######################################################################
# 
#	   A Runge--Kutta (13:8(7)) pair with minimal interpolation coefficients
#
#     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.)
#
#     The derivation of other Runge-Kutta pairs included 
#  on this webpage was motivated primarily to minimize the
#  2-norm of the local truncation error coefficients of 
#  the propagating method.  Some users of software to 
#  solve ordinary differential equations numerically will
#  be motivated to obtain approximations that are 
#  continuous over an independent variable interval and
#  minimize the maximum error of components over this 
#  interval.
#
#     The derivation of this procedure was motivated to 
#  keep the maximum interpolation coefficient less than
#  5,000 (those for other procedures have coefficients
#  more than 1,000,000), and for this constraint attempts
#  to minimize the 2-norm of the local truncation error
#  coefficients of the high order propagating formula.  
#  This 2-norm for this propagating formula is 
#
#        T_92 ~ .00000046975
#
#  While this is a factor of 40 times larger than that of 
#  the "considerably more efficient" procedure above, this
#  procedure may be expected to give more accurate inter-
#  polation approximations when high precision is required.
#  
#  Values of the 2-norm of the interpolant coefficients are
#  provided for the interval [0,2].  The maximum of this 
#  on [0,1] is 
#
#        Ti_92 ~ .000002415
#
#  and has only one local maximum value near  x= 0.4 .
#
#     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  
#
#          T_82 ~ .00001021
#
#  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 accuracy.
# 
#     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] = .354e-1
c[3] = .1056755182072829131652661064425770308123
c[4] = .1585132773109243697478991596638655462185
c[5] = .3868
c[6] = .4612
c[7] = .154
c[8] = .8808
c[9] = .9212245793872493661390228417549480165118
c[10] = .854
c[11] = .9449
c[12] = 1
c[13] = 1
c[14] = 1
c[15] = .3110177634953863863927417318829099695921
c[16] = .2
c[17] = .6
c[18] = .25
c[19] = .45
c[20] = .7
c[21] = .938
# 
***************************************************
# COUPLING COEFFICIENTS 
# --------------------- 
# for c[1] = 0
# # 
# for c[2] = .354e-1
# a[2,1] = .354e-1
# 
# for c[3] = .1056755182072829131652661064425770308123
# a[3,1] = -.5205492174155582250624853907443114611525e-1
a[3,2] = .1577304399488387356715146455170081769276
# 
# for c[4] = .1585132773109243697478991596638655462185
# a[4,1] = .3962831932773109243697478991596638655462e-1
a[4,2] = 0
a[4,3] = .1188849579831932773109243697478991596639
# 
# for c[5] = .3868
# a[5,1] = .3585664007048663722700408046702786531405
a[5,2] = 0
a[5,3] = -1.331088237627357372619884128428629872187
a[5,4] = 1.359321836922491000349843323758351219046
# 
# for c[6] = .4612
# a[6,1] = .4863397766051500662868971073965726345101e-1
a[6,2] = 0
a[6,3] = 0
a[6,4] = .2331621253041885501826067015246258166113
a[6,5] = .1794038970352964431887035877357169199377
# 
# for c[7] = .154
# a[7,1] = .6118497640210165277295953703456628794025e-1
a[7,2] = 0
a[7,3] = 0
a[7,4] = .1117131892666270495569792886237203510882
a[7,5] = -.3851895506598177670962113729853246953415e-1
a[7,6] = .1962078939725307437968231164024583050565e-1
# 
# for c[8] = .8808
# a[8,1] = -1.081633649266948801474301557132298129782
a[8,2] = 0
a[8,3] = 0
a[8,4] = -37.9
a[8,5] = -4.430899713554547935491657281685281870607
a[8,6] = 4.193542322306062035809874761192916371001
a[8,7] = 40.09979104051543470115608407762466362939
