Jsun Yui Wong

The following computer program seeks to solve the Troesch problem on page 29 of La Cruz et al. [5, page 29, Test function 43, Troesch problem].–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 20000 equations with 20000 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 h = 1 / (20000 + 1)

16 M = -1D+50

91 FOR KK = 1 TO 20000

94 A(KK) = FIX(RND * 6)

95 NEXT KK

128 FOR I = 1 TO 480000 STEP 1

129 FOR K = 1 TO 20000

131 X(K) = A(K)

132 NEXT K

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

181 B = 1 + FIX(RND * 20003)

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) = 2 * X(1) + 10 * h ^ 2 * SIN((h) * (10 * X(1)))

605 FOR J44 = 2 TO 19999

609 X(J44 + 1) = 2 * X(J44) + 10 * h ^ 2 * SIN((h) * (10 * X(J44))) - X(J44 - 1)

611 NEXT J44

699 P1 = 2 * X(20000) + 10 * h ^ 2 * (SIN((h)) * (10 * X(20000))) - X(19999)

999 P = -ABS(P1)

1451 IF P <= M THEN 1670

1657 FOR KEW = 1 TO 20000

1658 A(KEW) = X(KEW)

1659 NEXT KEW

1661 M = P

1666 PRINT A(1), A(2), A(20000), M, JJJJ

1668 IF M > -.0000000001 THEN 1912

1670 NEXT I

1890 IF M < -5 THEN 1999

1912 PRINT A(1), A(2), A(3)

1917 PRINT A(19997), A(19998), A(19999)

1939 PRINT A(20000), 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= -32000 is summarized below.

.

.

.

.9734582262498015 1.94691645251177 19470.2238435021

-19471.19740224601 -32000

0 0 0 0 -32000

0 0 0

0 0 0

0 0 -32000

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

Of the 20000 unknowns, only the 7 A’s of line 1912 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= -32000 was 16 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] 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

## Comments

You can follow this conversation by subscribing to the comment feed for this post.