Substance

Finding by Computer Integer Solutions of Nonlinear Systems of Equations, Second Edition

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system:

- 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2=-106,

13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)=127,

- 2 * X(1) * X(2) * X(3) - X(4) - X(5) - X(6)=-70,

1 * X(1) + 1 * X(2) + 2 * X(3) - 7 * X(4) - 1 * X(5) + 1 * X(6)=-29,

1 * X(1) + 1 * X(2) + 1 * X(3) + 2 * X(4) - 9 * X(5) - 3 * X(6)=21,

1 * X(1) + 1 * X(2) + 1 * X(3) + 1 * X(4) + 2 * X(5) - 7 * X(6)=-13.

The first three of these equations are based on page 177 of Conley [4]. The last three come from Greenspan and Casulli [5, page 41].

Whereas the starting search interval of the earlier edition is A(KK) = -30 + FIX(RND * 61), here it is
A(KK) = -50 + FIX(RND * 101), which is shown below in line 94.

0 DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333)

12 FOR JJJJ = -32000 TO 31000

14 RANDOMIZE JJJJ

16 M = -1D+317

77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91

88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128

91 FOR KK = 1 TO 6

94 A(KK) = -50 + FIX(RND * 101)

95 NEXT KK

128 FOR I = 1 TO 200000

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 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

996 N83 = 106 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2

998 N82 = -127 + 13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)

1000 N81 = 70 - 2 * X(1) * X(2) * X(3) - X(4) - X(5) - X(6)
1004 N84 = 29 + 1 * X(1) + 1 * X(2) + 2 * X(3) - 7 * X(4) - 1 * X(5) + 1 * X(6)
1005 N85 = -21 + 1 * X(1) + 1 * X(2) + 1 * X(3) + 2 * X(4) - 9 * X(5) - 3 * X(6)

1006 N86 = 13 + 1 * X(1) + 1 * X(2) + 1 * X(3) + 1 * X(4) + 2 * X(5) - 7 * X(6)

1335 P = -ABS(N81) - ABS(N82) - ABS(N83) - ABS(N84) - ABS(N85) - ABS(N86)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 6

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1664 NN81 = N81: NN82 = N82

1665 NN83 = N83: NN84 = N84

1666 NN85 = N85: NN86 = N86

1670 NEXT I
1888 IF M < 0 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, NN81, NN82, NN83, NN84, NN85, NN86

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-19312 is shown below:

2       3       5       7       -1
4       0       -31947       0       0
0       0       0       0

2 3 5 7 -1
4 0 -31946 0 0
0 0 0 0

2 3 5 7 -1
4 0 -31945 0 0
0 0 0 0

2 3 5 7 -1
4 0 -31944 0 0
0 0 0 0

2 3 5 7 -1
4 0 -31943 0 0
0 0 0 0

2 3 5 7 -1
4 0 -31942 0 0
0 0 0 0

2 3 5 7 -1
4 0 -28653 0 0
0 0 0 0

2 3 5 7 -1
4 0 -28652 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25332 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25331 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25330 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25329 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25328 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25327 0 0
0 0 0 0
2 3 5 7 -1
4 0 -25326 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25325 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25324 0 0
0 0 0 0
2 3 5 7 -1
4 0 -25323 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25322 0 0
0 0 0 0
2 3 5 7 -1
4 0 -25321 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25320 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25319 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25318 0 0
0 0 0 0

2 3 5 7 -1
4 0 -25317 0 0
0 0 0 0

2 3 5 7 -1
4 0 -19314 0 0
0 0 0 0

2 3 5 7 -1
4 0 -19313 0 0
0 0 0 0

2 3 5 7 -1
4 0 -19312 0 0
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 [11], the wall-clock time through JJJJ=-19312 was 50 minutes.

Incidentally, one can stop the computer run as soon as the first 0 (for M) appears on the screen. For the present problem, that happens at JJJJ=-31947.

