The following computer program seeks to solve the nonlinear system of equations on page 25 of Remani [13, page 25]. This system comes from the boundary value problem of nonlinear ordinary differential equation on page 23 of Remani [13, page 23] and on page 710 of Burden and Faires [1, page 710]. The present problem has 19 nonlinear equations with 19 unknowns.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), C(22), P(22)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
31 C(1) = .432: C(2) = .5495: C(3) = .686: C(4) = .84375: C(5) = 1.024: C(6) = 1.22825: C(7) = 1.458: C(8) = 1.71475: C(9) = 2
35 C(10) = 2.31525: C(11) = 2.662: C(12) = 3.04175: C(13) = 3.456: C(14) = 3.90625: C(15) = 4.394: C(16) = 4.92075: C(17) = 5.488
91 FOR KK = 1 TO 19
94 A(KK) = 10 + RND * 10
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 19
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 19)
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 ^ 2 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 5 * R
191 NEXT IPP
222 X(1) = (17 + X(2) - .0433275) / (2 + (.01 * (X(2) - 17)) / 1.6)
605 FOR J44 = 1 TO 17
608 X(J44 + 2) = (X(J44) - 2 * X(J44 + 1) - .01 * (4 + C(J44)) + (.01 * X(J44 + 1) * (X(J44)) / (1.6))) / (-1 + (.01 * X(J44 + 1)) / 1.6)
611 NEXT J44
615 FOR J46 = 1 TO 19
617 IF X(J46) < 0 THEN 1670
619 NEXT J46
624 PNEW = ABS(-X(18) + 2 * X(19) + .01 * (4 + 6.09725 + X(19) * (14.333333 - X(18)) / (1.6)) - 14.333333)
1111 P = -PNEW
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 19
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.00001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(4), A(5), A(6)
1939 PRINT A(7), A(8), A(9)
1940 PRINT A(10), A(11), A(12)
1941 PRINT A(13), A(14), A(15)
1942 PRINT A(16), A(17), A(18), A(19)
1946 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.
16.76053994626924 16.51343842564263 16.25888121383656
15.9973779663645 15.72983918588467 15.45767932518904
15.18292147311006 14.90831516541902 14.63746486371606
14.37496937262407 14.1265735915981 13.8993361360806
13.70181997224838 13.54431892112646 13.43914162426204
13.40098778053801 13.44747162587545 13.59987913254232
13.88429641641846
-1.844179150882475D-08 -32000
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 [14], the wall-clock time for obtaining the output through JJJJ= -32000 was thirty seconds, not including “Creating .EXE file” time.
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] 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
[5] 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
[6] 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
[7] 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.
[8] 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.
[9] 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
[10] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
[11] 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.
[12] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[13] Courtney Remani. Numerical Methods for Solving Systems of Nonlinear Equations. https://www.lakeheadu.ca/sites/default/files/updates/77/docs/RemaniFinal.pdf.
[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[15] 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
No comments:
Post a Comment