1 | 2 | 2 2 Y Y | 1 1 Y Y Y | 1 1e-40
% SDIRK-2-2 | 2 | 2 2 Y Y | 1 1 | 1 1e-40
% TRBDF2-ESDIRK | 3 | 2 2 Y Y | 3 3 | 2 1e-40
% TRX2-ESDIRK | 3 | 2 2 Y | 3 3 | 2 1e-40
% Billington-SDIRK | 3 | 2 2 Y | 3 3 | 1 1e-11
% ARK3(2)4L[2]SA-ESDIRK | 4 | 3 3 Y Y | 2 2 Y | 2 1e-25
% Kvaerno(4,2,3)-ESDIRK | 4 | 3 3 Y | 2 2 Y | 2 1e-14
% ESDIRK3(2)4L[2]SA | 4 | 3 3 Y Y | 2 2 Y | 2 1e-40
% ESDIRK3(2)5L[2]SA | 5 | 3 3 Y Y | 2 2 Y Y | 2 1e-25
% ESDIRK3(2I)5L[2]SA | 5 | 3 3 Y Y | 2 2 Y Y | 2 1e-40
% SDIRK-5-4 | 5 | 4 4 Y Y | 3 3 Y | 1 1e-40
% Cash(5,3,4)-SDIRK | 5 | 4 4 Y Y | 3 3 Y | 1 1e-10
% Kvaerno(5,3,4)-ESDIRK | 5 | 4 4 Y | 3 3 Y | 2 1e-14
% Cash(5,2,4)-SDIRK | 5 | 4 4 Y Y | 2 2 Y | 1 1e-10
% ESDIRK4(3)6L[2]SA | 6 | 4 4 Y Y | 3 3 Y Y | 2 1e-26
% ESDIRK4(3)7L[2]SA | 7 | 4 4 Y Y | 3 3 Y Y | 2 1e-25
% ESDIRK4(3I)6L[2]SA | 6 | 4 4 Y Y | 3 3 Y Y | 2 1e-25
% QESDIRK4(3)6L[2]SA | 6 | 4 4 Y Y | 3 3 Y Y | 2 1e-25
% ARK4(3)6L[2]SA-ESDIRK | 6 | 4 4 Y Y | 3 3 Y | 2 1e-40
% ARK4(3)7L[2]SA-ESDIRK | 7 | 4 4 Y Y | 3 3 Y Y | 2 1e-25
% ESDIRK5(3)6L[2]SA | 6 | 5 5 Y Y | 3 3 Y Y | 2 1e-25
% ESDIRK5(4)7L[2]SA | 7 | 5 5 Y Y | 4 4 Y Y | 2 1e-26
% ESDIRK5(4)7L[2]SA2 | 7 | 5 5 Y Y | 4 4 Y | 2 1e-26
% ESDIRK5(4)8L[2]SA | 8 | 5 5 Y Y | 1 1 | 2 1e-25
% ARK5(4)8L[2]SA-ESDIRK | 8 | 5 5 Y Y | 4 4 Y | 2 1e-25
% ARK5(4)8L[2]SAb-ESDIRK | 8 | 5 5 Y Y | 4 4 Y | 2 1e-25
% Kvaerno(7,4,5)-ESDIRK | 7 | 5 5 Y Y | 4 4 Y | 2 1e-15
% ESDIRK6(5)9L[2]SA | 9 | 6 6 Y Y | 0 0 | 2 1e-29
% ESDIRK5(4I)8L[2]SA | 8 | 0 0 | 0 0 | 0 -----
% ESDIRK6(4)7A[2] | 7 | 0 0 | 0 0 | 0 -----
%
% Note: both ESDIRK5(4I)8L[2]SA and ESDIRK6(4)7A[2] seem to have typos in the 7th row
% in the paper where they are presented, as both fail the row-sum condition. Similarly,
% both ESDIRK5(4)8L[2]SA and ESDIRK6(5)9L[2]SA seem to have typos in their embedding
% coefficients, since their base methods show the appropriate order of accuracy, but the
% embeddings are 1st and 0th order accurate, respectively.
%
% Fully-implicit, non-embedded methods:
%
% Name | s | q lq A B L | qs tol
% --------------------------------------------------------------------
% IRK-1-1 | 1 | 1 1 Y Y Y | 1 1e-40
% LobattoIIIC-2-2-IRK | 2 | 2 2 Y Y Y | 1 1e-40
% Crank-Nicolson-2-2-IRK | 2 | 2 2 Y | 2 1e-40
% SIRK-2-2 | 2 | 2 2 Y Y | 2 1e-40
% LobattoIIIA-2-2-IRK | 2 | 2 2 Y | 2 1e-40
% LobattoIII-2-2-IRK | 2 | 2 2 | 1 1e-40
% RadauIIA-2-3-IRK | 2 | 3 3 Y Y Y | 2 1e-40
% Gauss-2-4-IRK | 2 | 4 4 Y Y | 2 1e-40
% LobattoIIIC-3-4-IRK | 3 | 4 4 Y Y Y | 2 1e-40
% LobattoIIIA-3-4-IRK | 3 | 4 4 Y | 3 1e-40
% LobattoIIIB-3-4-IRK | 3 | 4 4 Y | 1 1e-40
% LobattoIII-3-4-IRK | 3 | 4 4 | 2 1e-40
% RadauIA-3-5-IRK | 3 | 5 5 Y Y Y | 2 1e-40
% RadauIIA-3-5-IRK | 3 | 5 5 Y Y Y | 3 1e-40
% RadauI-3-5-IRK | 3 | 5 5 | 3 1e-40
% RadauII-3-5-IRK | 3 | 5 5 | 2 1e-40
% Gauss-3-6-IRK | 3 | 6 6 Y Y | 3 1e-40
% LobattoIIIC-4-6-IRK | 4 | 6 6 Y Y Y | 3 1e-40
% LobattoIIIA-4-6-IRK | 4 | 6 6 Y | 4 1e-40
% LobattoIIIB-4-6-IRK | 4 | 6 6 Y | 2 1e-40
% LobattoIII-4-6-IRK | 4 | 6 6 | 3 1e-40
% RadauIIA-4-7-IRK | 4 | 7 7 Y Y Y | 4 1e-40
% LobattoIIIC-5-8-IRK | 5 | 8 8 Y Y Y | 4 1e-40
% LobattoIIIA-5-8-IRK | 5 | 8 8 Y | 5 1e-40
% LobattoIIIB-5-8-IRK | 5 | 8 8 Y | 3 1e-40
% LobattoIII-5-8-IRK | 5 | 8 8 | 4 1e-40
% RadauIIA-5-9-IRK | 5 | 9 8 Y Y Y | 5 1e-15
% Gauss-6-12-IRK | 6 | 12 8 Y Y | 6 1e-16
%
%
%------------------------------------------------------------
% Programmer(s): Daniel R. Reynolds @ SMU
%------------------------------------------------------------
% Copyright (c) 2018, Southern Methodist University.
% All rights reserved.
% For details, see the LICENSE file.
