Jsun Yui Wong
Suppose the problem is to find an integer solution to Problem 4 in Cao [2, p. 7]--see also La Cruz et al. [5] and Han and Han [4]. Here the case of 1010 nonlinear equations with 1010 integer variables is considered. The following computer program and its output illustrate.
0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
22 h = 1 / (1011)
91 FOR KK = 1 TO 1010
93 IF RND < .5 THEN A(KK) = 0 ELSE A(KK) = 1
94 REM A(KK) = 2 * RND * (h * (KK * h - 1))
95 NEXT KK
128 FOR I = 1 TO 1000000 STEP 1
129 FOR K = 1 TO 1010
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 1013)
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 IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
191 NEXT IPP
555 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
566 X(1009) = 2 * X(1010) + .5 * h ^ 2 * (X(1010) + h * (1010)) ^ 3
605 FOR J49 = 2 TO 1009
609 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * (J49)) ^ 3 - X(J49 - 1) + X(J49 + 1)
611 NEXT J49
711 P = 0
714 FOR J44 = 2 TO 1009
722 P = P - ABS(P(J44))
733 NEXT J44
999 P = P
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 1010
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.5 THEN 1999
1912 PRINT A(1), A(2), A(3)
1913 PRINT A(4), A(5), A(6)
1914 PRINT A(7), A(8), A(9)
1915 PRINT A(10), A(11), A(12)
1917 PRINT A(1001), A(1002), A(1003)
1919 PRINT A(1004), A(1005), A(1006)
1920 PRINT A(1007), A(1008), A(1009)
1939 PRINT A(1010), M, JJJJ
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= -32000 is shown below.
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 -1.229078E-04 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 1010 unknowns, only the 22 A's of line 1912 through line 1939 are shown above.
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 for obtaining the output through JJJJ= -32000 was 6 minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] Huiping Cao, Global Convergence of Schubert's Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587
[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[4] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am
[5] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
www.ime.unicamp.br/~martinez/lmrreport.pdf
[6] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[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] 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
[9] 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.
[10] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[12] M. Ziani, F. Guyomarc'h, An Autoadaptive Limited Memory Broyden's Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf
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