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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