Jsun Yui Wong

The computer program listed below deals with the following nonlinear equations from p. 628 of Moreno, Lopez, and Martinez [65, Example 1, Expression 70]:

(6 * X(2) ^ 2 + 20 * X(2) + 2 * X(1) + 44*X(3) - 170) =0,

(3 * X(2) ^ 2 - 43 * X(2) - 7 * X(1) - 6 * X(3) + 100) =0,

(X(3) ^ 2 - 79 * X(3) + 6 * X(1) ^ 2 - 10 * X(2) + 4) =0.

One notes the vicinity of line 1299, which is 1299 P = -ABS(3 * X(2) ^ 2 - 43 * X(2) - 7 * X(1) - 6 * X(3) + 100) - ABS(X(3) ^ 2 - 79 * X(3) + 6 * X(1) ^ 2 - 10 * X(2) + 4) .

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

99 E = 2.718281828

111 PI = 3.141592654

120 FOR J44 = 1 TO 3

121 A(J44) = FIX(RND * 6)

122 REM A(J44) = (RND * 5)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 100000)

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 j = 1 + FIX(RND * 3)

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 * 15)) * R

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 2.3) ELSE X(j) = A(j) + FIX(1 + RND * 2.3)

169 NEXT IPP

1178 GOTO 1184

1179 FOR J44 = 1 TO 3

1181 X(J44) = INT(X(J44))

1183 NEXT J44

1184 REM

1195 X(3) = (-6 * X(2) ^ 2 - 20 * X(2) - 2 * X(1) + 170) / 44

1299 P = -ABS(3 * X(2) ^ 2 - 43 * X(2) - 7 * X(1) - 6 * X(3) + 100) - ABS(X(3) ^ 2 - 79 * X(3) + 6 * X(1) ^ 2 - 10 * X(2) + 4)

1231 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1557 GOTO 128

1670 NEXT I

1894 IF M < -.000001 THEN 1999

1923 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [105]. The complete output of of a single run through JJJJ= -30228 is shown below:

6.263973713436613 .9459296379944783 3.026926421204978

-3.149396447035824D-07 -31978

6.263973719882112 .9459296259432056 3.02692642949885

-3.998544041705632D-08 -31358

6.263973719659795 .9459296263588767 3.026926429212778

-2.774341045092D-08 -30439

-3.128316384150188 3.796574387015103 .3145746038900835

-3.475213056734761D-07 -30228

Two distinct solutions are shown above.

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 [105], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -30228 was 170 seconds, counting from "Starting program...". One can compare the computational results above with those in Moreno, Lopez, and Martinez [65, p. 629, Example 1, Table 1].

**Acknowledgement**

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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