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] Thomas L. Saaty, Optimization in Integers and Related Extremal Problems. McGraw-Hill, 1970.
[10] Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

How To Find Integer Solutions of a Nonlinear System of Equations--yet Another System, Second Edition

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system from page 163 of Conley [4]:

X(1) + 6 * X(2) + 4 * X(3) + 5 * X(4)=155,

2 * X(1) ^ 2 + X(2) + X(3) + X(4)=229,

X(1) + 3 * X(2) * X(3) + X(4) ^ 2=391,

X(1) + 3 * X(2) + X(3) * X(4)=130.

Whereas line 94 of the earlier edition is 94 A(KK) = -20 + FIX(RND * 40), here line 94 is 94 A(KK) = -50 + FIX(RND * 100).

0 DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333)

12 FOR JJJJ = -32000 TO -29747

14 RANDOMIZE JJJJ

16 M = -1D+317

77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91

88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128

91 FOR KK = 1 TO 4

94 A(KK) = -50 + FIX(RND * 100)

95 NEXT KK

128 FOR I = 1 TO 100000

129 FOR K = 1 TO 4

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 4)

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

1005 N88 = -155 + X(1) + 6 * X(2) + 4 * X(3) + 5 * X(4)

1006 N89 = -229 + 2 * X(1) ^ 2 + X(2) + X(3) + X(4)

1009 N91 = -391 + X(1) + 3 * X(2) * X(3) + X(4) ^ 2

1117 N93 = -130 + X(1) + 3 * X(2) + X(3) * X(4)

1335 P = -ABS(N91) - ABS(N93) - ABS(N88) - ABS(N89)

1337 GOTO 1499

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1664 NN91 = N91: NN93 = N93

1665 NN88 = N88: NN89 = N89

1670 NEXT I
1888 IF M < 0 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), M, JJJJ, NN88, NN89, NN91, NN93

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-29758 is shown below:

10       10       10       9       0
-31295       0       0       0       0

10       10       10       9       0
-31294       0       0       0       0

10       10       10       9       0
-31293       0       0       0       0

and continues with no break until

10       10       10       9       0
-31264       0       0       0       0

10       10       10       9       0
-31147       0       0       0       0

10       10       10       9       0
-31146       0       0       0       0

10       10       10       9       0
-31145       0       0       0       0

10       10       10       9       0
-31144       0       0       0       0

10       10       10       9       0
-29758       0       0       0       0

and continues

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=-29758 was five minutes.

Incidentally, one can stop the computer run as soon as the first 0 for M appears on the screen. For the present problem, that happens at JJJJ=-31295.

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.

How To Find Integer Solutions (If Any) Of A Nonlinear System Of Equations--Now More Automatic

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 94 of the preceding paper 94 A(KK) = 50; here line 94 is 94 A(KK) = FIX(RND * 201). The latter is more automatic.

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

77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91

88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128

91 FOR KK = 1 TO 6

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

95 NEXT KK

128 FOR I = 1 TO 1250000

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 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

1005 N88 = 5492 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2

1006 N89 = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6) - X(5)

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(N93) - ABS(N94) - ABS(N88) - ABS(N89)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 6

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1664 NN88 = N88: NN89 = N89: NN91 = N91: NN93 = N93: NN94 = N94

1670 NEXT I
1888 IF M < 0 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, NN88, NN89, NN91, NN93, NN94

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-4230 is shown below:

58       47       117       152       102
75       0       -31721       0       0
0       0       0