%------------------------------------------------------------
% set anonymous function to handle behavior of double-vs-symbolic output
if ~exist('symbolic','var')
symbolic = false;
end
if (symbolic)
v = @(x) sym(x);
else
v = @(x) double(x);
end
z = v(0);
% set the butcher table
if (strcmp(method_name,'DBM-5-3-ERK'))
c = [z; v(0.1030620881159184); v(0.72139131281753662); v(1.28181117351981733); v(1) ];
b = [ v(0.87795339639076672), v(-0.72692641526151549), v(0.7520413715737272), ...
v(-0.22898029400415090), v(0.32591194130117246) ];
A = [ z, z, z, z, z;
v(0.10306208811591838), z, z, z, z;
v(-0.94124866143519894), v(1.6626399742527356), z, z, z;
v(-1.3670975201437765), v(1.3815852911016873), v(1.2673234025619065), z, z;
v(-0.81287582068772448), v(0.81223739060505738), v(0.90644429603699305), ...
v(0.094194134045674111), z ];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'DBM-5-3-ESDIRK'))
c = [z; v(0.1030620881159184); v(0.72139131281753662); v(1.28181117351981733); v(1) ];
b = [ v(0.87795339639076672), v(-0.72692641526151549), v(0.7520413715737272), ...
v(-0.22898029400415090), v(0.32591194130117246) ];
A = [ z, z, z, z, z;
v(-0.22284985318525410), v(0.32591194130117247), z, z, z;
v(-0.46801347074080545), v(0.86349284225716961), v(0.32591194130117247), z, z;
v(-0.46509906651927421), v(0.81063103116959553), v(0.61036726756832357), v(0.32591194130117245), z;
v(0.87795339639076675), v(-0.72692641526151547), v(0.75204137157372720), ...
v(-0.22898029400415088), v(0.32591194130117247) ];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'ARK3(2)4L[2]SA-ERK'))
c = [z; v(1767732205903)/v(2027836641118); v(3)/v(5); v(1)];
b = [v(1471266399579)/v(7840856788654), v(-4482444167858)/v(7529755066697), ...
v(11266239266428)/v(11593286722821), v(1767732205903)/v(4055673282236)];
b2 = [v(2756255671327)/v(12835298489170), v(-10771552573575)/v(22201958757719), ...
v(9247589265047)/v(10645013368117), v(2193209047091)/v(5459859503100)];
A = [z, z, z, z;
v(1767732205903)/v(2027836641118), z, z, z;
v(5535828885825)/v(10492691773637), v(788022342437)/v(10882634858940), z, z;
v(6485989280629)/v(16251701735622), v(-4246266847089)/v(9704473918619), ...
v(10755448449292)/v(10357097424841), z];
q = 3;
p = 2;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK3(2)4L[2]SA-ESDIRK'))
c = [z; v(1767732205903)/v(2027836641118); v(3)/v(5); v(1)];
b = [v(1471266399579)/v(7840856788654), v(-4482444167858)/v(7529755066697), ...
v(11266239266428)/v(11593286722821), v(1767732205903)/v(4055673282236)];
b2 = [v(2756255671327)/v(12835298489170), v(-10771552573575)/v(22201958757719), ...
v(9247589265047)/v(10645013368117), v(2193209047091)/v(5459859503100)];
gamma = v(1767732205903)/v(4055673282236);
A = [z, z, z, z;
v(1767732205903)/v(4055673282236), gamma, z, z;
v(2746238789719)/v(10658868560708), v(-640167445237)/v(6845629431997), gamma, z;
v(1471266399579)/v(7840856788654), v(-4482444167858)/v(7529755066697), ...
v(11266239266428)/v(11593286722821), gamma];
q = 3;
p = 2;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK4(3)6L[2]SA-ERK'))
c = [z; v(1)/v(2); v(83)/v(250); v(31)/v(50); v(17)/v(20); v(1)];
b = [v(82889)/v(524892), z, v(15625)/v(83664), v(69875)/v(102672), v(-2260)/v(8211), v(1)/v(4)];
b2 = [v(4586570599)/v(29645900160), z, v(178811875)/v(945068544), ...
v(814220225)/v(1159782912), v(-3700637)/v(11593932), v(61727)/v(225920)];
A = [z, z, z, z, z, z;
v(1)/v(2), z, z, z, z, z;
v(13861)/v(62500), v(6889)/v(62500), z, z, z, z;
v(-116923316275)/v(2393684061468), v(-2731218467317)/v(15368042101831), ...
v(9408046702089)/v(11113171139209), z, z, z;
v(-451086348788)/v(2902428689909), v(-2682348792572)/v(7519795681897), ...
v(12662868775082)/v(11960479115383), v(3355817975965)/v(11060851509271), z, z;
v(647845179188)/v(3216320057751), v(73281519250)/v(8382639484533), ...
v(552539513391)/v(3454668386233), v(3354512671639)/v(8306763924573), ...
v(4040)/v(17871), z];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK4(3)6L[2]SA-ESDIRK'))
c = [z; v(1)/v(2); v(83)/v(250); v(31)/v(50); v(17)/v(20); v(1)];
b = [v(82889)/v(524892), z, v(15625)/v(83664), v(69875)/v(102672), v(-2260)/v(8211), v(1)/v(4)];
b2 = [v(4586570599)/v(29645900160), z, v(178811875)/v(945068544), ...
v(814220225)/v(1159782912), v(-3700637)/v(11593932), v(61727)/v(225920)];
A = [z, z, z, z, z, z;
v(1)/v(4), v(1)/v(4), z, z, z, z;
v(8611)/v(62500), v(-1743)/v(31250), v(1)/v(4), z, z, z;
v(5012029)/v(34652500), v(-654441)/v(2922500), v(174375)/v(388108), v(1)/v(4), z, z;
v(15267082809)/v(155376265600), v(-71443401)/v(120774400), v(730878875)/v(902184768), ...
v(2285395)/v(8070912), v(1)/v(4), z;
v(82889)/v(524892), z, v(15625)/v(83664), v(69875)/v(102672), v(-2260)/v(8211), v(1)/v(4)];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ESDIRK4(3)6L[2]SA'))
% From: Table 16 of Kennedy & Carpenter, NASA/TM-2016-219173
c = [z; v(1)/v(2); (v(2)-sqrt(v(2)))/v(4); v(5)/v(8); v(26)/v(25); v(1)];
b = [(v(1181)-v(987)*sqrt(v(2)))/v(13782), (v(1181)-v(987)*sqrt(v(2)))/v(13782), v(47)*(-v(267)+v(1783)*sqrt(v(2)))/v(273343), -v(16)*(-v(22922)+v(3525)*sqrt(v(2)))/v(571953), -v(15625)*(v(97)+v(376)*sqrt(v(2)))/v(90749876), v(1)/v(4)];
b2 = [-v(480923228411)/v(4982971448372), -v(480923228411)/v(4982971448372), v(6709447293961)/v(12833189095359), v(3513175791894)/v(6748737351361), -v(498863281070)/v(6042575550617), v(2077005547802)/v(8945017530137)];
A = [z, z, z, z, z, z; ...
v(1)/v(4), v(1)/v(4), z, z, z, z; ...
(v(1)-sqrt(v(2)))/v(8), (v(1)-sqrt(v(2)))/v(8), v(1)/v(4), z, z, z; ...
(v(5)-v(7)*sqrt(v(2)))/v(64), (v(5)-v(7)*sqrt(v(2)))/v(64), v(7)*(v(1)+sqrt(v(2)))/v(32), v(1)/v(4), z, z; ...
