Substance

Seeking an Integer Solution to a Large Freundenstein and Roth System of 32760 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to find an integer solution to the Freundenstein and Roth system on page 11 of Cao [2, page 11, Problem 14, extended Freundenstein and Roth function (n is even)]--http://dx.doi.org/10.1155/2014/251587. See also La Cruz et al. [5, page 28, Test function 37]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 32760 equations with 32760 general integer variables. One notes the starting vectors, 94 A(KK) = 4 + FIX(RND * 2.9); one also notes lines 426 and line 428, which are 426 IF X(J44) < 3 THEN X(J44) = 3 and 428 IF X(J44) > 7 THEN X(J44) = 7, respectively.

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+200

91 FOR KK = 1 TO 32760

94 A(KK) = 4 + FIX(RND * 2.9)

96 NEXT KK

128 FOR I = 1 TO 960000 STEP 1

129 FOR K = 1 TO 32760

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 32763)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

188 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

393 FOR J44 = 1 TO 16380

395 X(2 * J44 - 1) = -((X(2 * J44) + 1) * X(2 * J44) - 14) * X(2 * J44) + 29

397 NEXT J44

422 FOR J44 = 1 TO 32760
426 IF X(J44) < 3 THEN X(J44) = 3

428 IF X(J44) > 7 THEN X(J44) = 7

439 NEXT J44

773 FOR J44 = 1 TO 16380

775 P(2 * J44 - 1) = -ABS(X(2 * J44 - 1) + ((5 - X(2 * J44)) * X(2 * J44) - 2) * X(2 * J44) - 13)

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 16380

837 Pone = Pone + P(2 * J44 - 1)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 32760

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

1666 PRINT A(1), A(32760), M, JJJJ

1668 IF M > -.00001 THEN 1912

1670 NEXT I

1912 IF M < -8 THEN 1999
1891 PRINT A(1), A(2), A(3), A(4), A(5)
1894 PRINT A(11996), A(11997), A(11998), A(11999), A(12000)
1897 PRINT A(32756), A(32757), A(32758), A(32759), A(32760)

1939 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
3       4        -401108       -32000
3       4       -401076       -32000
.
.
.

5       4       -148       -32000
5       4       -134       -32000
5       4       -128       -32000
5       4       -122       -32000
5       4       -116       -32000
5       4       -102       -32000
5       4       -96         -32000
5       4       -90       -32000
5       4       -84       -32000
5       4       -78       -32000
5       4       -72       -32000
5       4       -66       -32000
5       4       -60       -32000
5       4       -54       -32000
5       4       -48       -32000
5       4       -42       -32000
5       4       -36       -32000
5       4       -30       -32000
5       4       -24       -32000
5       4       -18       -32000
5       4       -12       -32000
5       4       -6       -32000
5       4       0       -32000
5       4       5       4       5
4       5       4       5       4
4       5       4       5       4
0       -32000

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

Of the 32760 unknowns, only the 15 A’s of line 1891 and line 1897 are shown above.

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 for obtaining the output through JJJJ= -32000 was 40 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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 of Nonlinear Integer/Continuous/Discrete Programming Seeking a Solution to a Large Troesch System of 32760 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to solve the Troesch problem on page 29 of La Cruz et al. [5, page 29, Test function 43, Troesch problem].–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 32760 equations with 32760 variables. One notes the starting vectors, 94 A(KK) = FIX(RND * 6).

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), L(32768), K(32768)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
15 h = 1 / (32760 + 1)

16 M = -1D+50

91 FOR KK = 1 TO 32760

94 A(KK) = FIX(RND * 6)

95 NEXT KK

128 FOR I = 1 TO 600000 STEP 1

129 FOR K = 1 TO 32760

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 32763)

183 R = (1 - RND * 2) * A(B)

187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

191 NEXT IPP

555 X(2) = 2 * X(1) + 10 * h ^ 2 * SIN((h) * (10 * X(1)))

605 FOR J44 = 2 TO 32759

609 X(J44 + 1) = 2 * X(J44) + 10 * h ^ 2 * SIN((h) * (10 * X(J44))) - X(J44 - 1)

611 NEXT J44

699 P1 = 2 * X(32760) + 10 * h ^ 2 * (SIN((h)) * (10 * X(32760))) - X(32759)

999 P = -ABS(P1)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 32760

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(2), A(32760), M, JJJJ
1668 IF M > -.0000000001 THEN 1912

1670 NEXT I
1890 IF M < -5 THEN 1999

1912 PRINT A(1), A(2), A(3)

1917 PRINT A(32757), A(32758), A(32759)

1939 PRINT A(32760), M, JJJJ

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= -32000 is shown below.

