######################################################################
# 
#	   A considerably more `efficient' Runge--Kutta (13:8(7)) pair
#
#     These are approximate REAL coefficients 
#  computed using MAPLE with 40 digits for 
#
#     a THIRTEEN-stage conventional pair methods of 
#  orders  p=7  and  p=8, with dominant 
#  stage order = 4, 
#    
#     together with approximate coefficients for 
#  two interpolants of orders 7 and 8 that
#  require 4 and 4 extra stages respectively.
#
#  (Companion files list only the RATIONAL COEFFICIENTS 
#  of the pair, and 40-digit floating point approximations 
#  otherwise.)
#
#	   This procedure is considerably more "efficient" in 
#  the sense that for unrestricted values of coefficients
#  b  and  A, it has a propagating formula which yields  
#  the 2-norm of the local truncation error as
#
#          T_92 ~ .000000011182
#
#     This appears to be the minimum possible for this 
#  2-norm.  (Formulas with slightly different nodes  c_i
#  may have a marginally smaller error norm.)
#
#     Additional stages and interpolating weights allow
#  for the computation of an approximation at any 
#  point of the domain of solution of order up to  p.
#
#  These interpolants have continuous derivatives. 
#
#     Nodes  c[14]=1, c[15] - c[21] were selected in an 
#  attempt to minimize the maximum of the 2-norm of  
#  the local truncation error over the interval  [0,1]   
#  for the interpolant of order 8, and this value is 
#
#          Ti_92 ~ .0000011515
#
#  This 2-norm has two local maximum values on [0,1].
#
#     For the nodes c[14]=1, c[15] - c[17], coefficients 
#  of an interpolant of order 7 on the interval [0,1 are
#  provided.  For this interpolant, the maximum of the 
#  2-norm of the local truncation error on the interval  
#  [0,1]  for the interpolant of order 7,  is  
#
#          Ti_82 ~ .0000063833
#
#  This 2-norm has three local maximum values on [0,1].
#
#     The formulas scanned for this optimal formula are
#  those developed in J.H. Verner, SIAM NA 1978, 772-790,
#  "Explicit Runge--Kutta methods with estimates of the
#  Local Truncation Error". It is possible that the
#  pairs in J.H. Verner, Annals of Num. Math 1 1994,
#  225-244, "Strategies for deriving new explicit Runge--
#  Kutta pairs", or the 11-stage contemporary or 12-stage 
#  FSAL methods derived by Sharp and Verner, SIAM NA 31, 
#  1994, 1169--1190, "Completely imbedded Runge--Kutta 
#  pairs" may yield particular pairs of equivalant or 
#  more efficiency.
# 
#     Instructions for using the interpolants are contained 
#  in J.H. Verner, SIAM NA 30, 1993, 1446-1466, "Differentiable 
#  Interpolants for high-order Runge--Kutta methods".
# 
######################################################################
#
#  NODES
#  -----
c[1] = 0
c[2] = .92662e-1
c[3] = .1312230361754017604780747799402406525075
c[4] = .1968345542631026407171121699103609787612
c[5] = .427173
c[6] = .485972
c[7] = .161915
c[8] = .985468
c[9] = .9626977348604540392115664231947926180050
c[10] = .99626
c[11] = .997947
c[12] = 1
c[13] = 1
#
# ********************************************************
#  COUPLING COEFFICIENTS
#  ---------------------
# for c[1] = 0
#
# for c[2] = .92662e-1
a[2,1] = .92662e-1
#
# for c[3] = .1312230361754017604780747799402406525075
a[3,1] = .3830746548250284242039554953085778876548e-1
a[3,2] = .9291557069289891805767923040938286374199e-1
#
# for c[4] = .1968345542631026407171121699103609787612
a[4,1] = .4920863856577566017927804247759024469030e-1
a[4,2] = 0
a[4,3] = .1476259156973269805378341274327707340709
#
# for c[5] = .427173
a[5,1] = .2743076085702486894953667699331084997155
a[5,2] = 0
a[5,3] = -.9319887203102656329703895298805780048911
a[5,4] = 1.084854111740016943475022759947469505176
#
# for c[6] = .485972
a[6,1] = .6461852970939692178473977038055776659235e-1
a[6,2] = 0
a[6,3] = 0
a[6,4] = .2687629213368923356417418162215129378455
a[6,5] = .1525905489537107425735184133979292955622
#
# for c[7] = .161915
a[7,1] = .7189155819773216802747111989237192757241e-1
a[7,2] = 0
a[7,3] = 0
a[7,4] = .1221265783362549661095644262553368220549