-(v(13796)+v(54539)*sqrt(v(2)))/v(125000), -(v(13796)+v(54539)*sqrt(v(2)))/v(125000), (v(506605)+v(132109)*sqrt(v(2)))/v(437500), v(166)*(-v(97)+v(376)*sqrt(v(2)))/v(109375), v(1)/v(4), z; ...
b];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ESDIRK4(3)7L[2]SA'))
% From: Table 10 of Kennedy & Carpenter, Appl. Numer. Math., 2019
c = [z; v(1)/v(4); v(1200237871921)/v(16391473681546); v(1)/v(2); v(395)/v(567); v(89)/v(126); v(1)];
b = [z, -v(5649241495537)/v(14093099002237), v(5718691255176)/v(6089204655961), v(2199600963556)/v(4241893152925), v(8860614275765)/v(11425531467341), -v(3696041814078)/v(6641566663007), v(1)/v(8)];
b(1) = v(1)-sum(b);
d = [z, -v(1517409284625)/v(6267517876163), v(8291371032348)/v(12587291883523), v(5328310281212)/v(10646448185159), v(5405006853541)/v(7104492075037), -v(4254786582061)/v(7445269677723), v(19)/v(140)];
d(1) = v(1)-sum(d);
A = [z, z, z, z, z, z, z; ...
z, v(1)/v(8), z, z, z, z, z; ...
z, -v(39188347878)/v(1513744654945), v(1)/v(8), z, z, z, z; ...
z, v(1748874742213)/v(5168247530883), -v(1748874742213)/v(5795261096931), v(1)/v(8), z, z, z; ...
z, -v(6429340993097)/v(17896796106705), v(9711656375562)/v(10370074603625), v(1137589605079)/v(3216875020685), v(1)/v(8), z, z; ...
z, v(405169606099)/v(1734380148729), -v(264468840649)/v(6105657584947), v(118647369377)/v(6233854714037), v(683008737625)/v(4934655825458), v(1)/v(8), z; ...
b];
A(2,1) = c(2)-sum(A(2,:));
A(3,1) = c(3)-sum(A(3,:));
A(4,1) = c(4)-sum(A(4,:));
A(5,1) = c(5)-sum(A(5,:));
A(6,1) = c(6)-sum(A(6,:));
q = 4;
p = 3;
B = [c, A; q, b; p, d];
elseif (strcmp(method_name,'ESDIRK4(3I)6L[2]SA'))
% From: Table 17 of Kennedy & Carpenter, NASA/TM-2016-219173
c = [z; v(1)/v(2); (v(2)-sqrt(v(2)))/v(4); v(2012122486997)/v(3467029789466); v(1); v(1)];
b = [v(657241292721)/v(9909463049845), v(657241292721)/v(9909463049845), v(1290772910128)/v(5804808736437), v(1103522341516)/v(2197678446715), -v(3)/v(28), v(1)/v(4)];
b2 = [-v(71925161075)/v(3900939759889), -v(71925161075)/v(3900939759889), v(2973346383745)/v(8160025745289), v(3972464885073)/v(7694851252693), -v(263368882881)/v(4213126269514), v(3295468053953)/v(15064441987965)];
A = [z, z, z, z, z, z; ...
v(1)/v(4), v(1)/v(4), z, z, z, z; ...
-v(1356991263433)/v(26208533697614), -v(1356991263433)/v(26208533697614), v(1)/v(4), z, z, z; ...
-v(1778551891173)/v(14697912885533), -v(1778551891173)/v(14697912885533), v(7325038566068)/v(12797657924939), v(1)/v(4), z, z; ...
-v(24076725932807)/v(39344244018142), -v(24076725932807)/v(39344244018142), v(9344023789330)/v(6876721947151), v(11302510524611)/v(18374767399840), v(1)/v(4), z; ...
b];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'QESDIRK4(3)6L[2]SA'))
% From: Table 20 of Kennedy & Carpenter, NASA/TM-2016-219173
c = [z; v(16)/v(75); v(8)*(v(2)-sqrt(v(2)))/v(25); v(1298154247449)/v(2478647287318); v(11563539331134)/v(11078645848867); v(1)];
b = [v(540088238697)/v(4693226184761), z, v(1094762490994)/v(7880993776667), v(4016564763003)/v(7185357966874), -v(411820258827)/v(3096789411938), v(8)/v(25)];
b2 = [-v(374484326677)/v(8442488809460), -v(41125091159938)/v(25963879779069), v(24025088270494)/v(12927594097169), v(5193917461301)/v(8985383982321), -v(1843122001830)/v(16078617943063), v(2439572212972)/v(7960792257433)];
A = [z, z, z, z, z, z; ...
v(8)/v(75), v(8)/v(75), z, z, z, z; ...
v(605497861978)/v(9136257189845), -v(2127848798551)/v(10702252975294), v(8)/v(25), z, z, z; ...
-v(3005106686955)/v(6150333508049), -v(68662668193799)/v(11091168490809), v(80786898110822)/v(11737001380747), v(8)/v(25), z, z; ...
-v(26162805558846)/v(8363194173203), -v(291987295964487)/v(9066074244437), v(384682892278670)/v(10959450712301), v(13555548451102)/v(14148104892819), v(8)/v(25), z; ...
b];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'SDIRK4()5L[1]SA'))
% From: Table 22 of Kennedy & Carpenter, NASA/TM-2016-219173
c = [v(1)/v(4); (v(2)-sqrt(v(2)))/v(4); (v(13)+v(8)*sqrt(v(2)))/v(41); (v(41)+v(9)*sqrt(v(2)))/v(49); v(1)];
b = [z, v(4)*(v(74)+v(273)*sqrt(v(2)))/v(5253), (v(19187)+v(5031)*sqrt(v(2)))/v(55284), (v(116092)-v(100113)*sqrt(v(2)))/v(334956), v(1)/v(4)];
A = [v(1)/v(4), z, z, z, z; ...
(v(1)-sqrt(v(2)))/v(4), v(1)/v(4), z, z, z; ...
(-v(1676)+v(145)*sqrt(v(2)))/v(6724), v(3)*(v(709)+v(389)*sqrt(v(2)))/v(6724), v(1)/v(4), z, z; ...
-(v(371435)+v(351111)*sqrt(v(2)))/v(470596), (v(98054928)+v(73894543)*sqrt(v(2)))/v(112001848), (v(56061972)+v(30241643)*sqrt(v(2)))/v(112001848), v(1)/v(4), z; ...
b];
q = 4;
B = [c, A; q, b];
elseif (strcmp(method_name,'ESDIRK5(3)6L[2]SA'))
% From: Table 23 of Kennedy & Carpenter, NASA/TM-2016-219173
c = [z; v(4024571134387)/v(7237035672548); v(14228244952610)/v(13832614967709); v(1)/v(10); v(3)/v(50); v(1)];
b = [-v(6225479754948)/v(6925873918471), v(6894665360202)/v(11185215031699), -v(2508324082331)/v(20512393166649), -v(7289596211309)/v(4653106810017), v(39811658682819)/v(14781729060964), v(3282482714977)/v(11805205429139)];
b2 = [-v(2512930284403)/v(5616797563683), v(5849584892053)/v(8244045029872), -v(718651703996)/v(6000050726475), -v(18982822128277)/v(13735826808854), v(23127941173280)/v(11608435116569), v(2847520232427)/v(11515777524847)];
A = [z, z, z, z, z, z; ...
v(3282482714977)/v(11805205429139), v(3282482714977)/v(11805205429139), z, z, z, z; ...
v(606638434273)/v(1934588254988), v(2719561380667)/v(6223645057524), v(3282482714977)/v(11805205429139), z, z, z; ...
