Jsun Yui Wong
The computer program listed below seeks to solve simultaneously the following nonlinear system of equations:
X(74) = -X(75) ^ 3 + 5 * X(75) ^ 2 + X(75) + 10,
-29 + X(74) + X(75) ^ 3 + X(75) ^ 2 - 14 * X(75) =0,
X(76) = -X(77) ^ 3 + 5 * X(77) ^ 2 + 2 * X(77) + 11,
-29 + X(76) + X(77) ^ 3 + X(77) ^ 2 - 14 * X(77)=0,
-5 + 10 * X(71) - 2 * X(72) ^ 2 + X(72) - 2 * X(73) ^ 2 = 0,
-12 + 8 * X(72) ^ 2 + 4 * X(73) ^ 2 = 0,
8 + 8 * X(72) * X(73) = 0,
X(67) = -10 + X(65) ^ 2 + 10 * X(66),
-12 + 15 * X(65) + X(66) - 4 * X(67) = 0,
24 + X(66) - 25 * X(67) = 0,
-7 + 10 * X(68) - 2 * X(69) ^ 2 + X(69) - 2 * X(70) ^ 2 = 0,
-12 + 8 * X(69) ^ 2 + 4 * X(70) ^ 2 = 0,
8 + 8 * X(69) * X(70) = 0,
X(54) = -X(52) ^ 2,
X(52) * COS(X(53)) - X(54) = 0,
(EXP(X(52) + X(54))) * SIN(X(53) / 2) + X(54) = 0,
X(55) ^ 5 + X(56) ^ 3 * X(57) ^ 4 + 1 = 0,
X(55) * X(56) * X(57) = 0,
X(57) ^ 4 - 1 = 0,
X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) = 1,
X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9,
2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) = 9,
X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) = 9,
X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) = 9,
X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) = 9,
X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) = 9,
X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) = 9,
-18 - X(61) * (X(61) + 1) + 2 * X(62) =0,
-25 + (X(61) - 1) ^ 2 + (X(62) - 6) ^ 2=0,
X(63) ^ 2 - 10 * X(63) + X(64) ^ 2 + 8=0,
X(63) * X(64) ^ 2 + X(63) - 10 * X(64) + 8=0,
X(41) ^ 2 + X(42) - 37 = 0,
X(41) - X(42) ^ 2 - 5 = 0,
X(41) + X(42) + X(43) - 3 = 0,
and
-10<= Integers X(i)<= 20 for i.= 44 through 77.
The first four equations are based on page 510 of Burden and Faires [3]. The next nine equations are based on page 604 of Burden and Faires [2]. The next six equations come from Johnson and Riess [4]-- the first three from p. 194 and the other three from p. 198. The next eight equations are based on pages 510 of Burden and Faires [3, p. 510]. The last seven equations come from p. 648 and p.649 of Burden, Faires, and Burden [1].
One notes line 13, which is 13 A(KNEW) = -10 + FIX(RND * 30.98).
0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
13 A(KNEW) = -10 + FIX(RND * 30.98)
14 RANDOMIZE JJJJ
16 M = -1D+317
91 FOR KK = 44 TO 77
94 A(KK) = A(KNEW)
95 NEXT KK
128 FOR I = 1 TO 20000
129 FOR K = 44 TO 77
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 44 + FIX(RND * 34)
182 IF RND < -.1 THEN 183 ELSE GOTO 189
183 R = (1 - RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = CINT(A(B))
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 2.99) ELSE X(B) = A(B) + FIX(1 + RND * 2.99)
191 NEXT IPP
255 FOR J45 = 44 TO 77
256 IF X(J45) < -10 THEN X(J45) = A(J45)
257 IF X(J45) > 20 THEN X(J45) = A(J45)
259 NEXT J45
451 X(54) = -X(52) ^ 2
453 N44 = X(52) * COS(X(53)) - X(54)
455 N45 = (EXP(X(52) + X(54))) * SIN(X(53) / 2) + X(54)
461 N46 = X(55) ^ 5 + X(56) ^ 3 * X(57) ^ 4 + 1
463 N47 = X(55) * X(56) * X(57)
465 N48 = X(57) ^ 4 - 1
499 X(44) = -2 * X(45) - X(46) - X(47) - X(48) - X(49) - X(50) - X(51) + 9
501 N35 = 2 * X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9
504 N38 = X(44) + X(45) + 2 * X(46) + X(47) + X(48) + X(49) + X(50) + X(51) - 9
505 N39 = X(44) + X(45) + X(46) + 2 * X(47) + X(48) + X(49) + X(50) + X(51) - 9
