Jsun Yui Wong
The following computer program seeks an integer solution to the tridiagonal system of equations on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 34, Tridiagonal system]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present case has 5019 nonlinear equations and 5019 general integer variables.
0 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
91 FOR KK = 1 TO 5019
94 A(KK) = RND * 3
95 NEXT KK
128 FOR I = 1 TO 500000 STEP 1
129 FOR K = 1 TO 5019
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 5022)
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
188 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 7 * R
199 NEXT IPP
577 X(1) = X(2) ^ 2
605 FOR J49 = 2 TO 5018
610 P(J49) = 8 * X(J49) * (X(J49) ^ 2 – X(J49 – 1)) – 2 * (1 – X(J49)) + 4 * (X(J49) – X(J49 + 1) ^ 2)
611 NEXT J49
615 PS = 0
617 FOR J49 = 2 TO 5018
619 PS = PS – ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 5019
633 IF ABS(X(J33)) > 5 THEN 1670
655 NEXT J33
666 PZ = 8 * X(5019) * (X(5019) ^ 2 – X(5018)) – 2 * (1 – X(5019))
999 P = -ABS(PZ) + PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 5019
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1663 PRINT A(1), A(2), A(3), A(4), A(5)
1664 PRINT A(5015), A(5016), A(5017), A(5018), A(5019)
1665 PRINT M, JJJJ
1668 IF M > -.000001 THEN 1890
1670 NEXT I
1890 IF M < -.000001 THEN 1999
1912 PRINT A(1), A(2), A(3), A(4), A(5)
1917 PRINT A(5015), A(5016), A(5017), A(5018), A(5019)
1939 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [16]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.
.
.
.
1 1 3 1 2
0 1 1 0 1
-246534 -32000
.
.
.
1 1 2 1 2
0 1 1 0 1
-215152 -32000
.
.
.
4 2 2 1 2
0 1 1 1 1
-161392 -32000
.
.
.
1 1 1 1 1
1 1 1 1 1
-126 -32000
1 1 1 1 1
1 1 1 1 1
-100 -32000
1 1 1 1 1
1 1 1 1 1
-74 -32000
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
0 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 5019 unknowns, only the ten A’s of line 1912 and line 1917 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 [16], the wall-clock time for obtaining the output through JJJJ= -32000 was ten minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] 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.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] 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.
[7] 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.
[8] 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.
[9] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207. .
[10] 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.
[11 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[12] 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.
[13] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[14] 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.
[15] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[17] 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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