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 et al. [11]. The present case has 2719 nonlinear equations and 2719 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 / (2719 + 1)

91 FOR KK = 1 TO 2719

93 A(KK) = (h * (KK * h - 1))

95 NEXT KK

128 FOR I = 1 TO 2000000 STEP 1

129 FOR K = 1 TO 2719

131 X(K) = A(K)

132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 2722)

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 < .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

566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3

605 FOR J49 = 2 TO 2718

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 2718

619 PS = PS - ABS(P(J49))

629 NEXT J49

622 FOR J33 = 1 TO 2719

633 IF ABS(X(J33)) > 3 THEN 1670

655 NEXT J33

677 PZ = 2 * X(2719) + .5 * h ^ 2 * (X(2719) + h * 2719) ^ 3 - X(2718)

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670

1657 FOR KEW = 1 TO 2719

1658 A(KEW) = X(KEW)

1659 NEXT KEW

1661 M = P

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)

1914 PRINT A(7), A(8), A(9)

1915 PRINT A(10), A(11), A(12)

1927 PRINT A(2711), A(2712), A(2713)

1937 PRINT A(2714), A(2715), A(2716)

1947 PRINT A(2717), A(2718), A(2719)

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 complete output through JJJJ= -32000 is shown below.

-1.19960417268234D-20 3.334354272574636D-18 -2.925819998435894D-18

-8.591990681672726D-17 -4.256224601882662D-17 -2.94393417772928D-16

-1.796171628930897D-16 -3.945869991029357D-16 -4.123734581215336D-16

-1.524667708237257D-15 -1.021015743561118D-15 -2.487127024791742D-15

-3.343101267297926D-08 -3.342896885699016D-08 -3.356462642178777D-08

-3.336033559667833D-08 -3.397986304410705D-08 -3.261254487807038D-08

-3.60413779066783D-08 -2.788851782937236D-08 -4.769825161337166D-08

-3.165101733156332D-09 -32000

Above there is no rounding by hand; it is just straight copying by hand from the screen.

Of the 2719 unknowns, only the 21 A’s of line 1912 and 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 30 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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