The computer program listed below seeks to solve simultaneously the following nonlinear system of equations from Burden, Faires,and Burden [1, p. 656, Exercise 7d]:
6*X(1) - 2 * COS(X(2) * X(3)) - 1 = 0,
9 * X(2) + (X(1) ^ 2 + SIN(X(3)) + 1.06) ^ .5 + .9 = 0,
60 * X(3) + 3 * EXP(-X(1) * X(2)) + 10 * 3.141592654 - 3 = 0.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+317
91 FOR KK = 91 TO 98
94 A(KK) = -20 + FIX(RND * 150)
95 NEXT KK
128 FOR I = 1 TO 8000
129 FOR K = 91 TO 98
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 91 + FIX(RND * 8)
182 REM IF RND < -.1 THEN 183 ELSE GOTO 189
183 R = (1 - RND * 2) * A(B)
186 X(B) = A(B) + (RND ^ 3) * R
188 GOTO 191
189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)
191 NEXT IPP
331 X(96) = (2 * COS(X(97) * X(98)) + 1) / 6
335 N98 = 9 * X(97) + (X(96) ^ 2 + SIN(X(98)) + 1.06) ^ .5 + .9
339 N99 = 3 * EXP(-X(96) * X(97)) + 60 * X(98) + (10 * 3.141592654 - 3)
1335 P = -ABS(N98) - ABS(N99)
1499 IF P <= M THEN 1670
1657 FOR KEW = 96 TO 98
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -.000001 THEN 1999
1917 PRINT A(96), A(97), A(98), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-30739 is shown below:
.4981446846090066 -.1996058948931147 -.5288259765133804
-7.301866214139618D-08 -31894
.4981446845080947 -.1996058989003793 -.5288259802917031
-1.850737475616996D-07 -30848
.4981446846603527 -.1996058928890903 -.5288259744982781
-2.096859361336691D-07 -30739
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time through
JJJJ=-30739 was 30 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
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