# 
# for c[9] = .9212245793872493661390228417549480165118
# a[9,1] = -1.683127213384659248195808464876869760043
a[9,2] = 0
a[9,3] = 0
a[9,4] = -58.99070001585455184226097607516379940003
a[9,5] = -6.364326940947622278441119463898305849240
a[9,6] = 5.814157368501575315595104687108858366436
a[9,7] = 62.16585327026794211217836972704932537691
a[9,8] = -.2063188919543469273654756846426071752123e-1
# 
# for c[10] = .854
# a[10,1] = -.7022332885769292791834206716602640863527
a[10,2] = 0
a[10,3] = 0
a[10,4] = -24.85139431715929854127298918097910633617
a[10,5] = -3.109457173853610471076636687844638482132
a[10,6] = 3.076033082857955922300498696129316036275
a[10,7] = 26.41757799084978771764283331388232836399
a[10,8] = .4764500846923696830646913489958681663911e-1
a[10,9] = -.2417130258714231671675460442722231225571e-1
# 
# for c[11] = .9449
# a[11,1] = -2.032504122713356499658033433423584836099
a[11,2] = 0
a[11,3] = 0
a[11,4] = -71.53151168815980723718752613229449284075
a[11,5] = -7.360175338036837759654544060248266963300
a[11,6] = 6.633835915929500034530518471949348711163
a[11,7] = 75.25129222714477720494199185149124615765
a[11,8] = -.1442949460156433049737080726263499667783
a[11,9] = .2033102508843710351230289071991541124719e-1
a[11,10] = .1079269267629304584889984844321843268650
# 
# for c[12] = 1
# a[12,1] = -3.611830767748946328739604332263986130391
a[12,2] = 0
a[12,3] = 0
a[12,4] = -128.3625660872269527131954724302863305148
a[12,5] = -11.91080830296098668168259934297173298604
a[12,6] = 10.41112839665637868088255929905137205105
a[12,7] = 134.5428902265549061101064842271540581842
a[12,8] = -1.211232187518601128197470357376701936564
a[12,9] = 1.160339439782001454374873116050415780647
a[12,10] = .5730741573967656367751349315066869169336
a[12,11] = -.5909948749345650303239051108637813649843
# 
# for c[13] = 1
# a[13,1] = -8.755719297038927955710129975421675404546
a[13,2] = 0
a[13,3] = 0
a[13,4] = -306.8325827611981076544761913698228773274
a[13,5] = -30.39817349138844407385058721538369714436
a[13,6] = 26.41593306465960318135672131697853231696
a[13,7] = 321.5895431471400678483369212810510620078
a[13,8] = 12.47261312047573686998422931498643976071
a[13,9] = -4.117625681166761466217581315794120575015
a[13,10] = -9.373988101483166749423382036593663634146
a[13,11] = 0
a[13,12] = 0
# 
# ********************************************************
# High order weights for c[14] = 1  
# i.e. This is the propagating stage, and stage 14 as well.
# ---------------------------------------------------------
b[1] = .4402358624577092145118715368145467710253e-1
b[2] = 0
b[3] = 0
b[4] = 0
b[5] = 0
b[6] = .3500087685916403979338531730927097088585
b[7] = .2461954575470813104475743797523742637935
b[8] = .2384674903100136612466301902940424332946
b[9] = -2.965427511436218238644852740056817413947
b[10] = .9001214140563761377479620941434090620185
b[11] = 2.411435215336631238363618312661595077035
b[12] = -.2248244206512954285459725635687678081555
b[13] = 0

# ********************************************************
# Low order weights c[extra] = 1 
# ---------------------------------------------------------
# bh[1] = .4388111285392309852461670621249633190603e-1
bh[2] = 0
bh[3] = 0
bh[4] = 0
bh[5] = 0
bh[6] = .3476421506877805618826884417439372521121
bh[7] = .2467316997475012279789241112902479938632
bh[8] = -2.741338801200780848667184890840849275847
bh[9] = 1.055249868079038465000633088075187166422
bh[10] = 2.080233638786108898631684139149911277938
bh[11] = 0
bh[12] = 0
bh[13] = -.3239966895357140335136159563093074639440e-1