2                                                4.000000000005688      65520.47716298586
-65522.47717219543       -32000

1.474635879184916       2.949271758374026         48309.7117800988
-48311.18644865046       -32000

1.47375489886832         2.947509797740831         48280.8509971798
-48282.3247846206       -32000

1.472006149411892       2.94401229882797           48223.56222909887
-48225.03426752713       -32000

1.451231022124283       2.902462044252693        47542.97313967141
-47544.4243996695       -32000

1.45049948026996        2.900998960544045       47519.00800457551
-47520.4585327924       -32000

1.432492348244362       2.864984696492798       46929.09950080921
-46930.53201856688       -32000

.3395539431503268       .6791078863016192         11126.94961386974
-11127.28934468647        -32000

0       0       0       0       -32000

0       0       0
0       0       0
0       0       -32000

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

Of the 32760 unknowns, only the 7 A’s of line 1912 through line 1939 are shown above.

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 for obtaining the output through JJJJ= -32000 was 31 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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 of Nonlinear Integer/Continuous/Discrete Programming Seeking a Solution to a Troesch System of 20000 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to solve the Troesch problem on page 29 of La Cruz et al. [5, page 29, Test function 43, Troesch problem].–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 20000 equations with 20000 variables. One notes the starting vectors, 94 A(KK) = FIX(RND * 6).

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), L(32768), K(32768)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
15 h = 1 / (20000 + 1)

16 M = -1D+50

91 FOR KK = 1 TO 20000

94 A(KK) = FIX(RND * 6)

95 NEXT KK

128 FOR I = 1 TO 480000 STEP 1

129 FOR K = 1 TO 20000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 20003)

183 R = (1 - RND * 2) * A(B)

187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

191 NEXT IPP

555 X(2) = 2 * X(1) + 10 * h ^ 2 * SIN((h) * (10 * X(1)))

605 FOR J44 = 2 TO 19999

609 X(J44 + 1) = 2 * X(J44) + 10 * h ^ 2 * SIN((h) * (10 * X(J44))) - X(J44 - 1)

611 NEXT J44

699 P1 = 2 * X(20000) + 10 * h ^ 2 * (SIN((h)) * (10 * X(20000))) - X(19999)

999 P = -ABS(P1)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 20000

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(2), A(20000), M, JJJJ
1668 IF M > -.0000000001 THEN 1912

1670 NEXT I
1890 IF M < -5 THEN 1999

1912 PRINT A(1), A(2), A(3)

1917 PRINT A(19997), A(19998), A(19999)

1939 PRINT A(20000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
.9734582262498015            1.94691645251177            19470.2238435021
-19471.19740224601           -32000
0          0          0          0          -32000
0          0          0
0          0          0
0          0          -32000

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

Of the 20000 unknowns, only the 7 A’s of line 1912 through line 1939 are shown above.

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 for obtaining the output through JJJJ= -32000 was 16 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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 of Nonlinear Integer/Continuous/Discrete Programming Seeking a Solution to a Troesch System of 10000 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to solve the Troesch problem on page 29 of La Cruz et al. [5, page 29, Test function 43, Troesch problem].–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 10000 equations with 10000 variables. One notes the starting vectors, 94 A(KK) = FIX(RND * 6).

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), L(32768), K(32768)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ

15 h = 1 / (10000 + 1)

16 M = -1D+50

91 FOR KK = 1 TO 10000

94 A(KK) = FIX(RND * 6)

95 NEXT KK

128 FOR I = 1 TO 480000 STEP 1

129 FOR K = 1 TO 10000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 10003)

183 R = (1 - RND * 2) * A(B)

187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

191 NEXT IPP

555 X(2) = 2 * X(1) + 10 * h ^ 2 * SIN((h) * (10 * X(1)))

605 FOR J44 = 2 TO 9999

609 X(J44 + 1) = 2 * X(J44) + 10 * h ^ 2 * SIN((h) * (10 * X(J44))) - X(J44 - 1)

611 NEXT J44

699 P1 = 2 * X(10000) + 10 * h ^ 2 * (SIN((h)) * (10 * X(10000))) - X(9999)

999 P = -ABS(P1)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10000

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(2), A(10000), M, JJJJ
1668 IF M > -.0000000001 THEN 1912

1670 NEXT I
1890 IF M < -5 THEN 1999

1912 PRINT A(1), A(2), A(3)

1917 PRINT A(9997), A(9998), A(9999)

1939 PRINT A(10000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
.947022837213985          1.894045674522644          9471.289137376827
-9472.236372197132          -32000
0       0       0       0       -32000
0       0       0
0       0       0
0       0       -32000

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

Of the 10000 unknowns, only the 7 A’s of line 1912 through line 1939 are shown above.

