Jsun Yui Wong
Rheinboldt [79, p. 3] wrote: "K. L. Hiebert [Hiebert 1982] included three chemical equilibrium problems in a computatioan perforrmance test of specific codes. This choice was prompted by the fact that systems of equations resulting from these types of problems are usually very poorly scaled and difficult to solve."
The computer program listed below seeks to solve simultaneously the following two equations:
X(2) - 10=0,
X(1) * X(2) - 5 * 10 ^ 4=0.
These equations are Hiebert's [41] two equations in Rheinboldt [79, p. 3].
One notes line 162, which is 162 IF RND < .16666 THEN X(j) = A(j) - FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(j) = A(j) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(j) = A(j) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(j) = A(j) - FIX(1 + RND * 200.3) ELSE X(j) = A(j) + FIX(1 + RND * 200.3).
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
83 RANDOMIZE JJJJ
87 M = -4E+299
120 FOR J44 = 1 TO 2
121 A(J44) = FIX(RND * 6)
122 REM A(J44) = (RND * 2)
123 NEXT J44
128 FOR I = 0 TO FIX(RND * 15000)
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND * .3)
143 j = 1 + FIX(RND * 2)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 REM
157 R = (1 - RND * 2) * (A(j))
160 X(j) = A(j) + (RND ^ (RND * 30)) * R
161 GOTO 169
162 IF RND < .16666 THEN X(j) = A(j) - FIX(1 + RND * 2.3) ELSE IF RND < .33333 THEN X(j) = A(j) + FIX(1 + RND * 2.3) ELSE IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 20.3) ELSE IF RND < .66666 THEN X(j) = A(j) + FIX(1 + RND * 20.3) ELSE IF RND < .83333 THEN X(j) = A(j) - FIX(1 + RND * 200.3) ELSE X(j) = A(j) + FIX(1 + RND * 200.3)
169 NEXT IPP
1111 X(2) = 10
1222 y1 = X(1) * X(2) - 5 * 10 ^ 4
1245 P = -ABS(y1)
1251 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1458 yyy1 = y1
1557 GOTO 128
1670 NEXT I
1894 IF M < -.0003 THEN 1999
1924 PRINT A(1), A(2), M, JJJJ, yyy1
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [107]. The complete output of a single run through JJJJ= -31993 is shown below:
5000 10 0 -32000 0
5000 10 0 -31999 0
5000 10 0 -31998 0
5000 10 0 -31997 0
5000 10 0 -31996 0
5000.000000000001 10 -9.094947017729282D-12
-31995 9.094947017729282D-12
5000 10 0 -31993 0
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Processor Intel (R) Core (TM) 2 Duo CPU E8400 @ 3.0 GHz 3.0 GHz, 4.00 GB of RAM (3.9 GB usable), 64-bit Operating System, and QB64v1000-win [107], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31993 was 1 second, counting from "Starting program...".
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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