Jsun Yui Wong
The computer program listed below seeks to solve simultaneously the following nonlinear system of equations from page 665 of Burden, Faires, and Burden [1, p. 665, Exercise 6]:
4 * X(1) -X( 2) +X(3) = X(1) * X(4),
-X(1) + 3 * X(2) - 2 * X(3) = X(2) * X(4),
X(1) - 2 * X(2) + 3 * X(3) = X(3) * X(4),
X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 = 1.
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 = 96 TO 99
94 A(KK) = -20 + FIX(RND * 150)
95 NEXT KK
128 FOR I = 1 TO 8000
129 FOR K = 96 TO 99
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 96 + FIX(RND * 9)
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(98) = X(96) * X(99) + X(97) - 4 * X(96)
335 REM N98 = X(96) ^ 2 - 625 * X(97) ^ 2 - 1 / 4
339 REM N99 = EXP(-X(96) * X(97)) + 20 * X(98) + (10 * 3.141592654 - 3) / 3
351 N97 = -X(97) * X(99) - X(96) + 3 * X(97) - 2 * X(98)
353 N98 = -X(98) * X(99) + X(96) - 2 * X(97) + 3 * X(98)
355 REM N98 = -X(98) * X(99) + X(97) - 4 * X(96)
359 N99 = -1 + X(96) ^ 2 + X(97) ^ 2 + X(98) ^ 2
1335 P = -ABS(N98) - ABS(N99) - ABS(N97)
1499 IF P <= M THEN 1670
1657 FOR KEW = 96 TO 99
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1888 IF M < -.0000001 THEN 1999
1917 PRINT A(96), A(97), A(98), A(99), 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=-30366 is shown below:
-2.493654401671402D-15 .7071068014058323 .7071068014058398
.9999999980999506 -5.98758606780458D-08 -31892
8.33475294629724D-18 .7071067794128311 .7071067794128311
1.000000018454071 -3.111482494474582D-08 -31647
7.720047542022925D-10 .707106789031514 .7071067867154998
1.000000005454302 -2.663362473018802D-08 -31561
3.038262892746562D-09 .7071067999755675 .707106790860779
1.00000002148609 -7063893669385927D-08 -30989
4.231662939811113D-10 .7071067906077655 .7071067893382665
.9999999970098467 -2.908354898334712D-08 -30691
0 -.7071067723044366 -.7071067723044366
1.00000000499632 -3.218826598465926D-08 -30366
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=-30366 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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