Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the first problem on page 177 of Conley [4], which is to solve

X(1)^2 + 3 * X(2) + 5 * X(4) + 6 * X(3) + 7 * X(6)) =5492,

2 * X(1) * X(2) * X(3) + X(4) + X(5) + X(6)= 1114 X(5) = 638213,

X(1) + X(2) + X(3) + 9 * X(4) + 11 * X(5) + X(6)=2787,

3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6)=1768,

13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)=844252,

where 0<=X(i)>=200 and X(i)'s are whole numbers.

Line 1114 and line 1009 of the earlier edition are 1114 X(5) = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6) and 1009 N91 = -1768 + 3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6), respectively, whereas line 1006 and line 1009 of the present edition are 1006 X(5) = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6) and 1009 N91 = -1768 + 3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6), respectively.

0 DEFDBL A-Z

3 DEFINT J, K, X

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 = 1 TO 6

94 A(KK) = FIX(RND * 201)

95 NEXT KK

128 FOR I = 1 TO 2000000

129 FOR K = 1 TO 6

131 X(K) = A(K)

132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 6)

182 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

1001 REM

1004 X(1) = (5492 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6)) ^ .5

1006 X(5) = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6)

1009 N91 = -1768 + 3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6)

1113 REM

1117 N93 = -2787 + X(1) + X(2) + X(3) + 9 * X(4) + 11 * X(5) + X(6)

1119 N94 = -844252 + 13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)

1335 P = -ABS(N91) - ABS(N92) - ABS(N93) - ABS(N94)

1499 IF P <= M THEN 1670

1657 FOR KEW = 1 TO 6

1658 A(KEW) = X(KEW)

1659 NEXT KEW

1661 M = P

1664 NN91 = N91: NN92 = N92: NN93 = N93: NN94 = N94

1670 NEXT I

1888 IF M < -1400 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, JJJJ, JJJJ, NN91, NN92, NN93, NN94

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer programâ€™s complete output through JJJJ=-29108 is shown below:

53 82 96 105 86

198 -1314 -30920 -30920 -30920

822 0 -467 25

55 139 46 129 111

169 -1029 -29865 -29865 -29865

977 0 4 48

58 47 117 152 102

75 0 -29108 -29108 -29108

0 0 0 0

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 [10], the wall-clock time through

JJJJ=-29108 was two hours.

**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] W. Conley, *Computer Optimization Techniques*, Revised Edition. Petrocelli Books, Inc., NY/Princeton, 1984.

[5] D. Greenspan, V. Casulli, *Numerical Analysis for Applied Mathematics, Science, and* *Engineering*. Addison-Wesley Publishing Company, 1988

[6] L. W. Johnson, R. D. Riess, *Numerical Analysis*, Second Edition. Addison-Wesley Publishing Company, 1982

[7] 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.

[8] William H. Mills, A System of Quadratic Diophantine Equations,* Pacific Journal of* *Mathematics*, 3 (1953), pp. 209-220.

[9] Terry E. Shoup,* Applied Numerical Methods for the Microcomputer*, Prentice-Hall, 1984.

[10] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

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