Jsun Yui Wong
The following computer program seeks to solve Broyden’s Case 5 [1, p. 587–Case 5; p. 590–Table 5]. See also 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 –and page 7 of Cao [3, p. 7]–http://dx.doi.org/10.1155/2014/251587. Here alpha=-0.1, beta=1, and n=50.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), P(999)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 50
94 A(KK) = -RND
95 NEXT KK
128 FOR I = 1 TO 20000000 STEP 1
129 FOR K = 1 TO 50
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 50)
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
555 X(2) = ((3 – .1 * X(1)) * X(1) + 1) / 2
605 FOR J44 = 2 TO 49
609 X(J44 + 1) = (-X(J44 – 1) + (3 – .1 * X(J44)) * X(J44) + 1) / 2
611 NEXT J44
651 FOR J47 = 1 TO 50
666 IF ABS(X(J47)) > 10 THEN 1670
688 NEXT J47
699 PZ = -X(49) + (3 – .1 * X(50)) * X(50) + 1
999 P = -ABS(PZ)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 50
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(2), A(50), M, JJJJ
1668 IF M > -.000001 THEN 1912
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1912 PRINT A(1), A(2), A(3)
1946 PRINT A(48), A(49), A(50)
1947 PRINT 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= -31997 is shown below.
-2.046688469543995 -2.77947938888421 -3.032150132215925
-1.933355399713024 -1.436534562035508 -.7913057205928774
-9.259125746119423D-07 -32000
9.083920214236214 9.999999998423855 5.95803989209382
3.162278049833678 3.162277420106777 3.162277181157467
-6.324554426318701 -31999
-2.046688469543996 -2.779479388884212 -3.032150132215928
-1.933355522512882 -1.436534725309781 -.7913059275592724
-4.355329195071533D-07 -31998
-2.046688469543996 -2.779479388884212 -3.032150132215928
-1.933355522512882 -1.436534725309781 -.7913059275592724
-4.355329195071533D-07 -31997
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 50 unknowns, only the six A’s of line 1912 and line 1946 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= -31997 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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