The following computer program seeks an integer solution to the five-diagonal system of nonlinear equations on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 35, Five-diagonal system]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present case has 32035 nonlinear equations and 32035 unknowns.
0 DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(32768), A(32768), L(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 32035
94 A(KK) = RND
95 NEXT KK
128 FOR I = 1 TO 1000000 STEP 1
129 FOR K = 1 TO 32035
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 32038)
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(1) = (4 * X(2) ^ 2 - X(2) + X(3) ^ 2) / 4
557 IF (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) < 0 THEN 1670
559 X(4) = (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) ^ .5
605 FOR J44 = 3 TO 32033
607 Xnew = 8 * X(J44) * (X(J44) ^ 2 - X(J44 - 1)) - 2 * (1 - X(J44)) + 4 * (X(J44) - X(J44 + 1) ^ 2) + X(J44 - 1) ^ 2 - X(J44 - 2) + X(J44 + 1)
618 IF Xnew < 0 THEN 1670
621 X(J44 + 2) = Xnew ^ .5
641 NEXT J44
651 FOR j47 = 1 TO 32035
666 IF ABS(X(j47)) > 30 THEN 1670
688 NEXT j47
699 P1 = 8 * X(32035) * (X(32035) ^ 2 - X(32034)) - 2 * (1 - X(32035)) + X(32034) ^ 2 - X(32033)
711 P2 = 8 * X(32034) * (X(32034) ^ 2 - X(32033)) - 2 * (1 - X(32034)) + 4 * (X(32034) - X(32035) ^ 2) + X(32033) ^ 2 - X(32032)
999 P = -ABS(P1) - ABS(P2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 32035
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(2), A(32035), M, JJJJ
1668 IF M > -.000001 THEN 1912
1670 NEXT I
1890 IF M < -1 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(4), A(5), A(6)
1943 PRINT A(32030), A(32031), A(32032)
1946 PRINT A(32033), A(32034), A(32035)
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= -31996 is shown below.
1 1 1
1 1 1
1 1 1
1 1 1
0 -32000
1 1 1
1 1 1
1 1 1
1 1 1
0 -31998
1 1 1
1 1 1
1 1 1
1 1 1
0 -31997
1 1 1
1 1 1
1 1 1
1 1 1
0 -31996
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 32035 unknowns, only the 12 A's of line 1912 through 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= -31996 was 16 minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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http://www.SciRP.org/journal/am
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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.
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[12] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
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[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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