a[7,5] = -.7943550859198561207449556225926957458888e-1
a[7,6] = .4733237205799847793746001611156082496161e-1
#
# for c[8] = .985468
a[8,1] = -6.073603893714328779581044654669389532243
a[8,2] = 0
a[8,3] = 0
a[8,4] = -73.8956
a[8,5] = 11.93985370695273926305714852083567777565
a[8,6] = -3.839251541405054537968455811084326767757
a[8,7] = 72.85406972816664405449235194491803852435
#
# for c[9] = .9626977348604540392115664231947926180050
a[9,1] = -4.868640079323569115532909137641679122100
a[9,2] = 0
a[9,3] = 0
a[9,4] = -59.18572799975646020086112235306397792921
a[9,5] = 9.230819319232425236436363261556755728972
a[9,6] = -2.676847914962525780976208768457285096420
a[9,7] = 58.45720009994685754009028962323039746457
a[9,8] = .5894309723726360055153797570581572198272e-2
#
# for c[10] = .99626
a[10,1] = -6.689861899320853351893486767784362717327
a[10,2] = 0
a[10,3] = 0
a[10,4] = -81.44271004053111286646869388606624375878
a[10,5] = 13.36778825698397107436768361456719138955
a[10,6] = -4.470777638416181156550300441220009175768
a[10,7] = 80.23321392161410397716314034458127569922
a[10,8] = -.1313638336212181564579824084323466067556e-1
a[10,9] = .1174378303219413902745537676538322377075e-1
#
# for c[11] = .997947
a[11,1] = -6.788841955800464091202279324616261399375
a[11,2] = -0
a[11,3] = -0
a[11,4] = -82.65639855934828888595908426400236814314
a[11,5] = 13.59973921874899036529372991555792312537
a[11,6] = -4.574464055350503753023585975406886448931
a[11,7] = 81.41943207216075927716035509139094411649
a[11,8] = -.1416248014826418017596237345824563213708e-1
a[11,9] = .1375441580835227405704608471667945126391e-1
a[11,10] = -.1111656070581006150219154181785069541859e-2
#
# for c[12] = 1.
a[12,1] = -6.910189846402485729483801895952800612193
a[12,2] = 0
a[12,3] = 0
a[12,4] = -84.14495154176748682468399818934446849309
a[12,5] = 13.88512122378983816888937012217322612857
a[12,6] = -4.702458788144493296650978281402083070191
a[12,7] = 82.87411451529241610017736751027256505872
a[12,8] = -.1645498337198780129448623133684652608689e-1
a[12,9] = .1644663972162521365153657249183964172873e-1
a[12,10] = .4275449370796530995842729680318146579263e-2
a[12,11] = -.5902668488222361600852336581750274034670e-2
#
# for c[13] = 1.
a[13,1] = -6.911973921198979615960353484856419992328
a[13,2] = 0
a[13,3] = 0
a[13,4] = -84.16635595878781036984379688648536865215
a[13,5] = 13.88834627565582007275122755367241993143
a[13,6] = -4.703463178409702575934171709526131267400
a[13,7] = 82.89518622207404915885492275044886922215
a[13,8] = -.1020345016228260287805622619295788864042e-1
a[13,9] = .1427900423230391471526298444108039935137e-1
a[13,10] = -.5814993403397981705034981501491752405078e-2
a[13,11] = 0
a[13,12] = 0
#
# ********************************************************
#  High order weights c[14] = 1
#  i.e. This is the propagating stage, and stage 14 as well.
#  -------------------------------------------------------
#       
b[1] = .4625543159712467285354070519930680076661e-1
b[2] = 0
b[3] = 0
b[4] = 0
b[5] = 0
b[6] = .3706666165521011182439275381303388440188
b[7] = .2590440824552746577195309846039127860157
b[8] = -679.9841468175039046601229652340421033215
b[9] = 49.89161129042053159104301060910837813887
b[10] = 10271.2352221373124138878446768864811886
b[11] = -14782.1966063568972805901057042269864062
b[12] = 5141.37795361606373932252398273750538432
b[13] = 0
# 
# ********************************************************
#  Low order weights c[extra] = 1
#  -------------------------------------------------------
# 
bh[1] = .4638504234365210644214797353760063769606e-1
bh[2] = 0
bh[3] = 0
bh[4] = 0
bh[5] = 0
bh[6] = .3725767681581196020016675337652919509451
bh[7] = .2585685495121687311414935155287094016045
bh[8] = -147.4950767589265301611858882107047892762
bh[9] = 23.84362712644587506762535723842314335392
bh[10] = 347.4264166730550802334952839143352280200
bh[11] = 0
bh[12] = 0
bh[13] = -223.4524974005883655795200619648851840879
# 
# ********************************************************
#  
# Largest coefficient in b or A has magnitude `, 14782.20
# 
# *******************************************************
# SUMMARY OF NORMS OF ERRORS: A91, A92, A9inf
# ----------------------------------------------------
#    A_[9, 1] = `, .1093494216e-6
#    A_[9, 2] = `, .1118129111e-7
#    A_[9,oo] = `, .3461291471e-8
#****************************************************
