Jsun Yui Wong
The following computer program seeks to solve simultaneously Brown’s almost linear system of forty equations; see Morgan [3, page 15], Floudas [1, page 660], and the preceding paper.
The following computer program uses qb64v1000-win [5, 6].
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(42), A(42), L(43), K(43)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 40
94 A(KK) = RND * 5
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 40
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 40)
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
391 IF (X(2) * X(3) * X(4) * X(5) * X(6) * X(7) * X(8) * X(9) * X(10) * X(11) * X(12) * X(13) * X(14) * X(15) * X(16) * X(17) * X(18) * X(19) * X(20) * X(21) * X(22) * X(23) * X(24) * X(25) * X(26) * X(27) * X(28) * X(29) * X(30) * X(31) * X(32) * X(33) * X(34) * X(35) * X(36) * X(37) * X(38) * X(39) * X(40)) < .001 THEN 1670
401 X(1) = (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) * X(14) * X(15) * X(16) * X(17) * X(18) * X(19) * X(20) * X(21) * X(22) * X(23) * X(24) * X(25) * X(26) * X(27) * X(28) * X(29) * X(30) * X(31) * X(32) * X(33) * X(34) * X(35) * X(36) * X(37) * X(38) * X(39) * X(40))
501 summ = 0
505 FOR j27 = 1 TO 40
511 summ = summ + X(j27)
521 NEXT j27
881 P1 = -ABS(X(1) + summ – 41) – ABS(X(2) + summ – 41) – ABS(X(3) + summ – 41) – ABS(X(4) + summ – 41) – ABS(X(5) + summ – 41) – ABS(X(6) + summ – 41) – ABS(X(7) + summ – 41) – ABS(X(8) + summ – 41) – ABS(X(9) + summ – 41) – ABS(X(10) + summ – 41) – ABS(X(11) + summ – 41) – ABS(X(12) + summ – 41) – ABS(X(13) + summ – 41) – ABS(X(14) + summ – 41) – ABS(X(15) + summ – 41) – ABS(X(16) + summ – 41) – ABS(X(17) + summ – 41) – ABS(X(18) + summ – 41) – ABS(X(19) + summ – 41) – ABS(X(20) + summ – 41) – ABS(X(21) + summ – 41) – ABS(X(22) + summ – 41) – ABS(X(23) + summ – 41) – ABS(X(24) + summ – 41) – ABS(X(25) + summ – 41) – ABS(X(26) + summ – 41) – ABS(X(27) + summ – 41) – ABS(X(28) + summ – 41) – ABS(X(29) + summ – 41) – ABS(X(30) + summ – 41)
885 P2 = -ABS(X(31) + summ – 41) – ABS(X(32) + summ – 41) – ABS(X(33) + summ – 41) – ABS(X(34) + summ – 41) – ABS(X(35) + summ – 41) – ABS(X(36) + summ – 41) – ABS(X(37) + summ – 41) – ABS(X(38) + summ – 41) – ABS(X(39) + summ – 41)
891 P = P1 + P2
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 40
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.00001 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), A(14), A(15)
1914 PRINT A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23), A(24), A(25)
1917 PRINT A(26), A(27), A(28), A(29), A(30), A(31), A(32), A(33), A(34), A(35)
1918 PRINT A(36), A(37), A(38), A(39), A(40), 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= -31991 is shown below.
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31998
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31997
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31995
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31994
1.000000073892815 1.00000003180202 1.000000026989913
1.000000035454121 1 1 1.000000000055708
1.00000004653717 .9999999999999302 1
1 1.0 1 1 1
1.00000000669753 1 1.000000033230602
1 1 1.000000016585487 1.000000019814557
1.000000000173704 1.000000000062483 1.00000000584586
1.00000001255953 1.000000016767502 1
1 1 1.000000015390592 1.000000004933208
1 1.000000025389344 1.00000000917817
1 1.00000001863862 1.00000001370827
1.000000028575267 .9999995577287065 -4.422744466214823D-07
-31993
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31992
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 -31991
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 [5, 6], the wall-clock time for obtaining the output through JJJJ= -31991 was one minute.
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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