The following computer program seeks a solution to the system of nonlinear equations of a 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.
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 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
22 h = 1 / (119 + 1)
91 FOR KK = 1 TO 119
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK
128 FOR I = 1 TO 500000 STEP 1
129 FOR K = 1 TO 119
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 122)
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 118
611 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * J49) ^ 3 - X(J49 - 1) + X(J49 + 1)
613 NEXT J49
615 P0 = 0
617 FOR j11 = 2 TO 118
619 P0 = P0 - ABS(P(j11))
629 NEXT j11
622 FOR j33 = 1 TO 119
633 IF ABS(X(j33)) > 3 THEN 1670
655 NEXT j33
677 P119 = 2 * X(119) + .5 * h ^ 2 * (X(119) + h * 119) ^ 3 - X(118)
999 P = -ABS(P119) + P0
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 119
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -9 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(117), A(118), A(119)
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= -31999 is shown below.
-2.68801128311508D-11 -3.36663472559238D-11 -1.202984329827381D-10
-1.688823992425807D-05 -1.318892470245795D-05 -2.352394767238498D-05
-8.339331381840622D-12 -32000
-2.687959843339338D-11 -3.366531846040526D-11 -1.202995204456031D-10
-1.688824009365936D-05 -1.318892476496033D-05 -2.35239477694988D-05
-8.327981091731493D-12 -31999
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 119 unknowns, only the six 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= -31999 was one minute.
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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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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