Jsun Yui Wong
The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system:
13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)=127,
– 2 * X(1) * X(2) * X(3) – X(4) – X(5) – X(6)=-70,
1 * X(1) + 2 * X(2) – 9 * X(3) – 1 * X(4) + 1 * X(5) + 3 * X(6)=-33,
1 * X(1) + 1 * X(2) + 2 * X(3) – 7 * X(4) – 1 * X(5) + 1 * X(6)=-29,
1 * X(1) + 1 * X(2) + 1 * X(3) + 2 * X(4) – 9 * X(5) – 3 * X(6)=21,
1 * X(1) + 1 * X(2) + 1 * X(3) + 1 * X(4) + 2 * X(5) – 7 * X(6)=-13.
The first two of these equations are based on page 177 of Conley [4]. The last four come from Greenspan and Casulli [5, page 41].
One notes that the starting search interval is A(KK) = -50 + FIX(RND * 101) of line 94 shown below.
0 DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+317
77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91
88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128
91 FOR KK = 1 TO 6
94 A(KK) = -50 + FIX(RND * 101)
95 NEXT KK
128 FOR I = 1 TO 200000
129 FOR K = 1 TO 6
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 6)
182 REM IF RND < -.1 THEN 183 ELSE GOTO 189
183 r = (1 – RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * r
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) – FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)
191 NEXT IPP
998 N82 = -127 + 13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)
1000 N81 = 70 – 2 * X(1) * X(2) * X(3) – X(4) – X(5) – X(6)
1003 N83 = 33 + 1 * X(1) + 2 * X(2) – 9 * X(3) – 1 * X(4) + 1 * X(5) + 3 * X(6)
1004 N84 = 29 + 1 * X(1) + 1 * X(2) + 2 * X(3) – 7 * X(4) – 1 * X(5) + 1 * X(6)
1005 N85 = -21 + 1 * X(1) + 1 * X(2) + 1 * X(3) + 2 * X(4) – 9 * X(5) – 3 * X(6)
1006 N86 = 13 + 1 * X(1) + 1 * X(2) + 1 * X(3) + 1 * X(4) + 2 * X(5) – 7 * X(6)
1335 P = -ABS(N81) – ABS(N82) – ABS(N83) – ABS(N84) – ABS(N85) – ABS(N86)
1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 6
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1664 NN81 = N81: NN82 = N82
1665 NN83 = N83: NN84 = N84
1666 NN85 = N85: NN86 = N86
1670 NEXT I
1888 IF M < 0 THEN 1999
1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, NN81, NN82, NN83, NN84, NN85, NN86
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s complete output through
JJJJ=-23740 is shown below:
2 3 5 7 -1
4 0 -31947 0 0
0 0 0 0
2 3 5 7 -1
4 0 -31946 0 0
0 0 0 0
2 3 5 7 -1
4 0 -31945 0 0
0 0 0 0
2 3 5 7 -1
4 0 -31944 0 0
0 0 0 0
2 3 5 7 -1
4 0 -31943 0 0
0 0 0 0
2 3 5 7 -1
4 0 -31942 0 0
0 0 0 0
2 3 5 7 -1
4 0 -28653 0 0
0 0 0 0
2 3 5 7 -1
4 0 -28652 0 0
0 0 0 0
2 3 5 7 -1
4 0 -28428 0 0
0 0 0 0
2 3 5 7 -1
4 0 -28427 0 0
0 0 0 0
2 3 5 7 -1
4 0 -28426 0 0
0 0 0 0
.
.
.
2 3 5 7 -1
4 0 -28399 0 0
0 0 0 0
2 3 5 7 -1
4 0 -23743 0 0
0 0 0 0
2 3 5 7 -1
4 0 -23742 0 0
0 0 0 0
2 3 5 7 -1
4 0 -23741 0 0
0 0 0 0
2 3 5 7 -1
4 0 -23740 0 0
0 0 0 0
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [11], the wall-clock time through
JJJJ=-23740 was 35 minutes.
Incidentally, one can stop the computer run as soon as the first 0 (for M) appears on the screen. For the present problem, that happens at JJJJ=-31947.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] W. Conley, Computer Optimization Techniques, Revised Edition. Petrocelli Books, Inc., NY/Princeton, 1984.
[5] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[6] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[7] 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.
[8] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[9] Thomas L.. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.
[10] Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
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