#
#  END OF GENERATION OF A PAIR OF RK METHODS`
#  
# *******************************************************`
# 
#  START OF GENERATION OF STABILITY INTERVALS `
# 
#############################################################`
#
# Stability Boundaries of High Order Method
# -----------------------------------------
# Real Stability Interval is nearly [ -4.825424354, 0]
#
# Stability Boundaries of Low Order Method
# ----------------------------------------
# Real Stability Interval is nearly [ -4.577481299, 0]`
#
#############################################################
#                                                           
#            START OF GENERATION OF INTERPOLANT             
#                                                           
#  ********************************************************
#
#  Coupling coefficients for j from 1 to 13
#  for node c[14] = 1.
#  ----------------------------------------------------
a[14,1] = .4625543159712467285354070519930680076661e-1
a[14,2] = 0
a[14,3] = 0
a[14,4] = 0
a[14,5] = 0
a[14,6] = .3706666165521011182439275381303388440188
a[14,7] = .2590440824552746577195309846039127860158
a[14,8] = -679.9841468175039046601229652340421033215
a[14,9] = 49.89161129042053159104301060910837813887
a[14,10] = 10271.23522213731241388784467688648118861
a[14,11] = -14782.19660635689728059010570422698640618
a[14,12] = 5141.377953616063739322523982737505384322
a[14,13] = 0
#
#  ********************************************************
#
#  FOUR ADDITIONAL STAGES FOR INTERPOLANT OF ORDER  7
#
#  Coupling coefficients for j from 1 to 14
#  for node c[15] = .3110177634953863863927417318829099695921
#  ----------------------------------------------------
a[15,1] = .4911138760562453021745624018512939494564e-1
a[15,2] = 0
a[15,3] = 0
a[15,4] = 0
a[15,5] = 0
a[15,6] = .3117373122649832951079085303758614870942e-1
a[15,7] = .2384923679553289916353395945129932253433
a[15,8] = -114.3180256262527071531290136810612300898
a[15,9] = 3.546408122969840402366032921479574656434
a[15,10] = 2460.597003929216132135942626992844507475
a[15,11] = -3738.924986857493946343920875863790949734
a[15,12] = 1389.104435743812477156452794999320497315
a[15,13] = 0
a[15,14] = -.1259503554386166268241032464519842162533e-1
#
#  ********************************************************
#
#  Coupling coefficients for j from 1 to 15
#  for node c[16] = .2
#  ----------------------------------------------------
a[16,1] = .5179349197485644585674743626479955978608e-1
a[16,2] = 0
a[16,3] = 0
a[16,4] = 0
a[16,5] = 0
a[16,6] = .3491876693023421912147307261172789499006e-1
a[16,7] = .1991958559185454797079748618106548367164
a[16,8] = -83.39793280532507836124705826791773425675
a[16,9] = 3.594107171680368155688005352337368373839
a[16,10] = 1608.071558743333488448763391096377128121
a[16,11] = -2405.179666082838573124443824955693898542
a[16,12] = 876.9134237687558313788496904957089183499
a[16,13] = 0
a[16,14] = -.7325540972670704140488857561266969491257e-2
a[16,15] = -.8007336945700193815591023393769736797741e-1
#
#  ********************************************************
#
#  Coupling coefficients for j from 1 to 16
#  for node c[17] = .6
#  ----------------------------------------------------
a[17,1] = .4436713671328350085533516489655998875304e-1
a[17,2] = 0
a[17,3] = 0
a[17,4] = 0
a[17,5] = 0
a[17,6] = .2341061441705438467404449942880480215796
a[17,7] = .3613912760091486818803425821815508521337
a[17,8] = 148.9884506482496936874751145110012899551
a[17,9] = -9.426031101789146984989508415038462154627
a[17,10] = -2394.447533524453400464298733938611210343
a[17,11] = 3478.192564092670645140476435107181157928
a[17,12] = -1223.319642639490730778566879958817950079
a[17,13] = 0
a[17,14] = .1180395533992265808766615770156321870980e-1
a[17,15] = .1349647037014828275755998285816810134371
a[17,16] = -.1744406911214421152358160333642284015360