# *******************************************************`
# 
# Largest coefficient in b or A has magnitude `, 321.5895
# ****************************************************`
# SUMMARY OF NORMS OF ERRORS: A91, A92, A9inf`
# ----------------------------------------------------`
#    A_[9, 1] = `, .4525191997e-5
#    A_[9, 2] = `, .4697540876e-6
#    A_[9,oo] = `, .1458047359e-6
# **************************************************** 
#  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 [ -5.785937206 ,0] 
# 
# Stability Boundaries of Low Order Method`
# ----------------------------------------`
# Real Stability Interval is nearly [ -6.105302877 ,0] 
#
#############################################################
#                                                           #
#            START OF GENERATION OF INTERPOLANT             #
#                                                           #
#############################################################
#
#  Coupling coefficients 
#  for node c[14] = 1
#  ----------------------------------------------------
a[14,1] = .4402358624577092145118715368145467710253e-1
a[14,2] = 0
a[14,3] = 0
a[14,4] = 0
a[14,5] = 0
a[14,6] = .3500087685916403979338531730927097088585
a[14,7] = .2461954575470813104475743797523742637935
a[14,8] = .2384674903100136612466301902940424332946
a[14,9] = -2.965427511436218238644852740056817413947
a[14,10] = .9001214140563761377479620941434090620185
a[14,11] = 2.411435215336631238363618312661595077035
a[14,12] = -.2248244206512954285459725635687678081555
a[14,13] = 0
# 
#  ********************************************************
#
#  Coupling coefficients 
#  for node c[15] = .3110177634953863863927417318829099695921
#  ----------------------------------------------------
a[15,1] = .4589100258691896762802118945662661781054e-1
a[15,2] = 0
a[15,3] = 0
a[15,4] = 0
a[15,5] = 0
a[15,6] = .4857081125622906117852713875683679564725e-1
a[15,7] = .2325420232049980029785187210102797336383
a[15,8] = 3.136304394664646557235215179687594745951
a[15,9] = -3.124357779292520685960312730734242961386
a[15,10] = -1.500181615000992013536841329736768872643
a[15,11] = 1.545587327077198433063434144327053892485
a[15,12] = -.6074336545723027351141025623927156028582e-1
a[15,13] = 0
a[15,14] = -.1259503554386166268241032464519842162533e-1
# 
#  ********************************************************
#
#  Coupling coefficients 
#  for node c[16] = .2
#  ----------------------------------------------------
a[16,1] = .4899332642240945529498066199109559545621e-1
a[16,2] = 0
a[16,3] = 0
a[16,4] = 0
a[16,5] = 0
a[16,6] = .4623614713014409969475827002201146576476e-1
a[16,7] = .1945878530338981497876735219105918606966
a[16,8] = 1.147700366663052225298265195354831468095
a[16,9] = -1.211058791990097122762589957755078877110
a[16,10] = -.5526120682745402827425718575338405666223
a[16,11] = .6518981306310910660186843379516424285475
a[16,12] = -.3834605318628494829280108044228903735901e-1
a[16,13] = 0
a[16,14] = -.7325540972670704140488857561266969491257e-2
a[16,15] = -.8007336945700193815591023393769736797741e-1
# 
#  ********************************************************
#
#  Coupling coefficients 
#  for node c[17] = .6
#  ----------------------------------------------------
a[17,1] = .4022215783658092885487104933344487123772e-1
a[17,2] = 0
a[17,3] = 0
a[17,4] = 0
a[17,5] = 0
a[17,6] = .2394865196246692778481611591728845874793
a[17,7] = .3492041237976048034287443180090472771373
a[17,8] = -.3721229506921462549653737059376443173945e-2
a[17,9] = .3770553242761762692615854767097170444870
a[17,10] = -.3674578198324505102773671917436327962691e-1
