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, Garbow, and Hillstrom [11]. The present case has 15719 nonlinear equations and 15719 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)). Also one notes that whereas line 128 of the earlier edition is 128 FOR I = 1 TO 6000000 STEP 1, here line 128 is 128 FOR I = 1 TO 3000000 STEP 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 / (15719 + 1)
91 FOR KK = 1 TO 15719
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK
128 FOR I = 1 TO 3000000 STEP 1
129 FOR K = 1 TO 15719
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15722)
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 < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 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
199 NEXT IPP
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 15718
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 15718
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15719
633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * h - 1))
655 NEXT J33
677 PZ = 2 * X(15719) + .5 * h ^ 2 * (X(15719) + h * 15719) ^ 3 - X(15718)
999 P = -ABS(PZ) + PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15719
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(5555), A(15718), M, JJJJ
1670 NEXT I
1890 REM IF M < -9 THEN 1999
1911 PRINT A(1), A(2), A(3), A(5555), A(15718), 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.
-3.043274626032177D-15 -6.086548731220798D-15 -2.317457533498823D-15
-2.543587536870807D-11 -8.38100480481299D-10 -5.675305127032603D-07
-32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 15719 unknowns, only the five A’s of line 1911 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 three hours and fifteen 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.
[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.
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http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.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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