The following computer program seeks to find an integer solution to the trigonometric system on page 22 of La Cruz et al. [5, page 22, Test function 8, Trigonometric function]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 10000 equations with 10000 integer variables. One notes the starting vector, 94 A(KK) = FIX(RND * 1.9); one also notes line 189, which is 189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1.
0 REM DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 10000
94 A(KK) = FIX(RND * 1.9)
96 NEXT KK
128 FOR I = 1 TO 2400000 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
189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1
191 NEXT IPP
211 nsum = 0
222 FOR J44 = 1 TO 10000
225 nsum = nsum + COS(X(J44))
228 NEXT J44
333 REM
774 FOR J44 = 1 TO 10000
775 P(J44) = -ABS(2 * (10000 + J44 * (1 - COS(X(J44))) - SIN(X(J44)) - nsum) * (2 * SIN(X(J44)) - COS(X(J44))))
777 NEXT J44
822 Pone = 0
833 FOR J44 = 1 TO 10000
837 Pone = Pone + P(J44)
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 REM PRINT A(1), A(10000), M, JJJJ
1668 IF M > -.00001 THEN 1891
1670 NEXT I
1891 PRINT A(1), A(2), A(3), A(4), A(5)
1892 PRINT A(6), A(7), A(8), A(9), A(10)
1897 PRINT A(9996), A(9997), A(9998), A(9999), A(10000)
1949 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 complete output through JJJJ= -31999 is shown below.
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-850.6778 -32000
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 -31999
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 10000 unknowns, only the 15 A’s of line 1891 through 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= -31999 was 2 hours and 20 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.
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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