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 for obtaining the output through JJJJ= -32000 was four minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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 of Nonlinear Integer/Continuous/Discrete Programming Seeking an Integer Solution to a Cragg and Levy System of 20000 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to find an integer solution to the Cragg and Levy on page 28 of La Cruz et al. [5, page 28, Test function 38, extended Cragg and Levy problem (n is a multiple of 4)].–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 20000 equations with 20000 general integer variables. One notes the starting vectors, 94 A(KK) = FIX(RND * 1.9). One also notes lines 395, 445, 495, 775, and 837.

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+50

91 FOR KK = 1 TO 20000

94 A(KK) = FIX(RND * 1.9)

96 NEXT KK

128 FOR I = 1 TO 360000 STEP 1

129 FOR K = 1 TO 20000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 20003)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

393 FOR J44 = 1 TO 5000

395 X(4 * J44) = 1

397 NEXT J44

443 FOR J44 = 1 TO 5000

445 X(4 * J44 - 1) = X(4 * J44)

447 NEXT J44

493 FOR J44 = 1 TO 5000

495 X(4 * J44 - 2) = X(4 * J44 - 1)

497 NEXT J44

773 FOR J44 = 1 TO 5000

775 P(4 * J44 - 3) = -ABS((EXP(X(4 * J44 - 3)) - X(4 * J44 - 2)) ^ 2)

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 5000

837 Pone = Pone + P(4 * J44 - 3)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 20000

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(20000), M, JJJJ

1668 IF M > -.00001 THEN 1891

1670 NEXT I

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)
1895 PRINT A(19994), A(19995), A(19996), A(19997), A(19998)

1939 PRINT A(19999), A(20000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
0       1       -32.47741       -32000
0       1       -29.52492       -32000
0       1       -26.57243       -32000
0       1       -23.61994       -32000
0       1       -20.66745       -32000
0       1       -17.71495       -32000
0       1       -14.76246       -32000
0       1       -11.80997       -32000
0       1       -8.857477       -32000
0       1       -5.904985       -32000
0       1       -2.952492       -32000
0       1        0        -32000
0       1       1       1       0
1       1       1       0       1
1       1       1       0       1
1       1       0       -32000

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

Of the 20000 unknowns, only the 17 A’s of line 1891 through line 1939 are shown above.

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 for obtaining the output through JJJJ= -32000 was seven minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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 of Nonlinear Integer/Continuous/Discrete Programming Seeking an Integer Solution to a Cragg and Levy System of 12000 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to find an integer solution to the Cragg and Levy on page 28 of La Cruz et al. [5, page 28, Test function 38, extended Cragg and Levy problem (n is a multiple of 4)].–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 12000 equations with 12000 general integer variables. One notes the starting vectors, 94 A(KK) = FIX(RND * 1.9). One also notes lines 395, 445, 495, 775, and 837.

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+50

91 FOR KK = 1 TO 12000

94 A(KK) = FIX(RND * 1.9)

96 NEXT KK

128 FOR I = 1 TO 240000 STEP 1

129 FOR K = 1 TO 12000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 12003)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

393 FOR J44 = 1 TO 3000

395 X(4 * J44) = 1

397 NEXT J44

443 FOR J44 = 1 TO 3000

445 X(4 * J44 - 1) = X(4 * J44)

447 NEXT J44

493 FOR J44 = 1 TO 3000

495 X(4 * J44 - 2) = X(4 * J44 - 1)

497 NEXT J44

773 FOR J44 = 1 TO 3000

775 P(4 * J44 - 3) = -ABS((EXP(X(4 * J44 - 3)) - X(4 * J44 - 2)) ^ 2)

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 3000

837 Pone = Pone + P(4 * J44 - 3)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 12000

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(12000), M, JJJJ

1668 IF M > -.00001 THEN 1891

1670 NEXT I

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)
1895 PRINT A(11994), A(11995), A(11996), A(11997), A(11998)

1939 PRINT A(11999), A(12000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
0       1       -23.61994       -32000
0       1       -20.66745       -32000
0       1       -17.71495        -32000
0       1       -14.76246       -32000
0       1       -11.80997       -32000
0       1       -8.857477       -32000
0       1       -5.904985       -32000
0       1       -2.952492       -32000
0       1       0       -32000
0       1       1       1       0
1        1       1       0       1
1        1       1       0       1
1        1       0       -32000

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

Of the 12000 unknowns, only the 17 A’s of line 1891 through line 1939 are shown above.

