Jsun Yui Wong
The following computer program seeks to solve the following system of thirteen equations; see Morgan [3, page 15] and Floudas [1, page 660]. The problem is to solve simultaneously
X(1) * X(2) * X(3) * X(4) * X(5) * X(6) * X(7) * X(8) * X(9) * X(10) * X(11) * X(12) * X(13) – 1 = 0
2 * X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14= 0
X(1) +2 * X(2) + X(3) + X(4) + X(5)+ X(6)+ X(7)+ X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14 = 0
X(1) + X(2) + 2 * X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + 2 * X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + 2 * X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + 2 * X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + 2 * X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + 2 * X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + 2 * X(9) + X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + 2 * X(10) + X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + 2 * X(11) + X(12) + X(13) – 14=0
X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + 2 * X(12) + X(13) – 14=0.
The following computer program uses qb64v1000-win [5, 6].
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(42), A(42), L(33), K(33)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 13
94 A(KK) = -2 + RND * 4
95 NEXT KK
128 FOR I = 1 TO 120000 STEP 1
129 FOR K = 1 TO 13
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 13)
183 R = (1 – RND * 2) * A(B)
187 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))
191 NEXT IPP
195 X(1) = -2## * X(2) – X(3) – X(4) – X(5) – X(6) – X(7) – X(8) – X(9) – X(10) – X(11) – X(12) – X(13) + 14##
222 N1 = 2## * X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
224 N3 = X(1) + X(2) + 2## * X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
225 N4 = X(1) + X(2) + X(3) + 2## * X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
226 N8 = X(1) + X(2) + X(3) + X(4) + 2## * X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
228 N9 = X(1) + X(2) + X(3) + X(4) + X(5) + 2## * X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
231 NZ = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + 2## * X(7) + X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
232 NL = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + 2## * X(8) + X(9) + X(10) + X(11) + X(12) + X(13) – 14##
236 NM = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + 2## * X(9) + X(10) + X(11) + X(12) + X(13) – 14##
238 NN = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + 2## * X(10) + X(11) + X(12) + X(13) – 14##
240 NP = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + 2## * X(11) + X(12) + X(13) – 14##
242 NQ = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) + 2## * X(12) + X(13) – 14##
244 N5 = X(1) * X(2) * X(3) * X(4) * X(5) * X(6) * X(7) * X(8) * X(9) * X(10) * X(11) * X(12) * X(13) – 1##
277 P = -ABS(N1) – ABS(N3) – ABS(N4) – ABS(N5) – ABS(N8) – ABS(N9) – ABS(NZ) – ABS(NL) – ABS(NM) – ABS(NN) – ABS(NP) – ABS(NQ)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 13
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.005 THEN 1999
1912 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), A(12), A(13), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [5, 6]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= 32000 is shown below.
.999697940738728 .999698002721541 .999697978711392
.9996980142792856 .9996980051894986 .9996979692564104
.9996979966422026 .9996980123474062 .9996979962335975
.9996980012819657 .999697948546023 .999697730050915
1.003926401279494 -2.942887415575626D-04 -26146
.9951717968297974 .9951718665348447 .9951718835083987
.9951713678410493 .9951717110787454 .9951718913304046
.995171893752952 .9951712689920377 .9951718989876394
.9951717484025024 .9951718129945611 .9951718602386652
1.062767132973558 -2.802201310676587D-03 -14624
.9864570074909738 .9864570036942828 .9864570010887167
.9864570142216348 .9864570109341728 .9864570074920177
.9864570038377301 .9864570334396015 .9864570052552653
.9864570044212241 .9864570036987831 .9864570029470811
1.176058897784234 -1.45648299738119D-03 4587
1 1 1 1 1
1 1 1 1 1
1 1 1 0 31159
Above there is no rounding by hand; it is just straight copying by hand from the screen.
The solutions at JJJJ= 4587 and at JJJJ= 31159 are distinct solutions.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with
qb64v1000-win [5, 6], the wall-clock time for obtaining the output through JJJJ= 32000 was five hours and ten minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[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] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf
[4] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[5] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY! issue70/#galleoninterview
[6] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
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