Seeking To Improve On Given Solutions

Jsun Yui Wong

The computer program listed below seeks to improve on the realized solutions of the preceding paper. One notes line 195 and line 211, which are 195 X(4) = INT(X(4)) and
211 X(1) = (-X(4) * X(2)) / X(3), respectively.

0 DEFDBL A-Z
1 DEFINT J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 4

112 A(J44) = -10 + RND * 20

115 NEXT J44

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 4)

150 R = (1 – RND * 2) * A(B)
160 X(B) = (A(B) + RND ^ 3 * R)

168 NEXT IPP
195 X(4) = INT(X(4))

198 X(3) = 2 – X(4)

210 IF X(3) = 0 THEN GOTO 1670

211 X(1) = (-X(4) * X(2)) / X(3)
215 REM N(6) = -ABS(X(3) * X(1) + X(4) * X(2))

216 N(7) = -ABS(X(3) * X(1) ^ 2 + X(4) * X(2) ^ 2 – 2 / 3)
217 N(8) = -ABS(X(3) * X(1) ^ 3 + X(4) * X(2) ^ 3)
322 PD1 = N(7) + N(8)

1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)

1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 IF M < -.000000001 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [18]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31918 is shown below:

.5773502691877946          -.5773502691877946          1
1          -4.228800415308903D-12          -31992

-.5773502691866187          -.5773502691866187          1
1          -6.944536525693712D-12          -31985

.5773502691915871          -.5773502691915871          1
1          -4.529397256564094D-12          -31974

.5773502692572973          -.5773502692572973          1
1          -1.562806386579997D-10          -31942

-.5773502691903677          .5773502691903677          1
1          -1.713410663350956D-12          -31939

-.5773502691900816          .5773502691900816          1
1          -1.052680729043964D-12          -31918

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer using a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and using qb64v1000-win [18], the wall-clock time for obtaining the output through JJJJ= -31918 was 11 seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] I.M.M. El-Emary, M.M.Abd El-Kareem, Towards Using Genetic Algorithmfor Solving Nonlinear Equation System. World Applied Sciences Journal 5 (3):282-289, 2008.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.388.158&rep=rep1&type=pdf

[4] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[5] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Angel Fernando Kuri-Morales, Solution of Simultaneous Non-Linear Equations Using Genetic Algorithm. http://www.wseas.us/e-library/conferences/brazil2002/papers/449-158.pdf.

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[8] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[9] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[10] 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.

[11] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[12] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[13] Anton Schigur, Solving a System of Diophantine Equations. Mathematics Stack Exchange. http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations.

[14] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[15] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview.

[16] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[17] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

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

[19] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing.

A Computer Program Solving a Nonlinear System of Equations with Continuous Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of equations from El-Emary and El-Kareem [3, p. 287].
See http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.388.158&rep=rep1&type.=pdf.

W(1)+W(2) – 2 = 0

W(1)*X(1)+W(2)*X(2) = 0

W(1)*X(1)^2+W(2)*X(2)^2 – 2/3 = 0

W(1)*X(1)^3+W(2)*X(2)^3 = 0.
0 DEFDBL A-Z
1 DEFINT J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 4
112 A(J44) = -10 + RND * 20
115 NEXT J44

128 FOR I = 1 TO 100000

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 4)
150 R = (1 – RND * 2) * A(B)
155 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 X(B) = (A(B) + RND ^ 3 * R)
168 NEXT IPP
195 X(3) = 2 – X(4)

215 N(6) = -ABS(X(3) * X(1) + X(4) * X(2))
216 N(7) = -ABS(X(3) * X(1) ^ 2 + X(4) * X(2) ^ 2 – 2 / 3)
217 N(8) = -ABS(X(3) * X(1) ^ 3 + X(4) * X(2) ^ 3)

322 PD1 = N(6) + N(7) + N(8)
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)

1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 IF M < -.00001 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4)

1905 PRINT M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [18]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31974 is shown below:

-.5773549463311843       .5773455920859565      .9999918989859991
1.000008101014001
-6.236163485953182D-06          -31995

.577350912241483       -.5773496261384886       .9999988862020528
1.000001113797947
-8.574020639171893D-07          -31990

-.5773502949987083       .5773502433804245       .9999999552970431
1.000000044702957
-3.441249075542658D-08          -31974

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer using a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and using qb64v1000-win [18], the wall-clock time for obtaining the output through JJJJ= -31974 was 28 seconds, not including “Creating .EXE file…” time–the total time was 40 seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] I.M.M. El-Emary, M.M.Abd El-Kareem, Towards Using Genetic Algorithmfor Solving Nonlinear Equation System. World Applied Sciences Journal 5 (3):282-289, 2008.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.388.158&rep=rep1&type.=pdf.