a[17,11] = -.3903899350263818482522984813951798810533
a[17,12] = .5322029071951441166697763858943615968930e-1
a[17,13] = 0
a[17,14] = .1174118080627468100699369632520307756337e-1
a[17,15] = .1352019177159245836247809775513421356206
a[17,16] = -.1752745682601965938624253780621555493606
# 
# --------------------------------------------------------
#  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^1
bi47[1,2] = -8.5463085947551434748257820923102837836 u^2
bi47[1,3] = 37.7069605099594878283460071886138480974 u^3
bi47[1,4] = -91.718050363671802030339799514699341582 u^4
bi47[1,5] = 122.295480481653102453474436765461961805 u^5
bi47[1,6] = -83.224943834632636305026283203959689597 u^6
bi47[1,7] = 22.5308853876927624498226080105749597369 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[2]
bi47[2,1] = 0 u^1
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^1
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^1
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^1
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^1
bi47[6,2] = 24.6224524697618454536172069617606293686 u^2
bi47[6,3] = -198.59331182299135763711729422807310961 u^3
bi47[6,4] = 651.102565372746880257729103582097126291 u^4
bi47[6,5] = -1032.0966128921668137829994022847229287 u^5
bi47[6,6] = 784.596575989390672858731608138292738157 u^6
bi47[6,7] = -229.28166034814958675202736899626174577 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[7]
bi47[7,1] = 0 u^1
bi47[7,2] = 30.3387979363442731924841404992209389951 u^2
bi47[7,3] = -243.60551209279393050404259110820441544 u^3
bi47[7,4] = 793.340387418361358528274145838637907487 u^4
bi47[7,5] = -1245.7006858245668225625916875338297775 u^5
bi47[7,6] = 935.831636286333494767243620146725419348 u^6
bi47[7,7] = -269.95842826613129211092005346279769854 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[8]
bi47[8,1] = 0 u^1
bi47[8,2] = -.44909212102981182619481893942101856550 u^2
bi47[8,3] = 2.17585645458280778297150342005155948259 u^3
bi47[8,4] = -.71552150368989143481347749531613802944 u^4
bi47[8,5] = -15.583457948389036804012250547327260388 u^5
bi47[8,6] = 26.5928795668729646662830366922065141165 u^6
bi47[8,7] = -12.426166311357921014319492685684138408 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[9]
bi47[9,1] = 0 u^1
bi47[9,2] = 26.2205014343745071040785087172378703137 u^2
bi47[9,3] = -191.76429606219678746288773785489940435 u^3
bi47[9,4] = 532.432122111781155692303216505287127310 u^4
bi47[9,5] = -629.98431856749181040484316009506283056 u^5
bi47[9,6] = 277.868955296501240393421109327274823159 u^6
bi47[9,7] = -17.738391724404523560716789339894403274 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[10]
bi47[10,1] = 0 u^1
bi47[10,2] = 2.45751049639930721614915978934413334669 u^2
bi47[10,3] = -24.931722487081058863219578779114199137 u^3
bi47[10,4] = 106.694731175280990634293073335062676458 u^4
bi47[10,5] = -224.59266112166118471619444448995415308 u^5
bi47[10,6] = 222.841316082201729865877919803460270039 u^6
bi47[10,7] = -81.569052731083407999158167564655318562 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[11]
bi47[11,1] = 0 u^1
bi47[11,2] = -31.150028967565225255567171471120945198 u^2
bi47[11,3] = 234.382046513333249184863315690699288256 u^3
bi47[11,4] = -686.11498898172533668121904711948296093 u^4
bi47[11,5] = 904.618015522706090535974530701122829090 u^5
bi47[11,6] = -515.78905880838662282136399726235803687 u^6
bi47[11,7] = 96.4654499369744762756759877738014207386 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[12]
bi47[12,1] = 0 u^1
bi47[12,2] = 5.03668041807914841297631687093116206791 u^2