-v(651839358321)/v(6893317340882), -v(1510159624805)/v(11312503783159), v(235043282255)/v(4700683032009), v(3282482714977)/v(11805205429139), z, z; ...
-v(5266892529762)/v(23715740857879), -v(1007523679375)/v(10375683364751), v(521543607658)/v(16698046240053), v(514935039541)/v(7366641897523), v(3282482714977)/v(11805205429139), z; ...
b];
q = 5;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'SDIRK5()5L[1]'))
c = [v(4024571134387)/v(14474071345096); v(5555633399575)/v(5431021154178); v(5255299487392)/v(12852514622453); v(3)/v(20); v(10449500210709)/v(14474071345096)];
b = [-v(2522702558582)/v(12162329469185), v(1018267903655)/v(12907234417901), v(4542392826351)/v(13702606430957), v(5001116467727)/v(12224457745473), v(1509636094297)/v(3891594770934)];
A = [v(4024571134387)/v(14474071345096), z, z, z, z; ...
v(9365021263232)/v(12572342979331), v(4024571134387)/v(14474071345096), z, z, z; ...
v(2144716224527)/v(9320917548702), -v(397905335951)/v(4008788611757), v(4024571134387)/v(14474071345096), z, z; ...
-v(291541413000)/v(6267936762551), v(226761949132)/v(4473940808273), -v(1282248297070)/v(9697416712681), v(4024571134387)/v(14474071345096), z; ...
-v(2481679516057)/v(4626464057815), -v(197112422687)/v(6604378783090), v(3952887910906)/v(9713059315593), v(4906835613583)/v(8134926921134), v(4024571134387)/v(14474071345096)];
q = 5;
B = [c, A; q, b];
elseif (strcmp(method_name,'ESDIRK5(4)7L[2]SA'))
% From: Table 25 of Kennedy & Carpenter, NASA/TM-2016-219173
c = [z; v(46)/v(125); v(1518047795759)/v(14084074382095); v(13)/v(25); v(5906118540659)/v(9042400211275); v(26)/v(25); v(1)];
b = [z, -v(1319096626979)/v(17356965168099), v(4356877330928)/v(10268933656267), v(922991294344)/v(3350617878647), v(4729382008034)/v(14755765856909), -v(308199069217)/v(5897303561678), v(23)/v(125)];
b(1) = v(1)-sum(b);
b2 = [z, -v(12068858301481)/v(111697653055985), v(30204157393951)/v(62440428688139), v(26156819792768)/v(110856972047457), v(33531609809941)/v(89326307438822), -v(18686091006953)/v(578397443530870), v(10582397456777)/v(69011126173064)];
b2(1) = v(1)-sum(b2);
A = [z, z, z, z, z, z, z; ...
z, v(23)/v(125), z, z, z, z, z; ...
z, -v(121529886477)/v(3189120653983), v(23)/v(125), z, z, z, z; ...
z, v(186345625210)/v(8596203768457), v(3681435451073)/v(12579882114497), v(23)/v(125), z, z, z; ...
z, -v(9898129553915)/v(11630542248213), v(19565727496993)/v(11159348038501), v(2073446517052)/v(4961027473423), v(23)/v(125), z, z; ...
z, -v(39752543191591)/v(7894275939720), v(52228808998390)/v(5821762529307), v(2756378382725)/v(8748785577174), v(17322065038796)/v(10556643942083), v(23)/v(125), z; ...
b];
A(2,1) = c(2)-sum(A(2,:));
A(3,1) = c(3)-sum(A(3,:));
A(4,1) = c(4)-sum(A(4,:));
A(5,1) = c(5)-sum(A(5,:));
A(6,1) = c(6)-sum(A(6,:));
q = 5;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ESDIRK5(4)7L[2]SA2'))
% From: Table 11 of Kennedy & Carpenter, Appl. Numer. Math., 2019
c = [z; v(46)/v(125); v(7121331996143)/v(11335814405378); v(49)/v(353); v(3706679970760)/v(5295570149437); v(347)/v(382); v(1)];
b = [z, -v(188593204321)/(4778616380481), v(2809310203510)/v(10304234040467), v(1021729336898)/v(2364210264653),...
v(870612361811)/v(2470410392208), -v(1307970675534)/v(8059683598661), v(23)/v(125)];
b(1) = v(1)-sum(b);
d = [z, -v(582099335757)/v(7214068459310), v(615023338567)/v(3362626566945), v(3192122436311)/v(6174152374399),...
v(6156034052041)/v(14430468657929), -v(1011318518279)/v(9693750372484), v(1914490192573)/v(13754262428401)];
d(1) = v(1)-sum(d);
A = [z, z, z, z, z, z, z; ...
z, v(23)/v(125), z, z, z, z, z; ...
z, v(791020047304)/v(3561426431547), v(23)/v(125), z, z, z, z; ...
z, -v(158159076358)/v(11257294102345), -v(85517644447)/v(5003708988389), v(23)/v(125), z, z, z; ...
z, -v(1653327111580)/v(4048416487981), v(1514767744496)/v(9099671765375), v(14283835447591)/v(12247432691556), v(23)/v(125), z, z; ...
z, -v(4540011970825)/v(8418487046959), -v(1790937573418)/v(7393406387169), v(10819093665085)/v(7266595846747), v(4109463131231)/v(7386972500302), v(23)/v(125), z; ...
b];
A(2,1) = c(2)-sum(A(2,:));
A(3,1) = c(3)-sum(A(3,:));
A(4,1) = c(4)-sum(A(4,:));
A(5,1) = c(5)-sum(A(5,:));
A(6,1) = c(6)-sum(A(6,:));
q = 5;
p = 4;
B = [c, A; q, b; p, d];
elseif (strcmp(method_name,'ESDIRK5(4)8L[2]SA'))
% From: Table 12 of Kennedy & Carpenter, Appl. Numer. Math., 2019
% NOTE: \hat{b} has typos in their paper!!
c = [z; v(2)/v(7); v(5779892736881)/v(11850239716711); v(150)/v(203); v(27)/v(46); v(473)/v(532); v(30)/v(83); v(1) ];
b = [z, v(2162042939093)/v(22873479087181), -v(4222515349147)/v(9397994281350), v(3431955516634)/v(4748630552535), -v(374165068070)/v(9085231819471), -v(1847934966618)/v(8254951855109), v(5186241678079)/v(7861334770480), v(1)/v(7)];
b(1) = v(1)-sum(b);
d = [z, v(701879993119)/v(7084679725724), -v(8461269287478)/v(14654112271769), v(6612459227430)/v(11388259134383), v(2632441606103)/v(12598871370240), -v(2147694411931)/v(10286892713802), v(4103061625716)/v(6371697724583), v(4103061625716)/v(6371697724583)];
d(1) = v(1)-sum(d);
A = [z, z, z, z, z, z, z, z; ...
z, v(1)/v(7), z, z, z, z, z, z; ...
z, v(1521428834970)/v(8822750406821), v(1)/v(7), z, z, z, z, z; ...
