The following computer program seeks to solve an enlarged version of Example 3 on page 290 of Ge, Liu, and Xu [4, page 290, modified Broyden tridiagonal function]– http://www.scirp.org/journal/ojapps The present paper considers the case of 4000 equations with 4000 variables. One notes the starting vectors, 94 A(KK) = FIX(RND * 6).
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
15 REM h = 1 / (32760 + 1)
16 M = -1D+50
91 FOR KK = 1 TO 4000
94 A(KK) = FIX(RND * 6)
95 NEXT KK
128 FOR I = 1 TO 12000 STEP 1
129 FOR K = 1 TO 4000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 4003)
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
191 NEXT IPP
555 X(2) = ((3 - 2 * X(1)) * X(1) + 1) / 2
605 FOR J44 = 2 TO 3999
609 X(J44 + 1) = ((3 - 2 * X(J44)) * X(J44) - X(J44 - 1) + 2) / 2
611 NEXT J44
699 P1 = (3 - 2 * X(4000)) * X(4000) - X(3999)
999 P = -ABS(P1)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(4000), M, JJJJ
1668 IF M > -.000000000001 THEN 1912
1670 NEXT I
1890 REM IF M < -5555 THEN 1999
1912 PRINT A(1), A(2), A(3)
1915 PRINT A(3997), A(3998), A(3999)
1917 PRINT A(4000), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [12]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31994 is shown below.
3.903018182689294 -8.879023660369297 -93.1071057432964
-1.#IND -1.#IND -1.#IND
-1.#IND -1.#IND -32000
0 .5 1.5
1 1 1
1 0 -31999
4.34741353824193 -11.87888416512631 -160.0999220252991
-1.#IND -1.#IND -1.#IND
-1.#IND -1.#IND -31998
1.653270079584755 .24660316332695 .482456585035189
1 1 1
1 0 -31997
3.02737948516394 -4.123957319445566 -23.70664969435698
-1.#IND -1.#IND -1.#IND
-1.#IND -1.#IND -31996
3.613930553710306 -7.139598216475422 -62.49022529426731
-1.#IND -1.#IND -1.#IND
-1.#IND -1.#IND -31995
1 1 1
1 1 1
1 0 -31994
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 4000 unknowns, only the 7 A’s of line 1912 through 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 [12], the wall-clock time for obtaining the output through JJJJ= -31994 was four 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] 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
[5] 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
[6] 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
[7] 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.
[8] 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.
[9] 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
[10] 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.
[11] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[13] 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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