bi47[12,3] = -38.872604914636222868291558434020240779 u^3
bi47[12,4] = 118.897099504699708079324070273647712164 u^4
bi47[12,5] = -169.46709667942150353153811682768459019 u^5
bi47[12,6] = 110.976141468333964233566864270869822013 u^6
bi47[12,7] = -26.795044217706389754583548717312633081 u^7
# -----------------------------------------
#
#  COEFFICIENTS OF bi47[13]
bi47[13,1] = 0 u^1
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^1
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^1
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^1
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^1
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 node c[18] = .25
#  ----------------------------------------------------
a[18,1] = .4723910302740347243521351759404200729408e-1
a[18,2] = 0
a[18,3] = 0
a[18,4] = 0
a[18,5] = 0
a[18,6] = .1489029860559623175080300209174637964623
a[18,7] = .1843176226515228278707879216160623043374
a[18,8] = -.3834252466005522093443562565798420110377e-2
a[18,9] = .1738145644783229443310556017080021627415
a[18,10] = .1090980349665939190384073181696094809214e-1
a[18,11] = -.2014152580039779653569324521065310699748
a[18,12] = .3181300250560556935772241536627167409812e-1
a[18,13] = 0
a[18,14] = .5449192848475831840535658876239056372696e-2
a[18,15] = -.1443696573210151335442515976554247569207
a[18,16] = .8054527122531925175504956241432927456193e-1
a[18,17] = -.8337237849827298600760781798161697695424e-1
# 
#  ********************************************************
#
#  Coupling coefficients 
#  for node c[19] = .45
#  ----------------------------------------------------
a[19,1] = .4421216268801324396509460939131318288392e-1
a[19,2] = 0
a[19,3] = 0
a[19,4] = 0
a[19,5] = 0
a[19,6] = .2017709278432031668902792581957402182437
a[19,7] = .2524617572009699042879088218269523985316
a[19,8] = -.8770696556893985323948333941153090097834e-2
a[19,9] = .2842679885691825163383012296159640300231
a[19,10] = .2150646641574965678745383394398849175791e-2
a[19,11] = -.3146025568238765059212901568954609645367
a[19,12] = .4743626196163973325539647685616327135465e-1
a[19,13] = 0
a[19,14] = .8909366379059747507640786880542915890034e-2
a[19,15] = .4792054797927460342806804123244705373170e-1
a[19,16] = -.2200021508937620126498016204528367567642e-1
a[19,17] = -.9375619079277118884121595451162418952355e-1
a[19,18] = 0
# 
#  ********************************************************
#
#  Coupling coefficients 
#  for node c[20] = .7
#  ----------------------------------------------------
a[20,1] = .4267763300289232230029877837703073551691e-1
a[20,2] = 0
a[20,3] = 0
a[20,4] = 0
a[20,5] = 0
a[20,6] = .2374158998735259190176751304349328459706
a[20,7] = .2928518820873850719555588943011548358798
a[20,8] = -.4750364164470648842484964073680382642525e-2
a[20,9] = .2585521389992642645772489598683801078129
a[20,10] = .2221148705771689147733930575181295327406e-1
a[20,11] = -.3052536016479972823600004913887386466064
a[20,12] = .4907352889635131418349676983457735436412e-1
a[20,13] = 0
a[20,14] = .8692802242096861325332906832399638874883e-2
a[20,15] = .9136095885233150535257055362459649774023e-1
a[20,16] = -.8846964382883308113343887470420217296149e-1
a[20,17] = .9563727862973686214640303114173623277688e-1
a[20,18] = 0
a[20,19] = 0
# 
#  ********************************************************
#
#  Coupling coefficients 
#  for node c[21] = .938
#  ----------------------------------------------------
a[21,1] = .4823720753000990412952135003326021199878e-1
a[21,2] = 0
a[21,3] = 0
a[21,4] = 0
a[21,5] = 0
a[21,6] = .2778342955127964929197438644338843458914
