Jsun Yui Wong
"Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations," Rice [7, 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 [6, 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 [8], the following computer program seeks to solve simultaneously Brown's almost linear system of 15000 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. While line 94 and line 128 of the preceding paper are 94 A(KK) = RND * 3 and 128 FOR I = 1 TO 2500000 STEP 1, respectively, here line 94 and line 128 are 94 A(KK) = RND * 2.75 and 128 FOR I = 1 TO 6000000 STEP 1, respectively.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(15003), A(15003), L(15003), K(15003)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 15000
94 A(KK) = RND * 2.75
95 NEXT KK
128 FOR I = 1 TO 6000000 STEP 1
129 FOR K = 1 TO 15000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15000)
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) = CINT(A(B))
191 NEXT IPP
301 prodd = 1
305 FOR j55 = 2 TO 15000
311 prodd = prodd * X(j55)
321 NEXT j55
371 IF prodd < .00001 THEN 1670
389 X(1) = (1) / prodd
501 summ = 0
505 FOR j27 = 1 TO 15000
511 summ = summ + X(j27)
521 NEXT j27
901 DIFF = 0
905 FOR J77 = 1 TO 14999
911 DIFF = DIFF - ABS(X(J77) + summ - 15001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.0001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1915 PRINT A(4), A(5), A(6)
1917 PRINT A(7), A(8), A(9)
1930 PRINT A(14991), A(14992), A(14993)
1931 PRINT A(14994), A(14995), A(14996)
1933 PRINT A(14997), A(14998), A(14999)
1939 PRINT A(15000), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.
.9999995760742063 1 1
1 1 1
.9999999999168048 1 1
1 1 1
1 1 1
1 1 1
1 -2.657079987766764D-05 -31999
1.000000002575082 1 1
1.000000511925313 1 1
1 1 1
1 1 1
1 1 1
1 1 1
.9999947204554103 -2.928759216125343D-05 -31997
Above there is no rounding by hand; it is just straight copying by hand from the screen.
One notes that of the 15000 values for the 15000 unknowns, only the 19 A's of line 1912 through line 1939 of the computer program above are shown above and that the M of line 1939 is essentially defined in line 911, line 995, and line 1661, which are 911 DIFF = DIFF - ABS(X(J77) + summ - 15001), 995 P = DIFF, and 1661 M = P, respectively. This M includes 14999 absolutes values.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time for obtaining the output through JJJJ= -31997 was eight hours.
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] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[3] 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
[4] 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.
[5] 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
[6] 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.
[7] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
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