Jsun Yui Wong

Based on the computer program of an earlier paper [13], the following computer program seeks to solve the following system of four nonlinear Diophantine equations:

X(1)^2+X(2) ^2 +X(3)^2 = X(5)^2

X(1)^2+X(3) ^2 +X(4)^2 = X(6)^4

X(1)^2+X(2) ^2 +X(4)^2 = X(7)^4

X(2)^2+X(3) ^2 +X(4)^2 = X(8)^2,

which is a modified version of the following system:

X(1)^2+X(2) ^2 +X(3)^2 = 113^2

X(1)^2+X(3) ^2 +X(4)^2 = 11^4

X(1)^2+X(2) ^2 +X(4)^2 = 11^4

X(2)^2+X(3) ^2 +X(4)^2 = 132^2

taken from Piezas [7].

While line 112 and line 214 of the computer program of the preceding paper are 112 A(J44)=1+FIX(RND*300) and 214 IF X(J44)>300 THEN X(J44)=A(J44), line 112 and line 214 here are

112 A(J44)=1+FIX(RND*500) and 214 IF X(J44)>500 THEN X(J44)=A(J44), respectively. Thus, the search region is bigger here.

0 REM DEFDBL A-Z

1 DEFINT I,J,K,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),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)

88 FOR JJJJ=-32000 TO 32000

89 RANDOMIZE JJJJ

90 M=-3D+30

110 FOR J44=1 TO 8

112 A(J44)=1+FIX( RND*500)

113 NEXT J44

128 FOR I=1 TO 3000

129 FOR KKQQ=1 TO 8

130 X(KKQQ)=A(KKQQ)

131 NEXT KKQQ

139 FOR IPP=1 TO FIX(1+RND*3)

140 B=1+FIX(RND*8)

144 REM GOTO 167

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 REM IF A(B)=0 THEN X(B)=1 ELSE X(B)=0

169 NEXT IPP

181 X(5)= ( X(1)^2#+X(2) ^2# +X(3)^2# ) ^.5#

183 X(6)= ( X(1)^2#+X(3) ^2# +X(4)^2# ) ^.25#

185 X(7)= ( X(1)^2#+X(2) ^2# +X(4)^2# ) ^.25#

187 X(8)= ( X(2)^2#+X(3) ^2# +X(4)^2# ) ^.5#

212 FOR J44=1 TO 8

213 IF X(J44)<1 THEN X(J44)=A(J44)

214 IF X(J44)>500 THEN X(J44)=A(J44)

215 NEXT J44

217 N(7)= X(1)^2#+X(2) ^2# +X(3)^2# -X(5)^2#

218 N(8)= X(1)^2#+X(3) ^2# +X(4)^2# -X(6)^4#

219 N(9)= X(1)^2#+X(2) ^2# +X(4)^2# -X(7)^4#

220 N(10)= X(2)^2#+X(3) ^2# +X(4)^2# -X(8)^2#

335 PD1=-ABS(N(7))-ABS(N(8))-ABS(N(9)) -ABS(N(10))

1111 IF PD1<=M THEN 1670

1452 M=PD1

1454 FOR KLX=1 TO 8

1455 A(KLX)=X(KLX)

1456 NEXT KLX

1558 GOTO 128

1670 NEXT I

1889 IF M<-1 THEN 1999

1904 PRINT A(1),A(2),A(3),A(4),A(5)

1927 PRINT A(6),A(7),A(8),M,JJJJ

1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. See [6]. The complete output through JJJJ=-18116 is copied by hand from the screen and shown below. Immediately below there is no rounding by hand.

49 72 72 84 113

11 11 132 0 -18116

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=-18116 was three hours.

**Acknowledgment**

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. *Operations Research*, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method.* Operations Research*, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. *Operations Research*, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[4] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. *Operations Research*, Vol. 15, No. 3 (May - June, 1967), p. 578.

[5] Duan Li, Xiaoling Sun, *Nonlinear Integer Programming*. Publisher: Springer Science+Business Media,LLC (2006).

[6] Microsoft Corp., *BASIC*, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C,Boca Raton, Floridda 33432, 1981.

[7] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[8] Harvey M. Salkin, *Integer Programming*. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[9] Harvey M. Salkin, Kamlesh Mathur, *Foundations of Integer Programming*. Publisher: Elsevier Science Ltd (1989).

[10] K. Schittkowski, *More Test Examples for Nonlinear Programming Codes*. Springer-Verlag, 1987.

[11] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[12] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

[13] Jsun Yui Wong (2014, February 1). Testing the Nonlinear Integer Programming Solver with Another System of Nonlinear Diophantine Equations. http://computerprogramsandresults.wordpress.com/2014/02/01/