#
# --------------------------------------------------------
#  COEFFICIENTS FOR INTERPOLANT  bi47 WITH 17 STAGES
#     for i from 1 to 17, bi47[i] = SUM_j=1^7 bi47[i,j]u^j
# 
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[1]
bi47[1,1] = 1 u
bi47[1,2] = -8.2121107129100617780076230844008138621 u^2
bi47[1,3] = 35.0263532076960479147409203407753760659 u^3
bi47[1,4] = -83.002133398763376272419522092021866843 u^4
bi47[1,5] = 108.640900005722667687740587007429270181 u^5
bi47[1,6] = -72.996471060209216617173388743495227181 u^6
bi47[1,7] = 19.5897173900610637379725672769125684398 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[2]
bi47[2,1] = 0 u
bi47[2,2] = 0 u^2
bi47[2,3] = 0 u^3
bi47[2,4] = 0 u^4
bi47[2,5] = 0 u^5
bi47[2,6] = 0 u^6
bi47[2,7] = 0 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[3]
bi47[3,1] = 0 u
bi47[3,2] = 0 u^2
bi47[3,3] = 0 u^3
bi47[3,4] = 0 u^4
bi47[3,5] = 0 u^5
bi47[3,6] = 0 u^6
bi47[3,7] = 0 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[4]
bi47[4,1] = 0 u
bi47[4,2] = 0 u^2
bi47[4,3] = 0 u^3
bi47[4,4] = 0 u^4
bi47[4,5] = 0 u^5
bi47[4,6] = 0 u^6
bi47[4,7] = 0 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[5]
bi47[5,1] = 0 u
bi47[5,2] = 0 u^2
bi47[5,3] = 0 u^3
bi47[5,4] = 0 u^4
bi47[5,5] = 0 u^5
bi47[5,6] = 0 u^6
bi47[5,7] = 0 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[6]
bi47[6,1] = 0 u
bi47[6,2] = 24.4950463267912465724825907428435993783 u^2
bi47[6,3] = -197.69843456849391580459731560756280877 u^3
bi47[6,4] = 648.816707921001459445153728378767199245 u^4
bi47[6,5] = -1029.9136145419018933308145029460724723 u^5
bi47[6,6] = 784.290278276683546509851622238788957049 u^6
bi47[6,7] = -229.61931679752834227383219526863413572 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[7]
bi47[7,1] = 0 u
bi47[7,2] = 31.5691895350570444845757378427451077040 u^2
bi47[7,3] = -253.50186515241579700159565129363509060 u^3
bi47[7,4] = 825.652427431980811818397517121702665576 u^4
bi47[7,5] = -1296.6226332937703261131065731647428702 u^5
bi47[7,6] = 974.262805803163104958561227817419957097 u^6
bi47[7,7] = -281.10088024155956348911272733888585677 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[8]
bi47[8,1] = 0 u
bi47[8,2] = 13532.4606248634951941828500394940369756 u^2
bi47[8,3] = -103993.69121081234474846999199837534298 u^3
bi47[8,4] = 315790.051792644179878356624335835227454 u^4
bi47[8,5] = -444650.94086356341512652926823805092789 u^5
bi47[8,6] = 285484.299040403666308333520507989065779 u^6
bi47[8,7] = -66842.163530353085410533857612126101428 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[9]
bi47[9,1] = 0 u
bi47[9,2] = -797.33739377481263091301915333452072383 u^2
bi47[9,3] = 6069.29990633718694913905839444165396031 u^3
bi47[9,4] = -18132.728486459505277117555298277484960 u^4
bi47[9,5] = 24818.0833744339912327022483860123469122 u^5
bi47[9,6] = -15179.252666931207554905667614022492519 u^6
bi47[9,7] = 3271.82687768476781268597829578960570891 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[10]
bi47[10,1] = 0 u
bi47[10,2] = -223490.90589817548449455805622984284737 u^2
bi47[10,3] = 1723137.28129041203797382781061794213126 u^3
bi47[10,4] = -5261550.3608712234346584362083446548535 u^4
bi47[10,5] = 7478225.62867121190764968571321428434311 u^5
bi47[10,6] = -4874996.1238445630538493688499789530452 u^6
bi47[10,7] = 1168945.71587447533979273743539811075305 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[11]
bi47[11,1] = 0 u
bi47[11,2] = 325937.354598091149456157360053808939544 u^2
bi47[11,3] = -2514175.0453666388313255353675449313401 u^3
bi47[11,4] = 7682914.24089272425146206737593468653339 u^4
bi47[11,5] = -10933909.132147210045971651656776945524 u^5
bi47[11,6] = 7142613.57384784863461432511573092320769 u^6
bi47[11,7] = -1718163.1884311720555159529331017688020 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[12]
bi47[12,1] = 0 u
bi47[12,2] = -115180.89354308167685332546785529115382 u^2
bi47[12,3] = 888954.826743313341003390564643378373806 u^3
bi47[12,4] = -2718988.1080155526063345244849358468516 u^4
bi47[12,5] = 3875443.74497592817378873641380949115203 u^5
bi47[12,6] = -2537848.3594877310621455766242293369375 u^6
bi47[12,7] = 612760.167280739894280622122550342922539 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[13]