z, v(5338711108027)/v(29869763600956), v(1483184435021)/v(6216373359362), v(1)/v(7), z, z, z, z; ...
z, v(2264935805846)/v(12599242299355), v(1330937762090)/v(13140498839569), -v(287786842865)/v(17211061626069), v(1)/v(7), z, z, z; ...
z, v(118352937080)/v(527276862197), -v(2960446233093)/v(7419588050389), -v(3064256220847)/v(46575910191280), v(6010467311487)/v(7886573591137), v(1)/v(7), z, z; ...
z, v(1134270183919)/v(9703695183946), v(4862384331311)/v(10104465681802), v(1127469817207)/v(2459314315538), -v(9518066423555)/v(11243131997224), -v(811155580665)/v(7490894181109), v(1)/v(7), z; ...
b];
A(2,1) = c(2)-sum(A(2,:));
A(3,1) = c(3)-sum(A(3,:));
A(4,1) = c(4)-sum(A(4,:));
A(5,1) = c(5)-sum(A(5,:));
A(6,1) = c(6)-sum(A(6,:));
A(7,1) = c(7)-sum(A(7,:));
q = 5;
p = 4;
B = [c, A; q, b; p, d];
elseif (strcmp(method_name,'ESDIRK6(4)7A[2]'))
% From: Table 27 of Kennedy & Carpenter, NASA/TM-2016-219173
% NOTE: row 7 of A has typos in their paper!!
c = [z; v(5)/v(8); v(5)*(v(2)-sqrt(v(2)))/v(16); v(81)/v(100); v(89)/v(100); v(3)/v(20); v(11)/v(16)];
b = [v(541976983222)/v(5570117184863), v(424517620289)/v(10281234581904), v(3004784109584)/v(2968823999583), -v(1080268266981)/v(2111416452515), v(3198291424887)/v(7137915940442), -v(6709580973937)/v(9894986011196), v(4328230890552)/v(7324362344791)];
b2 = [v(23807813993)/v(6613359907661), v(122567156372)/v(6231407414731), v(5289947382915)/v(9624205771537), -v(132784415823)/v(2592433009541), v(2055455363695)/v(9863229933602), -v(686952476184)/v(6416474135057), v(2766631516579)/v(7339217152243)];
A = [z, z, z, z, z, z, z; ...
v(5)/v(16), v(5)/v(16), z, z, z, z, z; ...
-v(6647797099592)/v(102714892273533), -v(6647797099592)/v(102714892273533), v(5)/v(16), z, z, z, z; ...
-v(87265218833)/v(1399160431079), -v(87265218833)/v(1399160431079), v(3230569391728)/v(5191843160709), v(5)/v(16), z, z, z; ...
-v(3742173976023)/v(7880396319491), -v(4537732256035)/v(9784784042546), v(32234033847818)/v(24636233068093), v(1995418204833)/v(9606020544314), v(5)/v(16), z, z; ...
-v(460973220726)/v(7579441323155), -v(113988582459)/v(8174956167569), -v(679076942985)/v(7531712581924), v(1946214040135)/v(12392905069014), -v(2507263458377)/v(16215886710685), v(5)/v(16), z; ...
2429030329867/4957732179206, -5124723475981/12913403568538, 3612624980699/11761071195830, 714493169479/5549220584147, -4586610949246/13858427945825, -4626134504839/7500671962341, 5/16];
q = 6;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ESDIRK6(5)9L[2]SA'))
% From: Table 13 of Kennedy & Carpenter, Appl. Numer. Math., 2019
% NOTE: \hat{b} has typos in their paper!!
c = [z; v(4)/v(9); v(376327483029687)/v(1335600577485745); v(433625707911282)/v(850513180247701); v(183)/v(200);...
v(62409086037595)/v(296036819031271); v(81796628710131)/v(911762868125288); v(97)/v(100); v(1)];
b = [z, z, z, v(281246836687281)/v(672805784366875), v(250674029546725)/v(464056298040646), v(88917245119922)/v(798581755375683), v(127306093275639)/v(658941305589808), -v(319515475352107)/v(658842144391777), v(2)/v(9)];
d = [-v(204006714482445)/v(253120897457864), z, -v(818062434310719)/v(743038324242217), v(3176520686137389)/v(1064235527052079), -v(574817982095666)/v(1374329821545869), -v(507643245828272)/v(1001056758847831), v(2013538191006793)/v(972919262949000), v(352681731710820)/v(726444701718347), -v(12107714797721)/v(746708658438760)];
A = [z, z, z, z, z, z, z, z, z;...
v(2)/v(9), v(2)/v(9), z, z, z, z, z, z, z;...
v(1)/v(9), -v(52295652026801)/v(1014133226193379), v(2)/v(9), z, z, z, z, z, z;...
v(37633260247889)/v(456511413219805), -v(162541608159785)/v(642690962402252), v(186915148640310)/v(408032288622937), v(2)/v(9), z, z, z, z, z;...
-v(37161579357179)/v(532208945751958), -v(211140841282847)/v(266150973773621), v(884359688045285)/v(894827558443789), v(845261567597837)/v(1489150009616527), v(2)/v(9), z, z, z, z;...
v(32386175866773)/v(281337331200713), v(498042629717897)/v(1553069719539220), -v(73718535152787)/v(262520491717733), -v(147656452213061)/v(931530156064788), -v(16605385309793)/v(2106054502776008), v(2)/v(9), z, z, z;...
-v(38317091100349)/v(1495803980405525), v(233542892858682)/v(880478953581929), -v(281992829959331)/v(709729395317651), -v(52133614094227)/v(895217507304839), -v(9321507955616)/v(673810579175161), v(79481371174259)/v(817241804646218), v(2)/v(9), z, z;...
-v(486324380411713)/v(1453057025607868), -v(1085539098090580)/v(1176943702490991), v(370161554881539)/v(461122320759884), v(804017943088158)/v(886363045286999), -v(15204170533868)/v(934878849212545), -v(248215443403879)/v(815097869999138), v(339987959782520)/v(552150039467091), v(2)/v(9), z;...
b];
q = 6;
p = 5;
B = [c, A; q, b; p, d];
elseif (strcmp(method_name,'ESDIRK5(4I)8L[2]SA'))
% From: Table 28 of Kennedy & Carpenter, NASA/TM-2016-219173
% NOTE: row 7 of A has typos in their paper!!
c = [z; v(1)/v(2); (v(2)+sqrt(v(2)))/v(4); v(53)/v(100); v(4)/v(5); v(17)/v(25); v(1); v(1)];
b = [v(3148564786223)/v(23549948766475), -v(4152366519273)/v(20368318839251), -v(143958253112335)/v(33767350176582), v(16929685656751)/v(6821330976083), v(37330861322165)/v(4907624269821), -v(103974720808012)/v(20856851060343), -v(93596557767)/v(4675692258479), v(1)/v(4)];
b2 = [v(1707112744407)/v(11125291145125), -v(34114578494)/v(9511465441463), -v(10730340352595)/v(5750059075211), v(16308974155447)/v(11154981868028), v(16015983083570)/v(4734398780449), -v(16745095336747)/v(7220642436550), -v(3941055932791)/v(7113931146175), v(9504350599766)/v(12768012822991)];
A = [z, z, z, z, z, z, z, z; ...
v(1)/v(4), v(1)/v(4), z, z, z, z, z, z; ...
v(1748874742213)/v(5795261096931), v(1748874742213)/v(5795261096931), v(1)/v(4), z, z, z, z, z; ...
v(2426486750897)/v(12677310711630), v(2426486750897)/v(12677310711630) -v(783385356511)/v(7619901499812), v(1)/v(4), z, z, z, z; ...
v(1616209367427)/v(5722977998639), v(1616209367427)/v(5722977998639), -v(211896077633)/v(5134769641545), v(464248917192)/v(17550087120101), v(1)/v(4), z, z, z; ...
v(1860464898611)/v(7805430689312), v(1825204367749)/v(7149715425471), -v(1289376786583)/v(6598860380111), v(55566826943)/v(2961051076052), v(1548994872005)/v(13709222415197), v(1)/v(4), z, z; ...