58 47 117 152 102
75 0 -29064 0 0
0 0 0

58 47 117 152 102
75 0 -23068 0 0
0 0 0

58 47 117 152 102
75 0 -23067 0 0
0 0 0

58 47 117 152 102
75 0 -20110 0 0
0 0 0

58 47 117 152 102
75 0 -19887 0 0
0 0 0

58 47 117 152 102
75 0 -19886 0 0
0 0 0

58 47 117 152 102
75 0 -19479 0 0
0 0 0

58 47 117 152 102
75 0 -19478 0 0
0 0 0

58 47 117 152 102
75 0 -15092 0 0
0 0 0

58 47 117 152 102
75 0 -10238 0 0
0 0 0

58 47 117 152 102
75 0 -10237 0 0
0 0 0

58 47 117 152 102
75 0 -10236 0 0
0 0 0

58 47 117 152 102
75 0 -10235 0 0
0 0 0

58       47       117       152       102
75       0       -4230       0       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=-4230 was 11 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.

 

How To Find Integer Solutions (If Any) Of A Nonlinear System Of Equations, Improved Edition

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.

One notes that line 94 of the computer program below is 94 A(KK) = 50.

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

77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91

88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128

91 FOR KK = 1 TO 6

94 A(KK) = 50

95 NEXT KK

128 FOR I = 1 TO 1250000

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 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

1005 N88 = 5492 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2

1006 N89 = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6) - X(5)

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(N93) - ABS(N94) - ABS(N88) - ABS(N89)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 6

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1664 NN88 = N88: NN89 = N89: NN91 = N91: NN93 = N93: NN94 = N94

1670 NEXT I
1888 IF M < 0 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, NN88, NN89, NN91, NN93, NN94

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-30037 is shown below:

58       47       117       152       102
75       0       -31994       0       0
0       0       0

58       47       117       152       102
75       0       -31992       0       0
0 0 0

58       47       117       152       102
75       0       -31991       0       0
0       0       0

58       47       117       152       102
75       0       -31795       0       0
0       0       0

58       47       117       152       102
75       0       -31794       0       0
0       0       0

58       47       117       152       102
75       0       -31793       0       0
0       0       0

58       47       117       152       102
75       0       -30040       0       0
0       0       0

58       47       117       152       102
75       0       -30037       0       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=-30037 was fifty minutes.

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.

How to Find Integer Solutions of a Nonlinear System of Equations, a Part of a Series

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system, which is based on the nonlinear system of Problem 2.8 of Shoup [9, pages 40-41].

5 * X(1) + 3 * X(2) + X(3)+ X(4)=15,

X(1) * X(2) + X(2) * X(3) + X(3) * X(4)=17,

X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 - X(4) ^ 2=9,

X(1) * X(3) + X(2) * X(4) + X(1) ^ 3=8.

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 = 1 TO 4

94 A(KK) = -500 + FIX(RND * 10001)

95 NEXT KK
128 FOR I = 1 TO 100000

129 FOR K = 1 TO 4

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 4)

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 X(4) = 15 - 5 * X(1) - 3 * X(2) - X(3)

1004 N96 = -17 + X(1) * X(2) + X(2) * X(3) + X(3) * X(4)

1111 N97 = -9 + X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 - X(4) ^ 2

1116 N98 = -8 + X(1) * X(3) + X(2) * X(4) + X(1) ^ 3

1335 P = -ABS(N96) - ABS(N97) - ABS(N98)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P

1670 NEXT I
1888 IF M < -1000 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31985 is shown below:

1       1       4       3       0
-32000

1       1       4       3       0
-31999

1       1       4       3       0
-31996

3       -6       8       10       -20
-31995

1       1       4       3        0
-31994

1       1       4       3        0
-31992

1       1       4       3       0
-31990

2       -2       5       6       -15
-31988

11       -36       24       44       -272
-31986

1       1       4       3       0
-31985

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=-31985 was 5 seconds, not including "Creating .EXE file..." time.

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] Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[9] Terry E. Shoup, A Practical Guide to Computer Methods for Engineers, Prentice-Hall, 1979.
[10] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

How To Find Integer Solutions (If Any) Of A Nonlinear System Of Equations--Yet Another System, Revised Edition

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 program’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.