bi47[13,1] = 0 u
bi47[13,2] = 0 u^2
bi47[13,3] = 0 u^3
bi47[13,4] = 0 u^4
bi47[13,5] = 0 u^5
bi47[13,6] = 0 u^6
bi47[13,7] = 0 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[14]
bi47[14,1] = 0 u
bi47[14,2] = .817197452808953918419304790251977356783 u^2
bi47[14,3] = -5.9298894939332396539035637575223058161 u^3
bi47[14,4] = 16.1704714222592133498393146952906014079 u^4
bi47[14,5] = -18.188640730913089542675554890384959858 u^5
bi47[14,6] = 6.49943790673672805935089677372745197447 u^6
bi47[14,7] = .631423443041433868969602388637234935741 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[15]
bi47[15,1] = 0 u
bi47[15,2] = -19.589917077606922465754841240579899704 u^2
bi47[15,3] = 128.442052186315593905263796845808034639 u^3
bi47[15,4] = -321.87562867926418328688632952811221637 u^4
bi47[15,5] = 394.572135120649164000513439500529661300 u^5
bi47[15,6] = -239.33600756073354143264887159705531349 u^6
bi47[15,7] = 57.7873660106398892795128060194097336408 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[16]
bi47[16,1] = 0 u
bi47[16,2] = -15.352746937526659326789451813442545859 u^2
bi47[16,3] = 180.168624955189542174072826639169022411 u^3
bi47[16,4] = -701.64764142189820781774998997308253816 u^4
bi47[16,5] = 1211.46882327266085511303356041942685289 u^5
bi47[16,6] = -961.90548741275195883516119840906945162 u^6
bi47[16,7] = 287.268427544326428692594253136998660353 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[17]
bi47[17,1] = 0 u
bi47[17,2] = -14.405046509284272948592572071872018337 u^2
bi47[17,3] = 120.821796254251916113944874377491922262 u^3
bi47[17,4] = -417.20951540820078758208641034511448025 u^4
bi47[17,5] = 702.659019366941049241858649282425195305 u^5
bi47[17,6] = -564.95144497986603545027470468011454720 u^6
bi47[17,7] = 173.085191276158130625150163437183928232 u^7
#  ********************************************************
#
#  FOUR ADDITIONAL STAGES FOR INTERPOLANT OF ORDER  8
#
#  Coupling coefficients for j from 1 to 17
#  for node c[18] = .251
#  ----------------------------------------------------
a[18,1] = .4927309335160545018837186692162194114006e-1
a[18,2] = 0
a[18,3] = 0
a[18,4] = 0
a[18,5] = 0
a[18,6] = .1478473900086014712887940970539529915367
a[18,7] = .1915391503780640342241981813756999876738
a[18,8] = 85.69949519051240679824481172082685765978
a[18,9] = -5.094095153569584154270332765058351241762
a[18,10] = -1410.985179005200593252911690612297856885
a[18,11] = 2056.873083845176325396897975581141270029
a[18,12] = -726.4906359530084743440935010911966625383
a[18,13] = 0
a[18,14] = .5440493103670101179960418796640615006313e-2
a[18,15] = -.1437912346504880100428763097368602714809
a[18,16] = .8130926198406332202245160436986436093968e-1
a[18,17] = -.8328707808559681272816269219617664899487e-1
#
#  ********************************************************
#
#  Coupling coefficients for j from 1 to 18
#  for node c[19] = .451
#  ----------------------------------------------------
a[19,1] = .4696791108973669580037521395572005624943e-1
a[19,2] = 0
a[19,3] = 0
a[19,4] = 0
a[19,5] = 0
a[19,6] = .2010207606761097437232392903607688964540
a[19,7] = .2636820858834097192664348219821625228752
a[19,8] = 129.6946428658473021706281203179680341347
a[19,9] = -7.848674084672579804338628962846012175606
a[19,10] = -2121.729757286215698398545896705499941278
a[19,11] = 3090.157418699412077487400584308334376641
a[19,12] = -1090.274673114349457503609053352481192228
a[19,13] = 0
a[19,14] = .8963759217746587577580254051724170346208e-2
a[19,15] = .4875201082359089303480235638217668769362e-1
a[19,16] = -.2352913684548179696659019693524338623585e-1
a[19,17] = -.9381447086675579397096734527257404140647e-1
a[19,18] = 0
#
#  ********************************************************
#
#  Coupling coefficients for j from 1 to 19
#  for node c[20] = .705
#  ----------------------------------------------------
a[20,1] = .4619987481300149103923582280519782866590e-1
a[20,2] = 0
a[20,3] = 0
a[20,4] = 0
a[20,5] = 0
a[20,6] = .2322836783017554180760864761340094394641
a[20,7] = .2982051978226913141175018152700660351379
a[20,8] = 123.6261700088200670668986465184100244666
