Jsun Yui Wong
The following computer program seeks only integer solutions to the tridiagonal system of equations on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 34, Tridiagonal system]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present case has 15019 nonlinear equations and 15019 general integer variables.
While line 94 and line 633 of the preceding paper are 94 A(KK) = RND * 3 and 633 IF ABS(X(J33)) > 5 THEN 1670, here line 94 and line 633 are 94 A(KK) = -1 + RND * 3
and 633 IF ABS(X(J33)) > 10 THEN 1670, respectively.
0 DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 15019
94 A(KK) = -1 + RND * 3
95 NEXT KK
128 FOR I = 1 TO 6000000 STEP 1
129 FOR K = 1 TO 15019
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15022)
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
188 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 7 * R
199 NEXT IPP
577 X(1) = X(2) ^ 2
605 FOR J49 = 2 TO 15018
610 P(J49) = 8 * X(J49) * (X(J49) ^ 2 - X(J49 - 1)) - 2 * (1 - X(J49)) + 4 * (X(J49) - X(J49 + 1) ^ 2)
611 NEXT J49
615 PS = 0
617 FOR J49 = 2 TO 15018
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15019
633 IF ABS(X(J33)) > 10 THEN 1670
655 NEXT J33
666 PZ = 8 * X(15019) * (X(15019) ^ 2 - X(15018)) - 2 * (1 - X(15019))
999 P = -ABS(PZ) + PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15019
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1663 PRINT A(1), A(2), A(3), A(4), A(5)
1664 PRINT A(5015), A(5016), A(5017), A(5018), A(5019)
1665 PRINT M, JJJJ
1668 IF M > -.000001 THEN 1890
1670 NEXT I
1890 REM IF M < -999999 THEN 1999
1912 PRINT A(1), A(2), A(3), A(4), A(5)
1917 PRINT A(15015), A(15016), A(15017), A(15018), A(15019)
1939 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [16]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.
.
.
.
0 0 2 0 1
-1 0 0 -1 0
-269528 -32000
.
.
.
0 0 2 0 1
-1 0 1 -1 0
-248060 -32000
.
.
.
1 1 2 0 1
0 1 1 -1 0
-202084 -32000
.
.
.
1 1 2 1 1
1 1 1 -1 0
-175232 -32000
.
.
.
4 2 1 1 1
1 1 1 1 1
-41504 -32000
.
.
.
1 1 1 1 1
1 1 1 1 1
-202 -32000
1 1 1 1 1
1 1 1 1 1
-184 -32000
1 1 1 1 1
1 1 1 1 1
-110 -32000
1 1 1 1 1
1 1 1 1 1
-92 -32000
1 1 1 1 1
1 1 1 1 1
-18 -32000
1 1 1 1 1
1 1 1 1 1
0 -32000
1 1 1 1 1
1 1 1 1 1
0 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 15019 unknowns, only the ten A's of line 1912 and line 1917 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 [16], the wall-clock time for obtaining the output through JJJJ= -32000 was fifty minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[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.
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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] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207. .
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[16] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[17] 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.