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 for obtaining the output through JJJJ= -32000 was three minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64Equations

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

Seeking an Integer Solution to a Freundenstein and Roth System of 22000 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to find an integer solution to the Freundenstein and Roth system on page 11 of Cao [2, page 11, Problem 14, extended Freundenstein and Roth function (n is even)]--http://dx.doi.org/10.1155/2014/251587. See also La Cruz et al. [5, page 28, Test function 37]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 22000 equations with 22000 general integer variables. One notes the starting vectors, 94 A(KK) = 4 + FIX(RND * 2.9); one also notes lines 426 and line 428, which are 426 IF X(J44) < 3 THEN X(J44) = 3 and 428 IF X(J44) > 7 THEN X(J44) = 7, respectively.

0 REM DEFDBL A-Z

3 DEFINT J, K

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+200

91 FOR KK = 1 TO 22000

94 A(KK) = 4 + FIX(RND * 2.9)

96 NEXT KK

128 FOR I = 1 TO 480000 STEP 1

129 FOR K = 1 TO 22000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 22003)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

188 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

393 FOR J44 = 1 TO 11000

395 X(2 * J44 - 1) = -((X(2 * J44) + 1) * X(2 * J44) - 14) * X(2 * J44) + 29

397 NEXT J44

422 FOR J44 = 1 TO 22000
426 IF X(J44) < 3 THEN X(J44) = 3

428 IF X(J44) > 7 THEN X(J44) = 7

439 NEXT J44

773 FOR J44 = 1 TO 11000

775 P(2 * J44 - 1) = -ABS(X(2 * J44 - 1) + ((5 - X(2 * J44)) * X(2 * J44) - 2) * X(2 * J44) - 13)

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 11000

837 Pone = Pone + P(2 * J44 - 1)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 22000

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

1666 PRINT A(1), A(22000), M, JJJJ

1668 IF M > -.00001 THEN 1912

1670 NEXT I

1912 IF M < -8 THEN 1999
1891 PRINT A(1), A(2), A(3), A(4), A(5)
1894 PRINT A(11996), A(11997), A(11998), A(11999), A(12000)
1897 PRINT A(21996), A(21997), A(21998), A(21999), A(22000)

1939 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.

5       4       -60       -32000
5       4       -54       -32000
5       4       -48       -32000
5       4       -42       -32000
5       4       -36       -32000
5       4       -30       -32000
5       4       -24        -32000
5       4       -18        -32000
5       4       -12        -32000
5       4       -6           -32000
5       4        0           -32000
5       4       5       4       5
4       5       4       5       4
4       5       4       5       4
0       -32000

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

Of the 22000 unknowns, only the 15 A’s of line 1891 and line 1897 are shown above.

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 for obtaining the output through JJJJ= -32000 was 17 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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

Seeking an Integer Solution to a Freundenstein and Roth System of 11000 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to find an integer solution to the Freundenstein and Roth system on page 11 of Cao [2, page 11, Problem 14, extended Freundenstein and Roth function (n is even)]--http://dx.doi.org/10.1155/2014/251587. See also La Cruz et al. [5, page 28, Test function 37]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 11000 equations with 11000 general integer variables. One notes the starting vectors, 94 A(KK) = 4 + FIX(RND * 2.9); one also notes lines 426 and line 428, which are 426 IF X(J44) < 3 THEN X(J44) = 3 and 428 IF X(J44) > 7 THEN X(J44) = 7, respectively.

0 REM DEFDBL A-Z

3 DEFINT J, K

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+200

91 FOR KK = 1 TO 11000

94 A(KK) = 4 + FIX(RND * 2.9)

96 NEXT KK

128 FOR I = 1 TO 240000 STEP 1

129 FOR K = 1 TO 11000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 11003)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

188 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

393 FOR J44 = 1 TO 5500

395 X(2 * J44 - 1) = -((X(2 * J44) + 1) * X(2 * J44) - 14) * X(2 * J44) + 29

397 NEXT J44

422 FOR J44 = 1 TO 11000
426 IF X(J44) < 3 THEN X(J44) = 3

428 IF X(J44) > 7 THEN X(J44) = 7

439 NEXT J44

773 FOR J44 = 1 TO 5500

775 P(2 * J44 - 1) = -ABS(X(2 * J44 - 1) + ((5 - X(2 * J44)) * X(2 * J44) - 2) * X(2 * J44) - 13)

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 5500

837 Pone = Pone + P(2 * J44 - 1)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 11000

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

1666 PRINT A(1), A(11000), M, JJJJ

1668 IF M > -.00001 THEN 1912

1670 NEXT I

1912 IF M < -8 THEN 1999
1891 PRINT A(1), A(2), A(3), A(4), A(5)

1897 PRINT A(10996), A(10997), A(10998), A(10999), A(11000)

1939 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.