a[20,9] = -7.231890372340911603936034207018277000416
a[20,10] = -2046.801418638051394033916845196316375171
a[20,11] = 2986.087035469917759873466009280884537880
a[20,12] = -1055.671796398942928677813090910742298972
a[20,13] = 0
a[20,14] = .7930343978313866522846570067876462025829e-2
a[20,15] = .8980932823961332375619194670691924357732e-1
a[20,16] = -.8017153994764179414644520644800061934373e-1
a[20,17] = .1026430473896737559358970902463204065417
a[20,18] = 0
a[20,19] = 0
#
#  ********************************************************
#
#  Coupling coefficients for j from 1 to 20
#  for node c[21] = .8795
#  ----------------------------------------------------
a[21,1] = .5329137345256786599202448371739175627057e-1
a[21,2] = 0
a[21,3] = 0
a[21,4] = 0
a[21,5] = 0
a[21,6] = .2043185802985482892225157926395741314696
a[21,7] = .1358728897801712829783843582299538835128
a[21,8] = -402.8079844608128209694219700870509710185
a[21,9] = 29.46110568417652997397169152745513583508
a[21,10] = 6093.592821606031634755087198810590639032
a[21,11] = -8771.855718519913765428348513348127981921
a[21,12] = 3051.798314820693282143366860169701133691
a[21,13] = 0
a[21,14] = -.3308969275605729485812518877458826604068e-1
a[21,15] = .2805382986735395987509175854573185663385e-1
a[21,16] = .1335953634219562765350679978518628190610
a[21,17] = .1689185257605991455997737252221182006005
a[21,18] = 0
a[21,19] = 0
a[21,20] = 0
#
# --------------------------------------------------------
#  COEFFICIENTS FOR INTERPOLANT  bi48 WITH 21 STAGES
#     for i from 1 to 21, bi48[i] = SUM_j=1^8 bi48[i,j]u^j
# 
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[1]
bi48[1,1] = 1 u
bi48[1,2] = -8.5605623175055209658932662325458643485 u^2
bi48[1,3] = 37.9112031979064521067325710639394768180 u^3
bi48[1,4] = -95.909979703870494595779288611712455045 u^4
bi48[1,5] = 143.955798956065258867896521876889926389 u^5
bi48[1,6] = -126.86452588300103629551470103061663374 u^6
bi48[1,7] = 60.6789750815661370149820585945855649158 u^7
bi48[1,8] = -12.164653899563671459570354955340708181 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[2]
bi48[2,1] = 0 u
bi48[2,2] = 0 u^2
bi48[2,3] = 0 u^3
bi48[2,4] = 0 u^4
bi48[2,5] = 0 u^5
bi48[2,6] = 0 u^6
bi48[2,7] = 0 u^7
bi48[2,8] = 0 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[3]
bi48[3,1] = 0 u
bi48[3,2] = 0 u^2
bi48[3,3] = 0 u^3
bi48[3,4] = 0 u^4
bi48[3,5] = 0 u^5
bi48[3,6] = 0 u^6
bi48[3,7] = 0 u^7
bi48[3,8] = 0 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[4]
bi48[4,1] = 0 u
bi48[4,2] = 0 u^2
bi48[4,3] = 0 u^3
bi48[4,4] = 0 u^4
bi48[4,5] = 0 u^5
bi48[4,6] = 0 u^6
bi48[4,7] = 0 u^7
bi48[4,8] = 0 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[5]
bi48[5,1] = 0 u
bi48[5,2] = 0 u^2
bi48[5,3] = 0 u^3
bi48[5,4] = 0 u^4
bi48[5,5] = 0 u^5
bi48[5,6] = 0 u^6
bi48[5,7] = 0 u^7
bi48[5,8] = 0 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[6]
bi48[6,1] = 0 u
bi48[6,2] = 31.5810587696607075581771118102632934366 u^2
bi48[6,3] = -180.51780064979361013403188210694629343 u^3
bi48[6,4] = 371.242868417558534142479116069288941062 u^4
bi48[6,5] = -219.95944326161329224006075923515928405 u^5
bi48[6,6] = -240.09383761577352115218389000974272013 u^6
bi48[6,7] = 371.162514909573396721681753426001945511 u^7
bi48[6,8] = -133.04469395306011377781752241557554353 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[7]
bi48[7,1] = 0 u
bi48[7,2] = 22.0707396529273170902565380937085992160 u^2
bi48[7,3] = -126.15667542749176945651636068917208873 u^3
bi48[7,4] = 259.446801850776449695703467880767328985 u^4
bi48[7,5] = -153.72086293362928871737294227060509716 u^5
bi48[7,6] = -167.79198635926149051024032975322192051 u^6
bi48[7,7] = 259.390645995842190392584437555629967213 u^7
bi48[7,8] = -92.979618696708133836695279832502876224 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[8]
bi48[8,1] = 0 u
bi48[8,2] = -57935.131852001300044517506965268893855 u^2
bi48[8,3] = 331158.073532549845930244284192463684446 u^3
bi48[8,4] = -681041.27501731301381922979848103699048 u^4
bi48[8,5] = 403513.366679671772407804393320416504379 u^5
bi48[8,6] = 440449.708813618399558265249503060897132 u^6
