Jsun Yui Wong
The following computer program seeks to solve the system of nonlinear equations of the discrete boundary value problem on page 29 of La Cruz, Martinez, and Raydan [7, p. 29, Test function 41, Discrete boundary value problem]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. Also see More, Garbow, and Hillstrom [11]. The present case has 15719 nonlinear equations and 15719 continuous variables.
One notes line 93, which comes from La Cruz, Martinez, and Raydan [7, p. 29] and is 93 A(KK) = (h * (KK * h – 1)).
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 31111
14 RANDOMIZE JJJJ
16 M = -1D+50
22 h = 1 / (15719 + 1)
91 FOR KK = 1 TO 15719
93 A(KK) = (h * (KK * h – 1))
95 NEXT KK
128 FOR I = 1 TO 5000000 STEP 1
129 FOR K = 1 TO 15719
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15722)
183 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 < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
199 NEXT IPP
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 15718
610 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * J49) ^ 3 – X(J49 – 1) + X(J49 + 1)
613 NEXT J49
615 PS = 0
617 FOR J49 = 2 TO 15718
619 PS = PS – ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15719
633 IF ABS(X(J33)) > 3 THEN 1670
655 NEXT J33
677 PZ = 2 * X(15719) + .5 * h ^ 2 * (X(15719) + h * 15719) ^ 3 – X(15718)
999 P = -ABS(PZ) + PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15719
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(15718), M, JJJJ
1670 NEXT I
1890 REM IF M < -9 THEN 1999
1912 PRINT A(1), A(2), A(3)
1913 PRINT A(4), A(5), A(6)
1937 PRINT A(15714), A(15715), A(15716)
1947 PRINT A(15717), A(15718), A(15719)
1949 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.
-2.15837432252094D-17 … …
… … …
… … …
… -8.376606895933208D-10 -1.43076820705172D-09
-5.6761614663639D-08 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 15719 unknowns, only three of the twelve A’s of line 1912 through line 1947 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 five hours.
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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