a[21,7] = .1847545321774465459316649512594701805203
a[21,8] = .2034148974613708614363828398526227691874
a[21,9] = -2.546453795180392984243798786311967166684
a[21,10] = .7644071771125479164933194093781188967612
a[21,11] = 2.078790054042179631850798090303419414930
a[21,12] = -.1955592873813456281783138690110781674538
a[21,13] = 0
a[21,14] = -.3399808573709679959957193450753880804121e-1
a[21,15] = .9939967824606758743711369390962005581104e-2
a[21,16] = .7136449270140573466140101865915001084083e-1
a[21,17] = .7526854393647156585514169651969630646794e-1
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^1
bi48[1,2] = -8.8281971494668228779958130601054431755 u^2
bi48[1,3] = 39.6008638281844341754488401426520782003 u^3
bi48[1,4] = -100.18925499828426004921058815688836292 u^4
bi48[1,5] = 149.018229820471628295997946944464273892 u^5
bi48[1,6] = -129.27553692275574265013119214805082773 u^6
bi48[1,7] = 60.5704568230785743714510709670861909512 u^7
bi48[1,8] = -11.852537814982040344109077535476454531 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[2]
bi48[2,1] = 0 u^1
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^1
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^1
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^1
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^1
bi48[6,2] = 40.4668506343597811882076016383923704343 u^2
bi48[6,3] = -245.56597480357003594264078803780847789 u^3
bi48[6,4] = 580.316024182145149684872039897063684353 u^4
bi48[6,5] = -587.50881828808739341840087419705356505 u^5
bi48[6,6] = 132.301831318564135038022911520720164959 u^6
bi48[6,7] = 164.487535858977927207097795704895472379 u^7
bi48[6,8] = -84.147440133797923359224833353116939468 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[7]
bi48[7,1] = 0 u^1
bi48[7,2] = 28.4643006159633514193712997267313750958 u^2
bi48[7,3] = -172.73060834454741336923716801049439623 u^3
bi48[7,4] = 408.193113762003788117342792663765233657 u^4
bi48[7,5] = -413.25251053963267544286939226948873887 u^5
bi48[7,6] = 93.0608396665426302963301126994184641684 u^6
bi48[7,7] = 115.700198925131222080115417484671078374 u^7
bi48[7,8] = -59.189138627913821790605487914850641919 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[8]
bi48[8,1] = 0 u^1
bi48[8,2] = 27.5708187265010205072070790100870046301 u^2
bi48[8,3] = -167.30867044437249091260302022837703594 u^3
bi48[8,4] = 395.380111276179531671762349201471836677 u^4
bi48[8,5] = -400.28069581199699324632398094704110571 u^5
bi48[8,6] = 90.1397007992281677719191102849010330826 u^6
bi48[8,7] = 112.068420518152958317829994069149395689 u^7
bi48[8,8] = -57.331217573382180448544901199897085983 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[9]
bi48[9,1] = 0 u^1
bi48[9,2] = -342.85287381562128357206334633291889772 u^2
bi48[9,3] = 2080.54244036602764552189400736362296055 u^3
bi48[9,4] = -4916.6914027939599436831205790661601942 u^4
bi48[9,5] = 4977.63190326108693912354275003636121775 u^5
bi48[9,6] = -1120.9190329262922356128039320839167412 u^6
bi48[9,7] = -1393.6104117827368434987868504176653495 u^7
bi48[9,8] = 712.933950180059503482693097760620187011 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[10]
bi48[10,1] = 0 u^1
bi48[10,2] = 104.069046504105393974482702296827062707 u^2
bi48[10,3] = -631.52472829105341918540724770168107706 u^3
bi48[10,4] = 1492.40512570084958373402285297246905203 u^4
bi48[10,5] = -1510.9029137068719400833552788472848015 u^5
bi48[10,6] = 340.242080128155933127855273367456735899 u^6
bi48[10,7] = 423.014411803991360240387599397655006345 u^7
