The following computer program seeks to find an integer solution to Problem 14 in Cao [2, page 11, Problem 14 (extended Freundenstein and Roth function (n is even))]–http://dx.doi.org/10.1155/2014/251587. See also La Cruz et al. [5, page 28, Test function 37]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 100 equations with 100 variables. One notes the starting vectors, 95 IF RND < .5 THEN A(KK) = 3 ELSE A(KK) = 4.
0 DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 100
95 IF RND < .5 THEN A(KK) = 3 ELSE A(KK) = 4
96 NEXT KK
128 FOR I = 1 TO 120000 STEP 1
129 FOR K = 1 TO 100
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 103)
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
393 FOR J44 = 1 TO 50
395 X(2 * J44 - 1) = -((X(2 * J44) + 1) * X(2 * J44) - 14) * X(2 * J44) + 29
397 NEXT J44
773 FOR J44 = 1 TO 50
775 P(2 * J44 - 1) = -ABS(X(2 * J44 - 1) + ((5 - X(2 * J44)) * X(2 * J44) - 2) * X(2 * J44) - 13)
777 NEXT J44
822 Pone = 0
833 FOR J44 = 1 TO 50
837 Pone = Pone + P(2 * J44 - 1)
855 NEXT J44
998 P = Pone
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 100
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1668 IF M > -.00001 THEN 1891
1670 NEXT I
1891 PRINT A(1), A(2), A(3), A(4), A(5)
1892 PRINT A(6), A(7), A(8), A(9), A(10)
1893 PRINT A(11), A(12), A(13), A(14), A(15)
1894 PRINT A(16), A(17), A(18), A(19), A(20)
1895 PRINT A(21), A(22), A(23), A(24), A(25)
1896 PRINT A(26), A(27), A(28), A(29), A(30)
1897 PRINT A(31), A(32), A(33), A(34), A(35)
1898 PRINT A(36), A(37), A(38), A(39), A(40)
1899 PRINT A(41), A(42), A(43), A(44), A(45)
1900 PRINT A(46), A(47), A(48), A(49), A(50)
1901 PRINT A(51), A(52), A(53), A(54), A(55)
1902 PRINT A(56), A(57), A(58), A(59), A(60)
1903 PRINT A(61), A(62), A(63), A(64), A(65)
1904 PRINT A(66), A(67), A(68), A(69), A(70)
1905 PRINT A(71), A(72), A(73), A(74), A(75)
1906 PRINT A(76), A(77), A(78), A(79), A(80)
1907 PRINT A(81), A(82), A(83), A(84), A(85)
1908 PRINT A(86), A(87), A(88), A(89), A(90)
1909 PRINT A(91), A(92), A(93), A(94), A(95)
1910 PRINT A(96), A(97), A(98), A(99), A(100)
1939 PRINT 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= -31852 is summarized below.
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 29
0 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 29 0 5 4
5 4 5 4 5
4 5 4 5 4
5 4 29 0 5
4 5 4 5 4
5 4 5 4 5
4 5 4 29 0
-64 -32000
.
.
.
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
0 -31919
.
.
.
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
5 4 5 4 5
4 5 4 5 4
0 -31852
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 [11], the wall-clock time for obtaining the output through JJJJ= -31852 was 4 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
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