[4] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[5] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[6] Angel Fernando Kuri-Morales, Solution of Simultaneous Non-Linear Equations Using Genetic Algorithm. http://www.wseas.us/e-library/conferences/brazil2002/papers/449-158.pdf.

[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[8] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[9] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[10] 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.

[11] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[12] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[13] Anton Schigur, Solving a System of Diophantine Equations. Mathematics Stack Exchange. http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations.

[14] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[15] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview.

[16] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[17] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

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

[19] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing.

A Computer Program Solving a System of Nonlinear Equations with Continuous Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following system of nonlinear equations from Kuri-Morales [5, Problem 4; http://www.wseas.us/e-library/conferences/brazil2002/papers/449-158.pdf%5D.

3 * X(1) ^ 2 + SIN(X(1) * X(2)) – X(3) ^ 2 + 2 = 0

2 * X(1) ^ 3 – X(2) ^ 2 -X(3) + 3 = 0

SIN(2 * X(1)) + COS(X(2) * X(3)) + X(2) – 1 = 0.
0 DEFDBL A-Z
1 DEFINT J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 3
112 A(J44) = -10 + RND * 20

113 NEXT J44
128 FOR I = 1 TO 1000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 3)
150 R = (1 – RND * 2) * A(B)
155 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 X(B) = (A(B) + RND ^ 3 * R)
165 REM GOTO 168

167 REM IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)
168 NEXT IPP

177 GOTO 195

185 FOR J44 = 1 TO 4
187 IF X(J44) < 1 GOTO 1670

188 NEXT J44
195 X(3) = 2 * X(1) ^ 3 – X(2) ^ 2 + 3
215 N(7) = -ABS(3 * X(1) ^ 2 + SIN(X(1) * X(2)) – X(3) ^ 2 + 2)
217 N(8) = -ABS(SIN(2 * X(1)) + COS(X(2) * X(3)) + X(2) – 1)
322 PD1 = N(7) + N(8)

1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 IF M < -.000001 THEN 1999

1904 PRINT A(1), A(2), A(3)

1905 PRINT M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [17]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31981 is shown below:

-3.275901167926566D-02       1.264628712643975       1.400643908303197
-3.830827227766127D-08          -31997

-3.275906423514454D-02       1.264628736846734       1.400643846749787
-1.148727933914396D-07          -31991

-.0327591672630557         1.264628749553351      1.400643813948088
-2.661112784670865D-07          -31981

Above there is no rounding by hand; it is just straight copying by hand from the screen.

On a personal computer using a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and using qb64v1000-win [17], the wall-clock time for obtaining the output through JJJJ= -31981 was 2 seconds, not including “Creating .EXE file…” time–the total time was 9 seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[4] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[5] Angel Fernando Kuri-Morales, Solution of Simultaneous Non-Linear Equations Using Genetic Algorithm. http://www.wseas.us/e-library/conferences/brazil2002/papers/449-158.pdf.

[6] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[7] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[8] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[9] 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.

[10] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[11] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021.

[12] Anton Schigur, Solving a System of Diophantine Equations. Mathematics Stack Exchange. http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations.

[13] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[14] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview.

[15] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[16] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

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

[18] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing.

Erratum: A Computer Program Solving a System of Diophantine Equations

Jsun Yui Wong

“JJJJ= -31998 was 3 seconds” of the last sentence should read “JJJJ= -31991 was 3 seconds.”

A Computer Program Solving a System of Nonlinear Diophantine Equations

Jsun Yui Wong

The computer program listed below seeks to solve a system of nonlinear Diophantine equations based on the following system:

“a^3+40033=d
b^3+39312=d
c^3+4104=d
where a, b, c>0 are all Distinct postive integers, …” Schigur [11, http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations%5D.
0 DEFDBL A-Z
1 DEFINT J, K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 4
112 A(J44) = 1 + FIX(RND * 30)

113 NEXT J44
128 FOR I = 1 TO 100

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 4)

150 REM R = (1 – RND * 2) * A(B)

155 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 REM X(B) = (A(B) + RND ^ 3 * R)

