######################################################################
# 
#	  A nullspace 'efficient' Runge--Kutta (13:8(7)) pair
#
#     These are exact REAL coefficients 
#	computed using MAPLE for 
#
#	a THIRTEEN-stage nullspace pair of methods of 
#	    orders  p=7  and  p=8, with dominant 
#	    stage order = 3. 
#
#	This procedure is "efficient" in the sense that
#	that for a specified maximum coefficient from  
#	b  and  A, it has a propagating formula which almost 
#	minimizes the 2-norm of the local truncation error as
#
#		   T92 ~ .000000367
#
#     The formulas scanned for this optimal formula are
#  those developed by J.H. Verner and described in the 
#  manuscript "Nullspaces yield new explicit Runge--Kutta
#  pairs" of 26 pages dated July 3, 2023. It is known 
#  that other pairs provided on this webpage have lower
#  2-norms of the local truncation error coefficients, 
#  but this pair has coefficients that require fewer
#  digits.  
#
#     This pair is provided more for the interest of 
#  researchers working on the derivation of new formulas.
# 
######################################################################
# 
#  NODES
#  -----
c[1] = 0
c[2] = 1/1000
c[3] = 1/9
c[4] = 1/6
c[5] = 5/12
c[6] = 1/2
c[7] = 5/6
c[8] = 1/6
c[9] = 2/3
c[10] = 19/25
c[11] = 21/25
c[12] = 1
c[13] = 1
# 
# ********************************************************
#  COUPLING COEFFICIENTS
#  ---------------------

a[2,1] =  1/1000

a[3,1] = -491/81
a[3,2] =  500/81

a[4,1] =  1/24
a[4,2] =  0
a[4,3] =  1/8

a[5,1] =  5/12
a[5,2] =  0
a[5,3] = -25/16
a[5,4] =  25/16

a[6,1] =  1/20
a[6,2] =  0
a[6,3] =  0
a[6,4] =  1/4
a[6,5] =  1/5

a[7,1] =  -43/180
a[7,2] =  0
a[7,3] =  0
a[7,4] =  127/108
a[7,5] = -329/135
a[7,6] =  7/3

a[8, 1] = 27931/240300
a[8, 2] = 0
a[8, 3] = 0
a[8, 4] = -631/16020
a[8, 5] = 2459/6675
a[8, 6] = -3572/12015
a[8, 7] = 5/267

a[9, 1] = 26114/12015
a[9, 2] = 0
a[9, 3] = 0
a[9, 4] = -7480/801
a[9, 5] = 67264/4005
a[9, 6] = -30640/2403
a[9, 7] = 1051/1335
a[9, 8] = 3

a[10, 1] = 33096587331/17382812500
a[10, 2] = 0
a[10, 3] = 0
a[10, 4] = -1828977848/173828125
a[10, 5] = 62801809904/4345703125
a[10, 6] = -9389764774/869140625
a[10, 7] = 6380757669/8691406250
a[10, 8] = 98417891/19531250
a[10, 9] = -1692691/39062500

a[11, 1] = -1456295425347/2642187500000
a[11, 2] = 0
a[11, 3] = 0
a[11, 4] = 110740056/34765625
a[11, 5] = -21221682384/4345703125
a[11, 6] = 58859060169/13906250000
a[11, 7] = -177381525069/1529687500000
a[11, 8] = -28942485159/27812500000
a[11, 9] = -1272297/312500000
a[11, 10] = 5151/297616

a[12, 1] = 844300798/137013275
a[12, 2] = 0
a[12, 3] = 0
a[12, 4] = -9997568/206035
a[12, 5] = 49636624/1030175
a[12, 6] = -3358834871/91067470
a[12, 7] = -40456983/1813108
a[12, 8] = 495817135/16647628
a[12, 9] = -149375/84266
a[12, 10] = 7470703125/1567431866
a[12, 11] = 1562500000/72342361

a[13, 1] = -26225423/37371100
a[13, 2] = 0
a[13, 3] = 0
a[13, 4] = 807744/98345
a[13, 5] = -205584/37825
a[13, 6] = 5882202/1278485
a[13, 7] = -84543/432718
a[13, 8] = -223415/39338
a[13, 9] = -3625/6188
a[13, 10] = 292968750/374084711
a[13, 11] = 0
a[13, 12] = 0

# ********************************************************
#  HIGH AND LOW ORDER WEIGHTS
#  --------------------------
b[1] =  4241/88200
b[2] =  0
b[3] =  0
b[4] =  0
b[5] =  0
b[6] =  9946/23205
b[7] = -10449/1925
b[8] =  415449/1573075
b[9] = -2025/5096
b[10] = 48828125/44900856
b[11] = 48828125/9843561
b[12] = 463/12600
b[13] =  0

bh[1] =  3799/79800
bh[2] =  0
bh[3] =  0
bh[4] =  0
bh[5] =  0
bh[6] =  538/1365
bh[7] =  351/1925
bh[8] =  4149/15575
bh[9] = -45/392
bh[10] =  48828125/284372088
bh[11] =  0
bh[12] =  0
bh[13] =  221/4200