1783640092711/14417713428467, -5781183663275/18946039887294, 57847255876685/10564937217081, 29339178902168/9787613280015, 122011506936853/12523522131766, -60418758964762/9539790648093, 1/4, z;...
b];
q = 5;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK4(3)7L[2]SA-ERK'))
c = [z; v(247)/v(1000); v(4276536705230)/v(10142255878289); v(67)/v(200); v(3)/v(40); v(7)/v(10); v(1)];
b = [z, z, v(9164257142617)/v(17756377923965), v(-10812980402763)/v(74029279521829), ...
v(1335994250573)/v(5691609445217), v(2273837961795)/v(8368240463276), v(247)/v(2000)];
b2 = [z, z, v(4469248916618)/v(8635866897933), v(-621260224600)/v(4094290005349), ...
v(696572312987)/v(2942599194819), v(1532940081127)/v(5565293938103), v(2441)/v(20000)];
A = [z, z, z, z, z, z, z;
v(247)/v(1000), z, z, z, z, z, z;
v(247)/v(4000), v(2694949928731)/v(7487940209513), z, z, z, z, z;
v(464650059369)/v(8764239774964), v(878889893998)/v(2444806327765), ...
v(-952945855348)/v(12294611323341), z, z, z, z;
v(476636172619)/v(8159180917465), v(-1271469283451)/v(7793814740893), ...
v(-859560642026)/v(4356155882851), v(1723805262919)/v(4571918432560), ...
z, z, z;
v(6338158500785)/v(11769362343261), v(-4970555480458)/v(10924838743837), ...
v(3326578051521)/v(2647936831840), v(-880713585975)/v(1841400956686), ...
v(-1428733748635)/v(8843423958496), z, z;
v(760814592956)/v(3276306540349), v(760814592956)/v(3276306540349), ...
v(-47223648122716)/v(6934462133451), v(71187472546993)/v(9669769126921), ...
v(-13330509492149)/v(9695768672337), v(11565764226357)/v(8513123442827), z];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK4(3)7L[2]SA-ESDIRK'))
gam = v(1235)/v(10000);
c = [z; v(247)/v(1000); v(4276536705230)/v(10142255878289); v(67)/v(200); v(3)/v(40); v(7)/v(10); v(1)];
b = [z, z, v(9164257142617)/v(17756377923965), v(-10812980402763)/v(74029279521829), ...
v(1335994250573)/v(5691609445217), v(2273837961795)/v(8368240463276), v(247)/v(2000)];
b2 = [z, z, v(4469248916618)/v(8635866897933), v(-621260224600)/v(4094290005349), ...
v(696572312987)/v(2942599194819), v(1532940081127)/v(5565293938103), v(2441)/v(20000)];
A = [z, z, z, z, z, z, z;
gam, gam, z, z, z, z, z;
v(624185399699)/v(4186980696204), v(624185399699)/v(4186980696204), gam, ...
z, z, z, z;
v(1258591069120)/v(10082082980243), v(1258591069120)/v(10082082980243), ...
v(-322722984531)/v(8455138723562), gam, z, z, z;
v(-436103496990)/v(5971407786587), v(-436103496990)/v(5971407786587), ...
v(-2689175662187)/v(11046760208243), v(4431412449334)/v(12995360898505), gam, z, z;
v(-2207373168298)/v(14430576638973), v(-2207373168298)/v(14430576638973), ...
v(242511121179)/v(3358618340039), v(3145666661981)/v(7780404714551), ...
v(5882073923981)/v(14490790706663), gam, z;
z, z, v(9164257142617)/v(17756377923965), v(-10812980402763)/v(74029279521829), ...
v(1335994250573)/v(5691609445217), v(2273837961795)/v(8368240463276), gam];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK5(4)8L[2]SA-ERK'))
c = [z; v(41)/v(100); v(2935347310677)/v(11292855782101); ...
v(1426016391358)/v(7196633302097); v(92)/v(100); v(24)/v(100); v(3)/v(5); v(1)];
b = [v(-872700587467)/v(9133579230613), z, z, v(22348218063261)/v(9555858737531), ...
v(-1143369518992)/v(8141816002931), v(-39379526789629)/v(19018526304540), ...
v(32727382324388)/v(42900044865799), v(41)/v(200)];
b2 = [v(-975461918565)/v(9796059967033), z, z, v(78070527104295)/v(32432590147079), ...
v(-548382580838)/v(3424219808633), v(-33438840321285)/v(15594753105479), ...
v(3629800801594)/v(4656183773603), v(4035322873751)/v(18575991585200)];
A = [z, z, z, z, z, z, z, z;
v(41)/v(100), z, z, z, z, z, z, z;
v(367902744464)/v(2072280473677), v(677623207551)/v(8224143866563), z, z, z, z, z, z;
v(1268023523408)/v(10340822734521), z, v(1029933939417)/ ...
v(13636558850479), z, z, z, z, z;
v(14463281900351)/v(6315353703477), z, v(66114435211212)/v(5879490589093), ...
v(-54053170152839)/v(4284798021562), z, z, z, z;
v(14090043504691)/v(34967701212078), z, v(15191511035443)/v(11219624916014), ...
v(-18461159152457)/v(12425892160975), v(-281667163811)/ ...
v(9011619295870), z, z, z;
v(19230459214898)/v(13134317526959), z, v(21275331358303)/v(2942455364971), ...
v(-38145345988419)/v(4862620318723), v(-1)/v(8), v(-1)/v(8), z, z;
v(-19977161125411)/v(11928030595625), z, v(-40795976796054)/v(6384907823539), ...
v(177454434618887)/v(12078138498510), v(782672205425)/v(8267701900261), ...
v(-69563011059811)/v(9646580694205), v(7356628210526)/v(4942186776405), z];
q = 5;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK5(4)8L[2]SA-ESDIRK'))
c = [z; v(41)/v(100); v(2935347310677)/v(11292855782101); ...
v(1426016391358)/v(7196633302097); v(92)/v(100); v(24)/v(100); v(3)/v(5); v(1)];
b = [v(-872700587467)/v(9133579230613), z, z, v(22348218063261)/v(9555858737531), ...
v(-1143369518992)/v(8141816002931), v(-39379526789629)/v(19018526304540), ...
v(32727382324388)/v(42900044865799), v(41)/v(200)];
b2 = [v(-975461918565)/v(9796059967033), z, z, v(78070527104295)/v(32432590147079), ...
v(-548382580838)/v(3424219808633), v(-33438840321285)/v(15594753105479), ...