165 REM GOTO 168

167 IF RND < .5 THEN X(B) = (A(B) – 1) ELSE X(B) = (A(B) + 1)

168 NEXT IPP
185 FOR J44 = 1 TO 4
187 IF X(J44) < 1 GOTO 1670

188 NEXT J44
195 X(4) = X(1) ^ 3 + 40033

215 N(7) = -ABS(X(2) ^ 3 + 39312 – X(4))
217 N(8) = -ABS(X(3) ^ 3 + 4104 – X(4))

322 PD1 = N(7) + N(8)

1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)

1456 NEXT KLX
1557 GOTO 128
1670 NEXT I

1889 IF M < -50 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [16]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31991 is shown below:

2       9       33       40041
0       -32000

15       16       34       43408
0       -31999

2       9       33       40041
0       -31997

15       16       34       43408
0       -31996

2       9       33       40041
0       -31994

2       9       33       40041
0       -31993

15       16       34       43408
0       -31992

2       9       33      40041
0       -31991

Above there is no rounding by hand; it is just straight copying by hand from the screen.

The two solutions shown above are the same two solutions shown by “coffeemath” in [11].

On a personal computer using a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and using qb64v1000-win [16], the wall-clock time for obtaining the output through JJJJ= -31998 was 3 seconds, not including “Creating .EXE file…” time–the total time was 11 seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[4] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[5] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.

[6] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[7] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[8] 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.

[9] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[10] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[11] Anton Schigur, Solving a System of Diophantine Equations. Mathematics Stack Exchange. http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations.

[12] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[13] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview

[14] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[15] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick

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

[17] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing

The Nonlinear Integer/Continuous Programming Solver Applied to a Nonlinear Diophantine Equation with 400-General-Integer Unknowns

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [18, p. 55; http://www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf%5D; also see Abraham, Sanyal, and Sanglikar [1; https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf%5D:

X(1)^2+ X(2)^2+X(3)^2+…+X(400)^2=400000.

0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+50

91 FOR KK = 1 TO 400
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK

128 FOR I = 1 TO 15000 STEP 1
129 FOR K = 1 TO 400
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 400)
183 REM R = (1 – RND * 2) * A(B)

188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)

191 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 400
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 400
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44

261 PZ = -ABS(SUML – 400000)

1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 400
1658 A(KEW) = X(KEW)

1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9), A(10)

1975 PRINT A(391), A(392), A(393), A(394), A(395)

1977 PRINT A(396), A(397), A(398), A(399), A(400)

1989 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [21]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.

29    26    46    3    37
49    37    50    35    37
11    11    31    38    39
26    19    32    36    14
0       -32000

33    40    25    13    1
36    9    27    35    25
32    18    7    37    51
36    28    46    47    26
-1       -31999

18    2    61    36    28
43    17    12    19    39
16    21    44    29    11
43    16    42    20    32
0      -31998

Above there is no rounding by hand; it is just straight copying by hand from the screen.

Of the 400 unknowns, only the 20 A’s of line 1911 through line 1977 are shown above.

On a personal computer using a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and using qb64v1000-win [21], the wall-clock time for obtaining the output through JJJJ= -31998 was seven seconds, not including “Creating .EXE file…” time–the total time was twenty seconds.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar, Particle Swarm Optimization Based Diophantine Equation Solver. https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf.

[2] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.

[3] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.

[4] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.

[5] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

[6] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[7] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.

[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

[9] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.

[10] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229. http://www.SciRP.org/journal/am.

[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

[12] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.

[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.

[14] 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.

[15 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[16] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.

[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.

[18] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adapttive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014. http://www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf.

[19] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[20] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

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

[22] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.

[23] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.

[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.

The Nonlinear Integer/Continuous Programming Solver Applied to a Nonlinear Diophantine Equation with 100-General-Integer Unknowns

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [18, p. 55; http://www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf%5D; also see Abraham, Sanyal, and Sanglikar [1; https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf%5D:

X(1)^2+ X(2)^2+X(3)^2+…+X(100)^2=100000.
0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+50

91 FOR KK = 1 TO 100
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK

128 FOR I = 1 TO 5000 STEP 1
129 FOR K = 1 TO 100
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 100)
183 REM R = (1 – RND * 2) * A(B)

188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)

191 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 100
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 100
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44

261 PZ = -ABS(SUML – 100000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 100
1658 A(KEW) = X(KEW)

1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9), A(10)
1915 PRINT A(11), A(12), A(13), A(14), A(15)

