The following computer program seeks to find an integer solution to the Freundenstein and Roth system on page 11 of Cao [2, page 11, Problem 14, extended Freundenstein and Roth function (n is even)]--http://dx.doi.org/10.1155/2014/251587. See also La Cruz et al. [5, page 28, Test function 37]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 10000 equations with 10000 general integer variables. One notes the starting vectors, 94 A(KK) = 4 + FIX(RND * 2.9); one also notes lines 426 and line 428, which are 426 IF X(J44) < 3 THEN X(J44) = 3 and 428 IF X(J44) > 7 THEN X(J44) = 7, respectively.
0 REM DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)
5 FOR JJJJ = -32000 TO -32000
14 RANDOMIZE JJJJ
16 M = -1D+200
91 FOR KK = 1 TO 10000
94 A(KK) = 4 + FIX(RND * 2.9)
96 NEXT KK
128 FOR I = 1 TO 240000 STEP 1
129 FOR K = 1 TO 10000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 10003)
183 REM R = (1 - RND * 2) * A(B)
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 IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
188 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1
191 NEXT IPP
393 FOR J44 = 1 TO 5000
395 X(2 * J44 - 1) = -((X(2 * J44) + 1) * X(2 * J44) - 14) * X(2 * J44) + 29
397 NEXT J44
422 FOR J44 = 1 TO 10000
426 IF X(J44) < 3 THEN X(J44) = 3
428 IF X(J44) > 7 THEN X(J44) = 7
439 NEXT J44
773 FOR J44 = 1 TO 5000
775 P(2 * J44 - 1) = -ABS(X(2 * J44 - 1) + ((5 - X(2 * J44)) * X(2 * J44) - 2) * X(2 * J44) - 13)
777 NEXT J44
822 Pone = 0
833 FOR J44 = 1 TO 5000
837 Pone = Pone + P(2 * J44 - 1)
855 NEXT J44
998 P = Pone
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(10000), M, JJJJ
1668 IF M > -.00001 THEN 1912
1670 NEXT I
1912 IF M < -8 THEN 1999
1891 PRINT A(1), A(2), A(3), A(4), A(5)
1897 PRINT A(9996), A(9997), A(9998), A(9999), A(10000)
1939 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.
.
.
.
5 4 -42 -32000
5 4 -36 -32000
5 4 -30 -32000
5 4 -24 -32000
5 4 -18 -32000
5 4 -12 -32000
5 4 -6 -32000
5 4 0 -32000
5 4 5 4 5
4 5 4 5 4
0 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 10000 unknowns, only the 10 A’s of line 1891 and line 1897 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 3 minutes and a half.
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.
http://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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