v(3629800801594)/v(4656183773603), v(4035322873751)/v(18575991585200)];
A = [z, z, z, z, z, z, z, z;
v(41)/v(200), v(41)/v(200), z, z, z, z, z, z;
v(41)/v(400), v(-567603406766)/v(11931857230679), v(41)/v(200), z, z, z, z, z;
v(683785636431)/v(9252920307686), z, v(-110385047103)/ ...
v(1367015193373), v(41)/v(200), z, z, z, z;
v(3016520224154)/v(10081342136671), z, v(30586259806659)/v(12414158314087), ...
v(-22760509404356)/v(11113319521817), v(41)/v(200), z, z, z;
v(218866479029)/v(1489978393911), z, v(638256894668)/v(5436446318841), ...
v(-1179710474555)/v(5321154724896), v(-60928119172)/ ...
v(8023461067671), v(41)/v(200), z, z;
v(1020004230633)/v(5715676835656), z, v(25762820946817)/v(25263940353407), ...
v(-2161375909145)/v(9755907335909), v(-211217309593)/v(5846859502534), ...
v(-4269925059573)/v(7827059040749), v(41)/v(200), z;
v(-872700587467)/v(9133579230613), z, z, v(22348218063261)/v(9555858737531), ...
v(-1143369518992)/v(8141816002931), v(-39379526789629)/v(19018526304540), ...
v(32727382324388)/v(42900044865799), v(41)/v(200)];
q = 5;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK5(4)8L[2]SAb-ERK'))
gam = v(2)/v(9);
c = [z; v(4)/v(9); v(6456083330201)/v(8509243623797); v(1632083962415)/v(14158861528103); ...
v(6365430648612)/v(17842476412687); v(18)/v(25); v(191)/v(200); v(1)];
b = [z, z, v(3517720773327)/v(20256071687669), v(4569610470461)/v(17934693873752), ...
v(2819471173109)/v(11655438449929), v(3296210113763)/v(10722700128969), ...
v(-1142099968913)/v(5710983926999), gam];
b2 = [z, z, v(520639020421)/v(8300446712847), v(4550235134915)/v(17827758688493), ...
v(1482366381361)/v(6201654941325), v(5551607622171)/v(13911031047899), ...
v(-5266607656330)/v(36788968843917), v(1074053359553)/v(5740751784926)];
A = [z, z, z, z, z, z, z, z;
v(4)/v(9), z, z, z, z, z, z, z;
v(1)/v(9), v(1183333538310)/v(1827251437969), z, z, z, z, z, z;
v(895379019517)/v(9750411845327), v(477606656805)/v(13473228687314), ...
v(-112564739183)/v(9373365219272), z, z, z, z, z;
v(-4458043123994)/v(13015289567637), v(-2500665203865)/v(9342069639922), ...
v(983347055801)/v(8893519644487), v(2185051477207)/v(2551468980502), ...
z, z, z, z;
v(-167316361917)/v(17121522574472), v(1605541814917)/v(7619724128744), ...
v(991021770328)/v(13052792161721), v(2342280609577)/v(11279663441611), ...
v(3012424348531)/v(12792462456678), z, z, z;
v(6680998715867)/v(14310383562358), v(5029118570809)/v(3897454228471), ...
v(2415062538259)/v(6382199904604), v(-3924368632305)/v(6964820224454), ...
v(-4331110370267)/v(15021686902756), v(-3944303808049)/v(11994238218192), ...
z, z;
v(2193717860234)/v(3570523412979), v(2193717860234)/v(3570523412979), ...
v(5952760925747)/v(18750164281544), v(-4412967128996)/v(6196664114337), ...
v(4151782504231)/v(36106512998704), v(572599549169)/v(6265429158920), ...
v(-457874356192)/v(11306498036315), z];
q = 5;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'ARK5(4)8L[2]SAb-ESDIRK'))
gam = v(2)/v(9);
c = [z; v(4)/v(9); v(6456083330201)/v(8509243623797); v(1632083962415)/v(14158861528103); ...
v(6365430648612)/v(17842476412687); v(18)/v(25); v(191)/v(200); v(1)];
b = [z, z, v(3517720773327)/v(20256071687669), v(4569610470461)/v(17934693873752), ...
v(2819471173109)/v(11655438449929), v(3296210113763)/v(10722700128969), ...
v(-1142099968913)/v(5710983926999), gam];
b2 = [z, z, v(520639020421)/v(8300446712847), v(4550235134915)/v(17827758688493), ...
v(1482366381361)/v(6201654941325), v(5551607622171)/v(13911031047899), ...
v(-5266607656330)/v(36788968843917), v(1074053359553)/v(5740751784926)];
A = [z, z, z, z, z, z, z, z;
gam, gam, z, z, z, z, z, z;
v(2366667076620)/v(8822750406821), v(2366667076620)/v(8822750406821), ...
gam, z, z, z, z, z;
v(-257962897183)/v(4451812247028), v(-257962897183)/v(4451812247028), ...
v(128530224461)/v(14379561246022), gam, z, z, z, z;
v(-486229321650)/v(11227943450093), v(-486229321650)/v(11227943450093), ...
v(-225633144460)/v(6633558740617), v(1741320951451)/v(6824444397158), ...
gam, z, z, z;
v(621307788657)/v(4714163060173), v(621307788657)/v(4714163060173), ...
v(-125196015625)/v(3866852212004), v(940440206406)/v(7593089888465), ...
v(961109811699)/v(6734810228204), gam, z, z;
v(2036305566805)/v(6583108094622), v(2036305566805)/v(6583108094622), ...
v(-3039402635899)/v(4450598839912), v(-1829510709469)/v(31102090912115), ...
v(-286320471013)/v(6931253422520), v(8651533662697)/v(9642993110008), ...
gam, z;
z, z, v(3517720773327)/v(20256071687669), v(4569610470461)/v(17934693873752), ...
v(2819471173109)/v(11655438449929), v(3296210113763)/v(10722700128969), ...
v(-1142099968913)/v(5710983926999), gam];
q = 5;
p = 4;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'Sayfy-Aburub-4-3-ERK'))
A = [z, z, z, z, z, z; ...