bi48[10,8] = -216.40290072512053567023793939129856929 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[11]
bi48[11,1] = 0 u^1
bi48[11,2] = 278.802125632784350203689185551018919701 u^2
bi48[11,3] = -1691.8617259577420808906719120032516749 u^3
bi48[11,4] = 3998.16982405277945592643958999736674093 u^4
bi48[11,5] = -4047.7256025366364259901477160966459380 u^5
bi48[11,6] = 911.512292617266297327638234094036562946 u^6
bi48[11,7] = 1133.26028387895621274754171332383534178 u^7
bi48[11,8] = -579.74576247207117808612547655369835726 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[12]
bi48[12,1] = 0 u^1
bi48[12,2] = -25.993452352809825064399081932116686323 u^2
bi48[12,3] = 157.736699680505833180832926737462394942 u^3
bi48[12,4] = -372.75984386446612942503778582058370421 u^4
bi48[12,5] = 377.380058878619750773883659111869753195 u^5
bi48[12,6] = -84.982678282569221872128944306142869995 u^6
bi48[12,7] = -105.65682426373686982337986804415070195 u^7
bi48[12,8] = 54.0512157838051668016831216900930465464 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[13]
bi48[13,1] = 0 u^1
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^1
bi48[14,2] = .729633294423675422841022635732266221545 u^2
bi48[14,3] = -12.283826812973510255881453903717439497 u^3
bi48[14,4] = 70.1234223731162724125788904917144598062 u^4
bi48[14,5] = -190.71288703913486451968612869739432954 u^5
bi48[14,6] = 270.200499584294180613118453491086094339 u^6
bi48[14,7] = -192.71469324532335857513294915265343910 u^7
bi48[14,8] = 54.6578518455976049021621651352323877831 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[15]
bi48[15,1] = 0 u^1
bi48[15,2] = -1.3313745760019298082305114501777122238 u^2
bi48[15,3] = 43.4425798816850987851081265621820627434 u^3
bi48[15,4] = -276.80912230073493262967908704303116682 u^4
bi48[15,5] = 754.558402248117213308722529595675232621 u^5
bi48[15,6] = -1022.5621058640148257897519746230739390 u^6
bi48[15,7] = 679.460842234203827095895144521402807122 u^7
bi48[15,8] = -176.75922162325445096206422756297728440 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[16]
bi48[16,1] = 0 u^1
bi48[16,2] = 3.72023512841485176415075286728495356863 u^2
bi48[16,3] = -121.39079013371437422715071703216619494 u^3
bi48[16,4] = 773.482563968822817814588353403959963238 u^4
bi48[16,5] = -2108.4484600222385589164377954938503797 u^5
bi48[16,6] = 2857.32620690787499907605582999788554882 u^6
bi48[16,7] = -1898.6047497262428321103026191729385581 u^7
bi48[16,8] = 493.914993877083096599096195429824667165 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[17]
bi48[17,1] = 0 u^1
bi48[17,2] = -6.4948770861295882987571153260212257602 u^2
bi48[17,3] = 211.926997647205943583532653268806540283 u^3
bi48[17,4] = -1350.3646968094726432792899596432266070 u^4
bi48[17,5] = 3680.98066858439523651428740284591844952 u^5
bi48[17,6] = -4988.3896765285546298458006164753909577 u^6
bi48[17,7] = 3314.63040882256193514077828859802764809 u^7
bi48[17,8] = -862.28882463000625381475065326811384741 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[18]
bi48[18,1] = 0 u^1
bi48[18,2] = -33.378977343233602519958857985213881117 u^2
bi48[18,3] = 352.857449915335191330862070427231523940 u^3
bi48[18,4] = -1472.2321301290180073387941428170175927 u^4
bi48[18,5] = 3143.93285193620241847684912236834305804 u^5
bi48[18,6] = -3637.0669404245810618912834815342392375 u^6
bi48[18,7] = 2167.25046003935255617263896300664533918 u^7
bi48[18,8] = -521.36271399405749423031367346574920967 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[19]
bi48[19,1] = 0 u^1
bi48[19,2] = -15.992824681148976716714898155699870115 u^2
