The computer program below seeks to solve simultaneously the system of eight equations taken from page 11381 of Abraham, Sanyal, and Sanglikar [1] and from page 741 of Perez, Amaya, and Correa [3]. These simultaneous equations are as follows:
X(1)^6 + X(2) ^6 = 47385
3* X(1) +18* X(3)-5* X(5)+17* X(6) = 153
X(2) – 5* X(3)+ 3* X(5) – X(6) )= -4
6* X(1) + X(3)-99*X(2) +(15*X(6))^2 = 1772
5* X(1) +10* X(2)-5*X(3) + X(5) ^3+8*X(6) = 1772
– X(1) +5* X(2) +8*X(3)-6* X(4) +15*X(5)+10*X(6) = 277
( X(1) + X(2) )^2-7*X(3)+5* X(4)+12* X(5)-8*X(6) = 150
X(1)^2 + X(2) ^2+ X(3)^2+ X(4)^2 + X(5) ^2+ X(6)^2+ X(7)^2 + X(8)^2 + X(9)^2 + X(10) ^2 = 956
0 DEFDBL A-Z
1 DEFINT I,J,K,A,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99)
5 DIM AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(99)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
111 FOR J44=1 TO 10
112 A(J44)= ( RND *20)
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*10)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
169 FOR J44=1 TO 10
170 IF X(J44)<0 THEN 1670
171 NEXT J44
401 N(2)=-47385#+ X(1)^6# + X(2) ^6#
411 N(81)= -153#+3#* X(1) +18#* X(3)-5#* X(5)+17#* X(6)
413 N(83)= 4# + X(2) – 5#* X(3)+ 3#* X(5) – X(6)
415 N(85)= -1772#+6#* X(1) + X(3)-99#*X(2) +(15#*X(6))^2
417 N(87)=-1772#+5#* X(1) +10#* X(2)-5#*X(3) + X(5) ^3+8#*X(6)
419 N(89)= -277#- X(1) +5#* X(2) +8#*X(3)-6#* X(4) +15#*X(5)+10#*X(6)
421 N(91)= -150#+( X(1) + X(2) )^2-7#*X(3)+5#* X(4)+12#* X(5)-8#*X(6)
892 N(10)= -956# + X(1)^2 + X(2) ^2+ X(3)^2+ X(4)^2 + X(5) ^2+ X(6)^2+ X(7)^2 + X(8)^2 + X(9)^2 + X(10) ^2
922 PD1=-ABS(N(10)) -ABS(N(2)) -ABS(N(81)) -ABS(N(83)) -ABS(N(85))-ABS(N(87)) -ABS(N(89)) -ABS(N(91))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1511 NN(10)=N(10)
1512 NN(2)=N(2)
1514 NN(81)=N(81)
1515 NN(83)=N(83)
1516 NN(85)=N(85)
1517 NN(87)=N(87)
1518 NN(89)=N(89)
1519 NN(91)=N(91)
1557 GOTO 128
1670 NEXT I
1889 IF M<-1 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT M,JJJJ
1910 PRINT NN(10),NN(2),NN(81),NN(83)
1911 PRINT NN(85),NN(87),NN(89),NN(91)
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft’s GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31994 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
6 3 8 1 12
3 20 13 11 2
-1 -32000
1 0 0 0
0 0 0 0
6 3 8 1 12
3 15 6 17 12
-1 -31996
1 0 0 0
0 0 0 0
6 3 8 1 12
3 5 15 9 19
-1 -31995
-1 0 0 0
0 0 0 0
6 3 8 1 12
3 4 15 16 14
0 -31994
0 0 0 0
0 0 0 0
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31994 was two seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), Pages 11376-11387.
[2] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[3] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.
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