The following computer program seeks to find an integer solution to the Cragg and Levy nonlinear system of equations in La Cruz et al. [5, page 28, Test function 38]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 500 equations with 500 general integer variables. One notes the starting vectors, 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 500
94 A(KK) = FIX(RND * 1.9)
96 NEXT KK
128 FOR I = 1 TO 2400000 STEP 1
129 FOR K = 1 TO 500
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 503)
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
393 FOR J44 = 1 TO 125
395 X(4 * J44) = 1
397 NEXT J44
773 FOR J44 = 1 TO 125
775 P(4 * J44 - 3) = -ABS((EXP(X(4 * J44 - 3)) - X(4 * J44 - 2)) ^ 2)
777 NEXT J44
822 Pone = 0
833 FOR J44 = 1 TO 125
837 Pone = Pone + P(4 * J44 - 3)
855 NEXT J44
873 FOR J44 = 1 TO 125
875 P(4 * J44 - 2) = -ABS(10 * (X(4 * J44 - 2) - X(4 * J44 - 1)) ^ 3)
877 NEXT J44
878 Ptwo = 0
879 FOR J44 = 1 TO 125
880 Ptwo = Ptwo + P(4 * J44 - 2)
881 NEXT J44
883 FOR J44 = 1 TO 125
885 P(4 * J44 - 1) = -ABS((TAN(X(4 * J44 - 1) - X(4 * J44))) ^ 2)
887 NEXT J44
892 Pthr = 0
893 FOR J44 = 1 TO 125
897 Pthr = Pthr + P(4 * J44 - 1)
899 NEXT J44
998 P = Pone + Ptwo + Pthr
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 500
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(500), 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)
1939 PRINT A(499), A(500), 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= -31962 is summarized below.
0 1 1 1 0
1 1 1 0 1
1 1 -8.824343 -32000
0 1 1 1 0
1 1 1 0 1
1 1 -7.546016 -31999
0 1 1 1 0
1 1 1 0 1
1 1 -5.882895 -31998
0 1 1 1 0
1 1 1 0 1
1 1 -2.941447 -31997
0 1 1 1 0
1 1 1 0 1
1 1 -2.941447 -31996
0 1 1 1 0
1 1 1 0 1
1 1 -2.941447 -31995
0 1 1 1 0
1 1 1 0 1
1 1 -2.941447 -31994
0 1 1 1 0
1 1 1 0 1
1 1 -2.941447 -31993
.
.
.
0 1 1 1 0
1 1 1 0 1
1 1 0 -31962
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 500 unknowns, only the 12 A’s of line 1891 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= -31962 was 2 hours.
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
No comments:
Post a Comment