bi48[19,3] = -65.600975830443169108188705042997381030 u^3
bi48[19,4] = 922.440717861781320454058102347809799344 u^4
bi48[19,5] = -2948.2295600334889428362651911715677402 u^5
bi48[19,6] = 4228.26599583397498187501468443110600728 u^6
bi48[19,7] = -2877.6443557754987112162332905895618652 u^7
bi48[19,8] = 756.761002624823497548329298180911050077 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[20]
bi48[20,1] = 0 u^1
bi48[20,2] = -29.352986675256620444111462048258357699 u^2
bi48[20,3] = 85.6560375463814073866765325529451084793 u^3
bi48[20,4] = 262.509747114906130896497423647392913189 u^4
bi48[20,5] = -1559.7976091126191488638418890216854074 u^5
bi48[20,6] = 2712.03882205464130753691989520273022657 u^6
bi48[20,7] = -2046.8860729114170063370177084051512798 u^7
bi48[20,8] = 575.832061983363929824877208072026796733 u^8
# -----------------------------------------
#
#  COEFFICIENTS OF bi48[21]
bi48[21,1] = 0 u^1
bi48[21,2] = -19.597446856883775177718557435561878220 u^2
bi48[21,3] = 136.504231753090939927425854905591008443 u^3
bi48[21,4] = -413.97419939664813430703025207610605521 u^4
bi48[21,5] = 683.356942361813756824044835839380021180 u^5
bi48[21,6] = -651.89229796177491500097436391852626475 u^6
bi48[21,7] = 344.674088800549048187117298708752913929 u^7
bi48[21,8] = -79.071318700146920452864816023529745367 u^8
#
# ********************************************************
#  
# Norms of low order INTERPOLANT coefficients on  [0,2] 
#     u           Max norm        2-norm   
# ------------------------------------------------- 
.1000000000,  .3159448925e-5, .7486358525e-5
.2000000000,  .4281339863e-5, .1020639545e-4
.3000000000,  .2028513663e-5, .4799569883e-5
.4000000000, -.5006198840e-6, .1174281261e-5
.5000000000, -.1441758755e-5, .4386312733e-5
.6000000000, -.2310332505e-5, .5886542431e-5
.7000000000, -.1757683738e-5, .4382381290e-5
.8000000000, -.2327601165e-5, .7363608939e-5
.9000000000,  .2794406939e-5, .7500674130e-5
1.000000000, -.1078110119e-57,.3250910980e-57
1.100000000,  .2080618285e-4, .5133478193e-4
1.200000000,  .1766502840e-3, .4286255171e-3
1.300000000,  .7669310992e-3, .1838415174e-2
1.400000000,  .2449316387e-2, .5815796667e-2
1.500000000,  .6499237367e-2, .1531364948e-1
1.600000000,  .1518419991e-1, .3554895216e-1
1.700000000,  .3228169209e-1, .7517020206e-1
1.800000000,  .6377261785e-1, .1478170223
1.900000000,  .1187453364,    .2741500804
2.000000000,  .2105485370,    .4844364876
#
# ******************************************************** 
# Norms of high order INTERPOLANT error coefficients on  [0,2] 
#     u           Max norm        2-norm   
# ------------------------------------------------- 
.1000000000, -.2111069234e-6, .7713513253e-6
.2000000000,  .5441759130e-6, .1489611278e-5
.3000000000,  .8723727549e-6, .2208197002e-5
.4000000000,  .9390405529e-6, .2414796010e-5
.5000000000,  .8546517419e-6, .2214762964e-5
.6000000000,  .7496690626e-6, .2028224139e-5
.7000000000,  .6357934509e-6, .2008728345e-5
.8000000000,  .4456679205e-6, .1514277153e-5
.9000000000,  .2220594148e-6, .6279727451e-6
1.000000000,  .1458047359e-6, .4697540876e-6
1.100000000,  .1349034967e-5, .4665002678e-5
1.200000000,  .1743038836e-4, .6048969204e-4
1.300000000,  .1027692792e-3, .3506221847e-3
1.400000000,  .4181520172e-3, .1404577172e-2
1.500000000,  .1355552614e-2, .4496357747e-2
1.600000000,  .3755488281e-2, .1233290192e-1
1.700000000,  .9259165950e-2, .3016456115e-1
1.800000000,  .2084884272e-1, .6748350551e-1
1.900000000,  .4364213737e-1, .1405139930
2.000000000,  .8601960440e-1, .2757403452
# ********************************************************