Jsun Yui Wong
“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 1993, p. 355].
“We make an extreme, but wholly defensible, statement: There are no good, general methods for solving systems of more than one nonlinear equation. Furthermore, it is not hard to see why (very likely) there never will be any good, general methods,” Press, Teukolsky, Vetterling, and Flannery [7, 2007, p. 473].
“Solving a system of nonlinear equations is a problem that is avoided where possible, customarily by approximating the nonlinear system by a system of linear equations. When this is unsatisfactory, the problem must be tackled directly,” Burden, Faires, and Burden [1, 2016, page 642].
Using qb64v1000-win [9], the following computer program seeks to solve the tridiagonal exponential problem in Cao [2, p. 7, Problem 3]; here the case of 155 simultaneous equations/unknowns is considered.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(2222), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
22 h = 1 / (156)
91 FOR KK = 1 TO 155
94 A(KK) = RND * (3)
95 NEXT KK
128 FOR I = 1 TO 3000000 STEP 1
129 FOR K = 1 TO 155
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 155)
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
770 P(1) = X(1) – EXP(COS(h * (X(1) + X(2))))
772 P(155) = X(155) – EXP(COS(h * (X(154) + X(155))))
773 FOR J44 = 2 TO 154
775 P(J44) = -ABS(X(J44) – EXP(COS(h * (X(J44 – 1) + X(J44) + X(J44 + 1)))))
777 NEXT J44
822 P = 0
833 FOR J44 = 2 TO 154
837 P = P + P(J44)
855 NEXT J44
999 P = -ABS(P(1)) + P – ABS(P(155))
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 155
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.5 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(152), A(153), A(154)
1939 PRINT A(155), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.
2.716635055455467 2.714579391239117 2.714581256299916
2.714581254608846 2.714581256299776 2.714579391238663
2.716635055455428 -1.887380030041186D-09 -32000
2.71663505544694 2.714579391238138 2.714581256315673
2.714581254610846 2.714581256301268 2.714579391234607
2.716635055457201 -1.343718913915382D-09 -31999
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 155 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 [9], the wall-clock time for obtaining the output through JJJJ= -31999 was 6 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] 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.
[6] 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
[7] 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.
[8] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[9] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[10] 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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