Jsun Yui Wong
The following computer program seeks to solve the system of nonlinear equations of the discrete boundary value problem on page 29 of La Cruz, Martinez, and Raydan [7, p. 29, Test function 41, Discrete boundary value problem]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. Also see More et al. [11]. The present case has 2719 nonlinear equations and 2719 continuous variables.
One notes line 93, which comes from La Cruz, Martinez, and Raydan [7, p. 29] and is 93 A(KK) = (h * (KK * h – 1)).
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 31111
14 RANDOMIZE JJJJ
16 M = -1D+50
22 h = 1 / (2719 + 1)
91 FOR KK = 1 TO 2719
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK
128 FOR I = 1 TO 2000000 STEP 1
129 FOR K = 1 TO 2719
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 2722)
183 R = (1 - RND * 2) * A(B)
187 REM 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 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
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 2718
610 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * J49) ^ 3 - X(J49 - 1) + X(J49 + 1)
613 NEXT J49
615 PS = 0
617 FOR J49 = 2 TO 2718
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 2719
633 IF ABS(X(J33)) > 3 THEN 1670
655 NEXT J33
677 PZ = 2 * X(2719) + .5 * h ^ 2 * (X(2719) + h * 2719) ^ 3 - X(2718)
999 P = -ABS(PZ) + PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 2719
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -9 THEN 1999
1912 PRINT A(1), A(2), A(3)
1913 PRINT A(4), A(5), A(6)
1914 PRINT A(7), A(8), A(9)
1915 PRINT A(10), A(11), A(12)
1927 PRINT A(2711), A(2712), A(2713)
1937 PRINT A(2714), A(2715), A(2716)
1947 PRINT A(2717), A(2718), A(2719)
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 complete output through JJJJ= -32000 is shown below.
-1.19960417268234D-20 3.334354272574636D-18 -2.925819998435894D-18
-8.591990681672726D-17 -4.256224601882662D-17 -2.94393417772928D-16
-1.796171628930897D-16 -3.945869991029357D-16 -4.123734581215336D-16
-1.524667708237257D-15 -1.021015743561118D-15 -2.487127024791742D-15
-3.343101267297926D-08 -3.342896885699016D-08 -3.356462642178777D-08
-3.336033559667833D-08 -3.397986304410705D-08 -3.261254487807038D-08
-3.60413779066783D-08 -2.788851782937236D-08 -4.769825161337166D-08
-3.165101733156332D-09 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 2719 unknowns, only the 21 A’s of line 1912 and line 1947 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 30 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.
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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.
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