v(1)/v(2), z, z, z, z, z;
v(-1), v(2), z, z, z, z;
v(1)/v(6), v(2)/v(3), v(1)/v(6), z, z, z;
v(0.137), v(0.226), v(0.137), z, z, z;
v(0.452), v(-0.904), v(-0.548), z, v(2), z];
b = [v(1)/v(6), v(1)/v(3), v(1)/v(12), z, v(1)/v(3), v(1)/v(12)];
b2 = [v(1)/v(6), v(2)/v(3), v(1)/v(6), z, z, z];
c = [z; v(1)/v(2); v(1); v(1); v(1)/v(2); v(1)];
q = 4;
p = 3;
B = [c, A; q, b; p, b2];
elseif (strcmp(method_name,'Ascher(2,3,3)-ERK'))
gamma = (v(3) + sqrt(v(3)))/v(6);
c = [z; gamma; v(1)-gamma];
b = [z, v(1)/v(2), v(1)/v(2)];
A = [z, z, z;
gamma, z, z;
gamma-v(1), v(2)*(v(1)-gamma), z];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(2,3,3)-SDIRK'))
gamma = (v(3) + sqrt(v(3)))/v(6);
c = [z; gamma; v(1)-gamma];
b = [z, v(1)/v(2), v(1)/v(2)];
A = [z, z, z;
z, gamma, z;
z, v(1)-v(2)*gamma, gamma];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(2,3,2)-ERK'))
gamma = (v(2)-sqrt(v(2)))/v(2);
delta = -v(2)*sqrt(v(2))/v(3);
c = [z; gamma; 1];
b = [z, v(1)-gamma, gamma];
A = [z, z, z;
gamma, z, z;
delta, v(1)-delta, z];
q = 2;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(2,3,2)-SDIRK'))
gamma = (v(2)-sqrt(v(2)))/v(2);
c = [z; gamma; v(1)];
b = [z, v(1)-gamma, gamma];
A = [z, z, z;
z, gamma, z;
z, v(1)-gamma, gamma];
q = 2;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(2,2,2)-ERK'))
gamma = (v(2)-sqrt(v(2)))/v(2);
delta = v(1)-v(1)/(v(2)*gamma);
c = [z; gamma; v(1)];
b = [delta, v(1)-delta, z];
A = [z, z, z;
gamma, z, z;
delta, v(1)-delta, z];
q = 2;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(2,2,2)-SDIRK'))
gamma = (v(2)-sqrt(v(2)))/v(2);
c = [z; gamma; v(1)];
b = [z, v(1)-gamma, gamma];
A = [z, z, z;
z, gamma, z;
z, v(1)-gamma, gamma];
q = 2;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(3,4,3)-ERK'))
gamma = v(0.4358665215084589994160194511935568425293);
gamma2 = gamma^2;
b1 = -v(1.5) * gamma2 + v(4) * gamma - v(0.25);
b2 = v(1.5) * gamma2 - v(5) * gamma + v(1.25);
a42 = v(0.5529291480359398193611887297385924764949);
a43 = v(0.5529291480359398193611887297385924764949);
a31 = (v(1) - v(4.5) * gamma + v(1.5) * gamma2) * a42 ...
+ (v(2.75) - v(10.5) * gamma + v(3.75) * gamma2) * a43 ...
- v(3.5) + v(13) * gamma - v(4.5) * gamma2;
a32 = (-v(1) + v(4.5) * gamma - v(1.5) * gamma2) * a42 ...
+ (-v(2.75) + v(10.5) * gamma - v(3.75) * gamma2) * a43 ...
+ v(4) - v(12.5) * gamma + v(4.5) * gamma2;
a41 = v(1) - a42 - a43;
c = [z; gamma; (v(1)+gamma)/v(2); v(1)];
b = [z, b1, b2, gamma];
A = [z, z, z, z;
gamma, z, z, z;
a31, a32, z, z;
a41, a42, a43, z];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(3,4,3)-SDIRK'))
gamma = v(0.4358665215084589994160194511935568425293);
gamma2 = gamma^2;
b1 = -v(1.5) * gamma2 + v(4) * gamma - v(0.25);
b2 = v(1.5) * gamma2 - v(5) * gamma + v(1.25);
c = [z; gamma; (v(1)+gamma)/v(2); v(1)];
b = [z, b1, b2, gamma];
A = [z, z, z, z;
z, gamma, z, z;
z, (v(1)-gamma)/v(2), gamma, z;
z, b1, b2, gamma];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(4,4,3)-ERK'))
c = [z; v(1)/v(2); v(2)/v(3); v(1)/v(2); v(1)];
b = [v(1)/v(4), v(7)/v(4), v(3)/v(4), v(-7)/v(4), z];
A = [ z, z, z, z, z;
v(1)/v(2), z, z, z, z;
v(11)/v(18), v(1)/v(18), z, z, z;
v(5)/v(6), v(-5)/v(6), v(1)/v(2), z, z;
v(1)/v(4), v(7)/v(4), v(3)/v(4), v(-7)/v(4), z];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Ascher(4,4,3)-SDIRK'))
c = [v(1)/v(2); v(2)/v(3); v(1)/v(2); v(1)];
b = [v(3)/v(2), v(-3)/v(2), v(1)/v(2), v(1)/v(2)];
A = [ v(1)/v(2), z, z, z;
v(1)/v(6), v(1)/v(2), z, z;
v(-1)/v(2), v(1)/v(2), v(1)/v(2), z;
v(3)/v(2), v(-3)/v(2), v(1)/v(2), v(1)/v(2)];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Knoth-Wolke-ERK'))
c = [z; v(1)/v(3); v(3)/v(4)];
b = [v(1)/v(6), v(3)/v(10), v(8)/v(15)];
A = [z, z, z;
v(1)/v(3), z, z;
v(-3)/v(16), v(15)/v(16), z];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'3/8-Rule-ERK'))
c = [z; v(1)/v(3); v(2)/v(3); v(1)];
b = [v(1)/v(8), v(3)/v(8), v(3)/v(8), v(1)/v(8)];
A = [z, z, z, z;
v(1)/v(3), z, z, z;
v(-1)/v(3), v(1), z, z;
v(1), v(-1), v(1), z];
q = 4;
B = [c, A; q, b];
elseif (strcmp(method_name,'Cooper4-ERK'))
c = [z; v(2)/v(3); v(2)/v(3); v(1)];
b = [v(1)/v(4), v(1)/v(4), v(1)/v(2), z];
A = [ z, z, z, z;
v(2)/v(3), z, z, z;
v(1)/v(6), v(1)/v(2), z, z;
v(1)/v(4), v(1)/v(4), v(1)/v(2), z];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Cooper4-ESDIRK'))
c = [z; v(2)/v(3); v(2)/v(3); v(1)];
b = [v(1)/v(4), v(1)/v(4), v(1)/v(2), z];
A = [z, z, z, z;
(v(1)-sqrt(v(3)))/v(6), (v(3)+sqrt(v(3)))/v(6), z, z;
(v(5)+sqrt(v(3)))/v(12), -(v(1)+sqrt(v(3)))/v(4), (v(3)+sqrt(v(3)))/v(6), z;
v(1)/v(4), v(1)/v(4), v(1)/v(2), z];
q = 3;
B = [c, A; q, b];
elseif (strcmp(method_name,'Cooper6-ERK'))
c = [z; v(1)/v(2); v(1)/v(2); v(1)/v(2); v(1); v(1)];
b = [v(1)/v(6), z, z, v(2)/v(3), v(1)/v(6), z];
A = [z, z, z, z, z, z;
v(1)/v(2), z, z, z, z, z;
v(1)/v(4), v(1)/v(4), z, z, z, z;
v(1)/v(4), v(1)/v(4), z, z, z, z;
z, v(-1), z, v(2), z, z;
v(1)/v(6), z, z, v(2)/v(3), v(1)/v(6), z];
q = 5;
B = [c, A; q, b];
elseif (strcmp(method_name,'Cooper6-ESDIRK'))
beta = v(1.0685790213);
c = [z; v(1)/v(2); v(1)/v(2); v(1)/v(2); 1; 1];
b = [v(1)/v(6), z, z, v(2)/v(3), v(1)/v(6), z];
A = [z, z, z, z, z, z;
(v(1)-v(2)*beta)/v(2), beta, z, z, z, z;
v(1)/v(4), (v(1)-v(4)*beta)/v(4), beta, z, z, z;
v(1)/v(4), beta/v(2), (v(1)-v(6)*beta)/v(4), beta, z, z;