The following computer program seeks to solve the Broyden case on page 23 of La Cruz, Martinez, and Raydan [7, p. 23, Test function 11, Broyden Tridiagonal function]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Broyden [1, p. 587], More, Garbow, Hillstrom [10, page 28], and Cao [3, p. 7]--http://dx.doi.org/10.1155/2014/251587. The present case has 40 nonlinear equations and 40 unknowns.
One notes line 605 through line 611 and line 611 through line 990; the 2016 January 11 edition does not have these lines 690, 693, and 694, which are 690 FOR J44 = 2 TO 39, 693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1), and 694 NEXT J44, respectively.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), P(99)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 40
94 A(KK) = -RND * 5
95 NEXT KK
128 FOR I = 1 TO 5000000 STEP 1
129 FOR K = 1 TO 40
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 40)
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 ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
191 NEXT IPP
555 X(2) = ((3 - .5 * X(1)) * X(1) + 1) / 2
605 REM FOR J44 = 2 TO 39
609 REM X(J44 + 1) = (-X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1) / 2
611 REM NEXT J44
651 FOR j47 = 1 TO 40
666 IF ABS(X(j47)) > 40 THEN 1670
688 NEXT j47
690 FOR J44 = 2 TO 39
693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1)
694 NEXT J44
695 PS = 0
696 FOR J55 = 2 TO 39
697 PS = PS + ABS(P(J55))
698 NEXT J55
699 P1 = -X(39) + (3 - .5 * X(40)) * X(40) + 1
999 P = -ABS(P1) - PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 40
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(2), A(40), M, JJJJ
1668 REM IF M > -.000001 THEN 1912
1670 NEXT I
1890 IF M < -1 THEN 1999
1912 PRINT A(1), A(2), A(3)
1914 GOTO 1950
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)
1943 PRINT A(19), A(20), A(21)
1944 PRINT A(22), A(23), A(24)
1945 PRINT A(25), A(26), A(27)
1946 PRINT A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33)
1949 PRINT A(34), A(35), A(36)
1950 PRINT A(37), A(38), A(39)
1952 PRINT A(40), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.
-1.03239202604849 -1.31504636293486 -1.388710265545094
-1.2907819914299 -1.17751196878769 -.9675105666261581
-.5965290396787168 -1.846300889951635D-12 -32000
-1.032392026048484 -1.31504636293485 -1.388710265545085
-1.290781991429954 -1.177511968787726 -.9675105666261871
-.5965290396787257 -2.658651077069862D-12 -31999
-1.032392026048488 -1.315046362934857 -1.388710265545091
-1.290781991429892 -1.177511968787691 -.9675105666261593
-.5965290396787167 -3.23663318368972D-12 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 40 unknowns, only the seven A's of line 1912, line 1950, and line 1952 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 [15], the wall-clock time for obtaining the output through JJJJ= -31998 was three minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] 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
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] 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
[6] 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
[7] 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
[8] 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.
[9] 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.
[10] J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[11] 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
[12] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
[13] 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.
[14] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[16] 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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