bi48[8,7] = -680893.86732228152168442184793580074137 u^7
bi48[8,8] = 244069.141018938313747195103400931497656 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[9]
bi48[9,1] = 0 u
bi48[9,2] = 4250.80068696816949617870093798105836378 u^2
bi48[9,3] = -24297.639816601049229688784493269857272 u^3
bi48[9,4] = 49969.1746122654055503610926693635873240 u^4
bi48[9,5] = -29606.472643654408293295497141135490609 u^5
bi48[9,6] = -32316.555860831862195471937517525829865 u^6
bi48[9,7] = 49958.3590550849204934860567043395854712 u^7
bi48[9,8] = -17907.774421940755289978588149143945032 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[10]
bi48[10,1] = 0 u
bi48[10,2] = 875116.529793378266310114348923755218798 u^2
bi48[10,3] = -5002179.0726770314020262099591199499396 u^3
bi48[10,4] = 10287203.2597026176488674982398922530359 u^4
bi48[10,5] = -6095113.7226376951838009344144177116130 u^5
bi48[10,6] = -6653041.2273871943630134445764091719260 u^6
bi48[10,7] = 10284976.6502789895935411658959747767822 u^7
bi48[10,8] = -3686691.1818509272474643016901670650771 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[11]
bi48[11,1] = 0 u
bi48[11,2] = -1259453.6408822165887300591288176518290 u^2
bi48[11,3] = 7199055.70394767708463918957841581207085 u^3
bi48[11,4] = -14805177.549310992829226080633807752480 u^4
bi48[11,5] = 8771989.68162524567177693630500314357236 u^5
bi48[11,6] = 9574949.97694842506337042293792042888252 u^6
bi48[11,7] = -14801973.048824556894227646220426243831 u^7
bi48[11,8] = 5305826.67989006159511664705600803662879 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[12]
bi48[12,1] = 0 u
bi48[12,2] = 438049.050169490963392094514591484781835 u^2
bi48[12,3] = -2503894.8722420290838814530818150934384 u^3
bi48[12,4] = 5149370.92763771115508209117872425727904 u^4
bi48[12,5] = -3050975.1398557976491765724888723511154 u^5
bi48[12,6] = -3330251.7906769824569695847641195878263 u^6
bi48[12,7] = 5148256.37419264139010980079334765842715 u^7
bi48[12,8] = -1845413.1712714182548170536278736306024 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[13]
bi48[13,1] = 0 u
bi48[13,2] = 0 u^2
bi48[13,3] = 0 u^3
bi48[13,4] = 0 u^4
bi48[13,5] = 0 u^5
bi48[13,6] = 0 u^6
bi48[13,7] = 0 u^7
bi48[13,8] = 0 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[14]
bi48[14,1] = 0 u
bi48[14,2] = -2.5044314708158259525970526018497418718 u^2
bi48[14,3] = 10.0205159344470025003414404359191756544 u^3
bi48[14,4] = 3.18831479166784493056254732702021885937 u^4
bi48[14,5] = -81.289190874560959096771678638747875917 u^5
bi48[14,6] = 166.501641547533286172953271304807575888 u^6
bi48[14,7] = -137.96496048539513156396658256994982650 u^7
bi48[14,8] = 42.0481105571237830094780547428004738900 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[15]
bi48[15,1] = 0 u
bi48[15,2] = -8.7817140784003458905005822549334266607 u^2
bi48[15,3] = 95.9268368131788169073900471365173972176 u^3
bi48[15,4] = -437.86840188220595540320971563321075323 u^4
bi48[15,5] = 1022.89375781768068832881313351214157318 u^5
bi48[15,6] = -1279.3067716511304733780880292529608441 u^6
bi48[15,7] = 814.461977782550694188628778349353555570 u^7
bi48[15,8] = -207.32568480167342475303363185690750192 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[16]
bi48[16,1] = 0 u
bi48[16,2] = 23.6836108877825281375583068252587157220 u^2
bi48[16,3] = -258.70733851004426226367965660428074245 u^3
bi48[16,4] = 1180.89757394178065280828078207675885064 u^4
bi48[16,5] = -2758.6661924329604875818863811210508301 u^5
bi48[16,6] = 3450.19246987482707402605981340827163916 u^6
bi48[16,7] = -2196.5416309943691540465351696905747725 u^7
bi48[16,8] = 559.141507232983648920202305105617139584 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[17]
bi48[17,1] = 0 u
bi48[17,2] = -40.604955901261884397106887787380774997 u^2
bi48[17,3] = 443.547232780800480314888429151835197355 u^3
bi48[17,4] = -2024.6192247039862155424716081891577010 u^4
bi48[17,5] = 4729.66388532531277008655288790061707692 u^5
bi48[17,6] = -5915.2683158801093170198557614876293997 u^6
bi48[17,7] = 3765.91544610379409075813847299514703593 u^7
bi48[17,8] = -958.63406772454992420014553258343143446 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[18]