1917 PRINT A(16), A(17), A(18), A(19), A(20)

1918 PRINT A(21), A(22), A(23), A(24), A(25)

1919 PRINT A(26), A(27), A(28), A(29), A(30)

1925 PRINT A(31), A(32), A(33), A(34), A(35)

1927 PRINT A(36), A(37), A(38), A(39), A(40)
1935 PRINT A(41), A(42), A(43), A(44), A(45)

1937 PRINT A(46), A(47), A(48), A(49), A(50)

1945 PRINT A(51), A(52), A(53), A(54), A(55)

1947 PRINT A(56), A(57), A(58), A(59), A(60)

1955 PRINT A(61), A(62), A(63), A(64), A(65)

1957 PRINT A(66), A(67), A(68), A(69), A(70)
1965 PRINT A(71), A(72), A(73), A(74), A(75)

1967 PRINT A(76), A(77), A(78), A(79), A(80)
1968 PRINT A(81), A(82), A(83), A(84), A(85)

1969 PRINT A(86), A(87), A(88), A(89), A(90)
1975 PRINT A(91), A(92), A(93), A(94), A(95)

1977 PRINT A(96), A(97), A(98), A(99), A(100)
1989 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [21]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.

24    40    43    20    24
44    36    47    21    50
41    43    35    29    2
11    34    15    29    21
8    50    26    41    23
17    48    40    21    36
52    41    18    38    22
22    19    24    27    10
23    16    6    48    36
15    28    25    22    33
43    30    40    17    14
28    56    15    7    3
40    39    13    20    13
43    23    5    30    42
20    41    38    16    26
39    7    24    29    54
21    2    28    13    36
26    23    44    55    48
29    43    34    47    39
23    36    9    35    12
0          -32000

40 50 6 23 38
50 29 26 17 12
12 40 6 24 30
36 41 23 32 42
23 35 37 43 27
22 53 20 12 40
28 30 6 31 6
6 28 11 46 22
28 35 31 56 3
8 3 21 25 13
30 41 9 17 42
40 9 40 41 43
16 10 56 28 35
28 45 47 14 25
53 6 38 31 39
25 36 22 29 31
34 40 40 32 11
37 7 9 32 23
41 42 28 11 44
34 12 48 24 47
0          -31999

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 [21], the wall-clock time for obtaining the output through JJJJ= -31999 was three seconds, not including “Creating .EXE file…” time.

Acknowledgment

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

References

[1] S. Abraham, S. Sanyal, M. Sanglikar, Particle Swarm Optimization Based Diophantine Equation Solver. https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf.

[2] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.

[3] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.

[4] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.

[5] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

[6] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[7] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.

[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

[9] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.

[10] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229. http://www.SciRP.org/journal/am.

[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

[12] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.

[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.

[14] 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.

[15 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[16] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.

[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.

[18] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adapttive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014. http://www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf.

[19] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[20] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

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

[22] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.

[23] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.

[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.

The Nonlinear Integer/Continuous Programming Solver Applied to a Nonlinear Diophantine Equation with 50 General Integer Variables

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [17, p. 55]:

X(1)^2+ X(2)^2+X(3)^2+…+X(50)^2=50000.
0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+50

91 FOR KK = 1 TO 50
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK

128 FOR I = 1 TO 5000 STEP 1
129 FOR K = 1 TO 50
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 50)
183 REM R = (1 – RND * 2) * A(B)
188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)

191 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 50

203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 50
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44
261 PZ = -ABS(SUML – 50000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 50
1658 A(KEW) = X(KEW)

1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -99999 THEN 1999

1911 PRINT A(1), A(2), A(3), A(4), A(5)

1913 PRINT A(6), A(7), A(8), A(9), A(10)

1915 PRINT A(11), A(12), A(13), A(14), A(15)

1916 PRINT A(16), A(17), A(18), A(19), A(20)
1917 PRINT A(21), A(22), A(23), A(24), A(25)

1918 PRINT A(26), A(27), A(28), A(29), A(30)
1920 PRINT A(31), A(32), A(33), A(34), A(35)

1921 PRINT A(36), A(37), A(38), A(39), A(40)
1922 PRINT A(41), A(42), A(43), A(44), A(45)

1923 PRINT A(46), A(47), A(48), A(49), A(50)

1928 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [20]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.

33       35       38       17       40
48       35       31       31       40
27       38       47       31       10
34       39       11       16       24
33       52       37       38       26
41       45       47       21       26
37       26       4       45       20
15       12       11       17       42
19       20       21       29       27
13       23       30       20       46
0       -32000

37       22       43       12       40
32       8       48       44       15
28       15       21       35       39
18       29       23       6       28
2       20       12       48       34
33       43       32       19       29
27       25       35       25       44
25       22       50       19       26
50       42       43       40       21
9       25       43       40       32
0       -31999

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 [20], the wall-clock time for obtaining the output through JJJJ= -31999 was three seconds, not including “Creating .EXE file…” time.