506 N40 = X(44) + X(45) + X(46) + X(47) + 2 * X(48) + X(49) + X(50) + X(51) - 9
507 N41 = X(44) + X(45) + X(46) + X(47) + X(48) + 2 * X(49) + X(50) + X(51) - 9
508 N42 = X(44) + X(45) + X(46) + X(47) + X(48) + X(49) + 2 * X(50) + X(51) - 9
509 N43 = X(44) * X(45) * X(46) * X(47) * X(48) * X(49) * X(50) * X(51) - 1
601 X(58) = X(59) ^ 2 + 5
602 N49 = X(58) ^ 2 + X(59) - 37
604 N50 = X(58) + X(59) + X(60) - 3
651 N51 = -18 - X(61) * (X(61) + 1) + 2 * X(62)
652 N52 = -25 + (X(61) - 1) ^ 2 + (X(62) - 6) ^ 2
661 N53 = X(63) ^ 2 - 10 * X(63) + X(64) ^ 2 + 8
663 N54 = X(63) * X(64) ^ 2 + X(63) - 10 * X(64) + 8
671 X(67) = -10 + X(65) ^ 2 + 10 * X(66)
673 N55 = -12 + 15 * X(65) + X(66) - 4 * X(67)
675 N56 = 24 + X(66) - 25 * X(67)
681 N57 = -7 + 10 * X(68) - 2 * X(69) ^ 2 + X(69) - 2 * X(70) ^ 2
683 N58 = -12 + 8 * X(69) ^ 2 + 4 * X(70) ^ 2
685 N59 = 8 + 8 * X(69) * X(70)
691 N60 = -5 + 10 * X(71) - 2 * X(72) ^ 2 + X(72) - 2 * X(73) ^ 2
693 N61 = -12 + 8 * X(72) ^ 2 + 4 * X(73) ^ 2
695 N62 = 8 + 8 * X(72) * X(73)
751 X(74) = -X(75) ^ 3 + 5 * X(75) ^ 2 + X(75) + 10
755 N63 = -29 + X(74) + X(75) ^ 3 + X(75) ^ 2 - 14 * X(75)
851 X(76) = -X(77) ^ 3 + 5 * X(77) ^ 2 + 2 * X(77) + 11
855 N64 = -29 + X(76) + X(77) ^ 3 + X(77) ^ 2 - 14 * X(77)
1287 P = -ABS(N44) - ABS(N45) - ABS(N46) - ABS(N47) - ABS(N48) - ABS(N49) - ABS(N50) - ABS(N51) - ABS(N52) - ABS(N53) - ABS(N54) - ABS(N55) - ABS(N56) - ABS(N57) - ABS(N58) - ABS(N59) - ABS(N60) - ABS(N61) - ABS(N62) - ABS(N63) - ABS(N64) - ABS(N65) - ABS(N33) - ABS(N34) - ABS(N35) - ABS(N36) - ABS(N37) - ABS(N38) - ABS(N39) - ABS(N40) - ABS(N41) - ABS(N42) - ABS(N43)
1451 IF P <= M THEN 1670
1657 FOR KEW = 44 TO 77
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -.00001 THEN 1999
1917 PRINT A(44), A(45), A(46), A(47), A(48), A(49), A(50), A(51), M, JJJJ
1918 PRINT A(52), A(53), A(54), A(55), A(56), A(57), A(58), A(59), A(60), A(61), A(62), A(63), A(64), A(65), A(66), A(67), A(68), A(69), A(70), A(71), A(72), A(73), A(74), A(75), A(76), A(77)
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [7]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-23926 is shown below:
1 1 1 1 1
1 1 1 0 -29730
0 0 0 0 -1
1 6 1 -4 -2
10 1 1 1 1
1 1 1 -1 1
-1 1 15 -1 35
3
1 1 1 1 1
1 1 1 0 -27552
0 0 0 0 -1
-1 6 1 -4 -2
10 1 1 1 1
1 1 1 -1 1
-1 1 15 -1 15
-1
1 1 1 1 1
1 1 1 0 -26175
0 0 0 -1 0
-1 6 1 -4 -2
10 1 1 1 1
1 1 1 -1 1
-1 1 15 -1 35
3
1 1 1 1 1
1 1 1 0 -24882
0 0 0 -1 0
1 6 1 -4 -2
10 1 1 1 1
1 1 1 -1 1
-1 1 15 -1 35
3
1 1 1 1 1
1 1 1 0 -23926
0 0 0 0 -1
-1 6 1 -4 -2
10 1 1 1 1
1 1 1 -1 1
-1 1 15 -1 15
-1
Above there is no rounding by hand; it is just straight copying by hand from the screen.
One can ignore the candidate solution at JJJJ=-29730 with A(76)=35 because this candidate solution does not satisfy the requirement -10<= Integers X(i)<= 20 for i.= 44 through 77--see the problem statement above. In contrast, the candidate solution at JJJJ=-27552, for example, satisfies all requirements.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [7], the wall-clock time through JJJJ=-23926 was 22 minutes.
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] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[5] 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.
[6] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[7] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
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