bi48[18,1] = 0 u
bi48[18,2] = -46.254485925330558373778425312028398395 u^2
bi48[18,3] = 439.193846604121715965570227408978645829 u^3
bi48[18,4] = -1719.4382536811392825503568581454131645 u^4
bi48[18,5] = 3520.66706309792621758962156493780680395 u^5
bi48[18,6] = -3962.6349891406635221536779931291783991 u^6
bi48[18,7] = 2322.57948624348029215473813885386620576 u^7
bi48[18,8] = -554.11266719839486263211665461403169347 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[19]
bi48[19,1] = 0 u
bi48[19,2] = 14.2377522101039340296994581596105349702 u^2
bi48[19,3] = -276.29327240896391068084971918357323920 u^3
bi48[19,4] = 1558.15194374205109397668936584054170231 u^4
bi48[19,5] = -3984.1343813182518749896481819350673606 u^5
bi48[19,6] = 5195.12783039572519183101943432863382395 u^6
bi48[19,7] = -3374.4204430207031853737999392540293889 u^7
bi48[19,8] = 867.330570400038751206889582043883927576 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[20]
bi48[20,1] = 0 u
bi48[20,2] = 6.17796276921059154259957639776474159149 u^2
bi48[20,3] = -157.49107011755248272311779168004916065 u^3
bi48[20,4] = 978.638089062525412616594131240142095768 u^4
bi48[20,5] = -2692.2995573259959182752122590014229358 u^5
bi48[20,6] = 3734.57931471599315526725245349565736762 u^6
bi48[20,7] = -2556.4247397316013418152531549339569570 u^7
bi48[20,8] = 686.820000627420583387137044481864848563 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[21]
bi48[21,1] = 0 u
bi48[21,2] = -18.652890215881366589343447398203795252 u^2
bi48[21,3] = 130.373777217996135381235515104311683323 u^3
bi48[21,4] = -398.26735612352449471857093693945638715 u^4
bi48[21,5] = 665.175955179823972089770201612740391829 u^5
bi48[21,6] = -644.55266703892009697463364507821790767 u^6
bi48[21,7] = 346.695348237773779184123541943704543394 u^7
bi48[21,8] = -80.772167257267928372581229244878528459 u^8
#
#********************************************************
# Norms of low order INTERPOLANT coefficients on  [0,2]
#      u           Max norm        2-norm  
# -------------------------------------------------
.1000000000,  .1082908299e-5, .2386660479e-5
.2000000000,  .1454414983e-5, .3227290092e-5
.3000000000,  .6673508830e-6, .1469529334e-5
.4000000000, -.2668522183e-6, .7356949545e-6
.5000000000, -.1466103135e-5, .3463625886e-5
.6000000000, -.2339540235e-5, .5343992338e-5
.7000000000, -.1003363510e-5, .2620052662e-5
.8000000000,  .2105506049e-5, .5051292258e-5
.9000000000,  .2796419219e-5, .6383301364e-5
1.000000000, -.2142651922e-50,.5080264237e-50
1.100000000,  .2084080734e-4, .4704545419e-4
1.200000000,  .1769473087e-3, .3986684275e-3
1.300000000,  .7681888172e-3, .1728436815e-2
1.400000000,  .2453179043e-2, .5514087323e-2
1.500000000,  .6509035914e-2, .1461879170e-1
1.600000000,  .1520603731e-1, .3412921019e-1
1.700000000,  .3232596237e-1, .7251464835e-1
1.800000000,  .6385605801e-1, .1431786761
1.900000000,  .1188937204,    .2664816099
2.000000000,  .2108001307,    .4723193003
#
# ********************************************************
# Norms of high order INTERPOLANT error coefficients on  [0,2]
#     u           Max norm        2-norm  
# -------------------------------------------------
.1000000000, -.1182206619e-6, .3751725459e-6
.2000000000,  .1859520387e-6, .4406682002e-6
.3000000000,  .2972578621e-6, .6805183823e-6
.4000000000,  .3169617153e-6, .7556756372e-6
.5000000000,  .2701931312e-6, .7019452286e-6
.6000000000, -.2801489924e-6, .8098292066e-6
.7000000000, -.4698700869e-6, .1151527544e-5
.8000000000, -.4165260700e-6, .9898915913e-6
.9000000000, -.1304922957e-6, .4055200732e-6
1.000000000, -.3461291471e-8, .1118129111e-7
1.100000000, .1588302971e-5,  .5045390959e-5
1.200000000, .1895542621e-4,  .6098488543e-4
1.300000000, .1082098842e-3,  .3484317105e-3
1.400000000, .4331798682e-3,  .1390702407e-2
1.500000000, .1391316301e-2,  .4449863077e-2
1.600000000, .3832468405e-2,  .1221252062e-1
1.700000000, .9413030013e-2,  .2989709188e-1
1.800000000, .2113902554e-1,  .6694847104e-1
1.900000000, .4416403554e-1,  .1395240625
2.000000000, .8692145286e-1,  .2740170658
#   ********************************************************