Acknowledgment

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

References

[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.

[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.

[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.

[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.

[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.

[9] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229. http://www.SciRP.org/journal/am.

[10] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.

[12] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.

[13] 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.

[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[15] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.

[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.

[17] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adapttive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014.

[18] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

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

[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.

[22] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.

[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.

The Nonlinear Integer/Continuous Programming Solver Applied to a Nonlinear Diophantine Equation from the Literature

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [17, p. 56]:

X(1)^15+ X(2)^15=1088090731.

0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ

16 M = -1D+50

91 FOR KK = 1 TO 2

94 A(KK) = FIX(1 + RND * 10)

95 NEXT KK

128 FOR I = 1 TO 500 STEP 1
129 FOR K = 1 TO 2

131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 2)
183 R = (1 – RND * 2) * A(B)

188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)

191 IF RND < .5 THEN X(B) = CINT(A(B) – 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 2

203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 2
231 SUML = SUML + X(J44) ^ 15
251 NEXT J44
261 PZ = -ABS(SUML – 1088090731)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 2
1658 A(KEW) = X(KEW)

1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -1 THEN 1999

1915 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [20]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31995 is shown below.

4       3       0       -32000
4       3       0       -31999
4       3       0       -31998
3       4       0       -31997
3       4       0       -31996
4       3       0       -31995

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 [20], the wall-clock time for obtaining the output through JJJJ= -31995 was two seconds, not including “Creating .EXE file…” time.

Acknowledgment

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

References

[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.

[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.

[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.

[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.

[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.

[9] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229. http://www.SciRP.org/journal/am.

[10] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.

[12] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.

[13] 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.

[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[15] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.

[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.

[17] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adaptive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014.

[18] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

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

[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.

[22] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.

[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.

The Domino Method Applied to Solving Another Ordinary Differential Equation

Jsun Yui Wong

The following computer program seeks to estimate the solution of the boundary value problem of the last problem in Exercise 8.1 of Jacques and Judd [10, p. 271] by solving the corresponding system of nonlinear equations [10, p. 321].

0 DEFDBL A-Z

3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO -31997
14 RANDOMIZE JJJJ
16 M = -1D+50

91 FOR KK = 1 TO 4
94 IF RND < .5 THEN A(KK) = 1 – RND ELSE A(KK) = 1 + RND

95 NEXT KK

128 FOR I = 1 TO 10000 STEP 1
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)
183 R = (1 – RND * 2) * A(B)

188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
199 NEXT IPP
582 X(1) = (-5 * X(2)) / (-10 + X(2))
586 FOR J44 = 3 TO 4

611 X(J44) = (10 * X(J44 – 1) – 5 * X(J44 – 2) + X(J44 – 2) * X(J44 – 1)) / (5 + X(J44 – 1))

613 NEXT J44

811 PNEW = -9 * X(4) + 5 * X(3) – X(3) * X(4) + 5

999 P = -ABS(PNEW)

1451 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
1890 IF M < -.1 THEN 1999

1911 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

.2844718507803812       .5383165220917432       .7428145083348983
.8944101865246701       -5.864362814800295D-13          -32000

.2844718507820187       .5383165220946751       .7428145083386302
.8944101865287636       -2.514779096768338D-11          -31999

.2844718507803468       .5383165220916815       .7428145083348198
.894410186524584       -7.031875082219585D-14          -31998

.284471850780342       .538316522091673       .742814508334809
.8944101865245722       -3.313321839115702D-16          -31997

Above there is no rounding by hand; it is just straight copying by hand from the screen. See Jacques and Judd [10, p. 321].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [20], the wall-clock time for obtaining the output through JJJJ= -31997 was three seconds, not including “Creating .EXE file…” time.

Acknowledgment

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

References

[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.

[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.

[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.

[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.

[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.

[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.

[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.

[9] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229. http://www.SciRP.org/journal/am.

[10] Ian Jacques, Colin Judd. Numerical Analysis. Chapman and Hall Ltd., London, 1987.

[11] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.

[12] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.

[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.

[14] 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.

[15] J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.

[16] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.

[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.

[18] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.

[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.

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

[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.

[22] Xin-She Yang. Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.

[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.