 

How to Find Integer Solutions (if any) of a Nonlinear System of Equations

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system, which is based on the first nonlinear system on page 238 of Shoup [8].

X(1) + X(2) + X(3) + X(4) =5,

X(1) ^ 2 + X(2) + X(3) ^ 2 + X(4)=13,

X(1) * X(2) + X(2) * X(3) + X(4)=2,

X(1) * X(3) + X(2) * X(4) + X(4) ^ 2=9.

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 = 1 TO 4

94 A(KK) = -100 + FIX(RND * 201)

95 NEXT KK
128 FOR I = 1 TO 3000000

129 FOR K = 1 TO 4

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 4)

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 X(4) = 5 - X(1) - X(2) - X(3)

1004 N96 = -13 + X(1) ^ 2 + X(2) + X(3) ^ 2 + X(4)

1111 N97 = -2 + X(1) * X(2) + X(2) * X(3) + X(4)

1116 N98 = -9 + X(1) * X(3) + X(2) * X(4) + X(4) ^ 2

1335 P = -ABS(N96) - ABS(N97) - ABS(N98)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1664 NN96 = N96: NN97 = N97: NN98 = N98

1670 NEXT I
1888 IF M < -6 THEN 1999

1917 PRINT A(1), A(2), A(3), A(4), M, JJJJ, NN96, NN97, NN98

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31991 is shown below:

2       0       3       0       -5
-32000       0       -2       -3

2       1       3       -1       -5
-31999       0       2       -3

-1       -1       3       4       0
-31998       0       0       0

-1       -1       3       4       0
-31997       0       0       0

-1       -1       3       4       0
-31996       0       0       0

-1       -1       3       4       0
-31995       0       0       0

2       0       3       0       -5
-31994       0       -2       -3

-1       -1       3       4       0
-31993 0 0 0

3       -1       -1       4       0
-31992       0       0       0

3       -1       -1       4       0
-31991       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 [9], the wall-clock time through
JJJJ=-31991 was 40 seconds, not including "Creating .EXE file..." time.

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.
[9] Terry E. Shoup, Applied Numerical Methods for the Microcomputer. Prentice-Hall, 1984.
[9] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

 

Computer Solutions/Integer Solutions of Nonlinear Systems of Equations, Part 2

Jsun Yui Wong

The two computer programs listed below seek to solve a continuous case and an integer case of two nonlinear systems of equations, respectively. The first case comes from page 665 of Burden, Faires, and Burden [1, Exercise 7] and is as follows:

3*X(1) - COS(X(2) * X(3)) - 1 / 2) = 0,

X(1) ^ 2 - 625 * X(2) ^ 2 - 1 / 4 = 0,

EXP(-X(1) * X(2)) + 20 * X(3) + (10 * 3.141592654 - 3) / 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 = 1 TO 3

94 A(KK) = -20 + FIX(RND * 150)

95 NEXT KK
128 FOR I = 1 TO 8000

129 FOR K = 1 TO 3

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 3)

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(1) = (COS(X(2) * X(3)) + 1 / 2) / 3

335 N98 = X(1) ^ 2 - 625 * X(2) ^ 2 - 1 / 4

339 N99 = EXP(-X(1) * X(2)) + 20 * X(3) + (10 * 3.141592654 - 3) / 3

1335 P = -ABS(N98) - ABS(N99)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 3

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P

1670 NEXT I
1888 IF M < -.000001 THEN 1999

1917 PRINT A(1), A(2), A(3), 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=-31971 is shown below:

.4999999999702333       2.55236688287822D-05       -.523598137579018
-4.071908160135142D-07       -31997

.4999999999997746       2.221226407568314D-06       -.5235987201360377
-3.083886979305837D-09       -31974

.5       4.747449517789774D-17       -.5235987751826406
-9.68052082939391526D-09       -31971

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=-31971 was 2 seconds, not including "Creating .EXE file..." time.

