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 discrete boundary value problem in Cao [2, p. 7] and in Han and Han [4, p. 228].
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
16 M = -1D+50
22 h = 1 / (21)
91 FOR KK = 1 TO 20
94 A(KK) = RND * (-3)
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 20
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 20)
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 X(B) = INT(A(B))
191 NEXT IPP
555 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
566 X(19) = 2 * X(20) + .5 * h ^ 2 * (X(20) + h * (20)) ^ 3
605 FOR J49 = 3 TO 18
609 X(J49) = -2 * X(J49 – 1) – .5 * h ^ 2 * (X(J49 – 1) + h * (J49 – 1)) ^ 3 + X(J49 – 2)
611 NEXT J49
655 P1 = 2 * X(18) + .5 * h ^ 2 * (X(18) + h * (18)) ^ 3 – X(17) + X(19)
688 P2 = 2 * X(19) + .5 * h ^ 2 * (X(19) + h * (19)) ^ 3 – X(18) + X(20)
999 P = -ABS(P1) – ABS(P2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 20
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.0000000005 THEN 1999
1912 PRINT A(1), A(2), A(3)
1915 PRINT A(4), A(5), A(6)
1916 PRINT A(7), A(8), A(9)
1917 PRINT A(10), A(11), A(12)
1918 PRINT A(13), A(14), A(15)
1919 PRINT A(16), A(17), A(18)
1939 PRINT A(19), A(20), 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= -31783 is shown below.
-1.63788436818447D-07 -2.051521733951551D-07 -7.328854689073154D-07
-2.044831377003442D-06 -4.478232039118847D-06 -8.39074925714514D-06
-1.413841191862338D-05 -2.210068781554012D-05 -3.26082208635711D-05
-4.611240338779846D-05 -6.277381530349364D-05 -8.345515337503525D-05
-1.073228977085603D-04 -1.376393319470796D-04 -1.677730490905885D-04
-2.149897776538251D-04 -2.388258657996598D-04 -3.382846166917472D-04
-2.753998249101887D-04 -6.264380791706676D-04 -1.187929732179757D-12
-31783
Above there is no rounding by hand; it is just straight copying by hand from the screen.
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= -31783 was 8 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
computerprogramsandresults 