The second case--shown below--comes from page 649 of Burden, Faires, and Burden [1, Exercise 6c] and is as follows:

X(1) ^ 2 + X(2) - 37 = 0,

X(1) - X(2) ^ 2 - 5 = 0,

X(1) + X(2) + X(3) - 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 = 1 TO 3

94 A(KK) = -50 + FIX(RND * 101)

95 NEXT KK
128 FOR I = 1 TO 1000

129 FOR K = 1 TO 3

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 1 + FIX(RND * 3)

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 X(2) = -X(1) ^ 2 + 37

1004 N96 = X(1) - X(2) ^ 2 - 5

1007 N97 = X(1) + X(2) + X(3) - 3

1009 REM

1335 P = -ABS(N96) - ABS(N97)

1499 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 3

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P

1670 NEXT I
1888 IF M < -50 THEN 1999

1917 PRINT A(1), A(2), A(3), 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=-31992 is shown below:

-6       1       8       -12       -32000
-6       1       8       -12       -31999
-6       1       8       -12       -31998
6       1       -4       0       -31997
6       1       -4       0       -31996
6       1       -4       0       -31995
-6       1       8       -12       -31994
6       1       -4       0       -31993
6       1       -4       0       -31992

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=-31992 was 2 seconds, not including "Creating .EXE file..." time.

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.

Simultaneously Solving a Nonlinear System of Equations

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following nonlinear system of equations from page 680 of Burden, Faires, and Burden [1]:

3*X(1) - COS(X(2) * X(3)) - 1 / 2 = 0,

4 * X(1) ^ 2 - 625 * X(2)^ 2 + 2 * X(2) - 1 = 0,

EXP(-X(1) * X(2)) + 20 * X(3) + (10 * 3.141592654 - 3) / 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) = (COS(X(97) * X(98)) + 1 / 2) / 3

335 N98 = 4 * X(96) ^ 2 - 625 * X(97) ^ 2 + 2 * X(97) - 1
339 N99 = EXP(-X(96) * X(97)) + 20 * X(98) + (10 * 3.141592654 - 3) / 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 < -.0000001 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=-31957 is shown below:

.49999953252393       3.199064792960778D-03       -.5235188599374786
-6.228648575531222D-08       -31992

.5       1.412340998167154D-20       -.5235987768582213
-2.383109265480243D-08       -31965

.5       -1.353065163688721D-20       -.5235987751558124
-1.021708495585472D-08       -31957

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=-31957 was 10 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.

Integer Solutions of a Nonlinear System of Equations

Jsun Yui Wong

The computer program listed below is about the following quotation from Sleziak [8]:

"Do there exist integers x, y, z, w that satisfy

(x+1)^2+y^2=(x+2)^2+z^2

(x+2)^2+z^2=(x+3)^2+w^2?"

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 = 91 TO 99

94 A(KK) = -75 + FIX(RND * 150)

95 NEXT KK
128 FOR I = 1 TO 8000

129 FOR K = 91 TO 99

131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)

181 B = 91 + FIX(RND * 9)

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

335 N98 = (X(96) + 1) ^ 2 + X(97) ^ 2 - (X(96) + 2) ^ 2 - X(98) ^ 2

339 N99 = (X(96) + 2) ^ 2 + X(98) ^ 2 - (X(96) + 3) ^ 2 - X(99) ^ 2

1335 P = -ABS(N98) - ABS(N99)

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 < 0 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 [9]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-30603 is shown below:

30       -12       9       -4       0
-30990

30       -12       9       4       0
-30913

-34       -4       9       12       0
-30809

2       -4       3       0       0
-30603

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 [9], the wall-clock time through
JJJJ=-30603 was one minute.

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] Martin Sleziak, Integer Solutions to Nonlinear System of Equations.... www.math.stackexchange.com/questions/2085253/integer-solutions-to-nonlinear-system-of-equations
[9] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.