5       4       -120       -32000
5       4       -82        -32000
5       4       -76        -32000
5       4       -70        -32000
5       4       -64        -32000
5       4       -58        -32000
5       4       -38        -32000
5       4       -32        -32000
5       4       -26        -32000
5       4       -6           -32000
5       4        0           -32000
5       4       5       4       5
4       5       4       5       4
0       -32000

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

Of the 11000 unknowns, only the 10 A’s of line 1891 and line 1897 are shown above.

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 for obtaining the output through JJJJ= -32000 was 5 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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

Seeking an Integer Solution to a System of 8000 Simultaneous Nonlinear Equations Involving Sines and Cosines

Jsun Yui Wong

The following computer program seeks to find an integer solution to the trigonometric system on page 22 of La Cruz et al. [5, page 22, Test function 8, Trigonometric function]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 8000 equations with 8000 integer variables. One notes the starting vector, 94 A(KK) = FIX(RND * 1.9); one also notes line 189, which is 189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1.

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+50

91 FOR KK = 1 TO 8000

94 A(KK) = FIX(RND * 1.9)

96 NEXT KK

128 FOR I = 1 TO 1200000 STEP 1

129 FOR K = 1 TO 8000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 8003)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

211 nsum = 0
222 FOR J44 = 1 TO 8000
225 nsum = nsum + COS(X(J44))

228 NEXT J44

333 REM

774 FOR J44 = 1 TO 8000

775 P(J44) = -ABS(2 * (8000 + J44 * (1 - COS(X(J44))) - SIN(X(J44)) - nsum) * (2 * SIN(X(J44)) - COS(X(J44))))

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 8000

837 Pone = Pone + P(J44)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 8000

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(8000), M, JJJJ

1668 IF M > -.00001 THEN 1891

1670 NEXT I

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1897 PRINT A(7996), A(7997), A(7998), A(7999), A(8000)

1949 PRINT M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
0       0       -41773.37       -32000
0       0       -34168.6         -32000
0       0       -33238.38       -32000
0       0       -32620.42       -32000
0       0       -23177.79       -32000
0       0       -15678.31       -32000
0       0       -7792.506       -32000
0       0       0       -32000
0       0       0       0       0
0       0       0       0       0
0       0       0       0       0
0       -32000

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

Of the 8000 unknowns, only the 15 A’s of line 1891 through line 1897 are shown above.

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 for obtaining the output through JJJJ= -32000 was 16 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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

Seeking an Integer Solution to a System of 7000 Simultaneous Nonlinear Equations Involving Sines and Cosines

Jsun Yui Wong

The following computer program seeks to find an integer solution to the trigonometric system on page 22 of La Cruz et al. [5, page 22, Test function 8, Trigonometric function]–http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present paper considers the case of 7000 equations with 7000 integer variables. One notes the starting vector, 94 A(KK) = FIX(RND * 1.9); one also notes line 189, which is 189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1.

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(32768), A(32768), P(32768), K(32768), Q(2222)

5 FOR JJJJ = -32000 TO -32000

14 RANDOMIZE JJJJ
16 M = -1D+50

91 FOR KK = 1 TO 7000

94 A(KK) = FIX(RND * 1.9)

96 NEXT KK

128 FOR I = 1 TO 1200000 STEP 1

129 FOR K = 1 TO 7000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 7003)

183 REM R = (1 - RND * 2) * A(B)

187 REM IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1

189 IF RND < .5 THEN X(B) = ABS(A(B) - 1) ELSE X(B) = A(B) + 1

191 NEXT IPP

211 nsum = 0
222 FOR J44 = 1 TO 7000
225 nsum = nsum + COS(X(J44))

228 NEXT J44

333 REM

774 FOR J44 = 1 TO 7000

775 P(J44) = -ABS(2 * (7000 + J44 * (1 - COS(X(J44))) - SIN(X(J44)) - nsum) * (2 * SIN(X(J44)) - COS(X(J44))))

777 NEXT J44

822 Pone = 0

833 FOR J44 = 1 TO 7000

837 Pone = Pone + P(J44)

855 NEXT J44

998 P = Pone

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 7000

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 PRINT A(1), A(7000), M, JJJJ

1668 IF M > -.00001 THEN 1891

1670 NEXT I

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1897 PRINT A(6996), A(6997), A(6998), A(6999), A(7000)

1949 PRINT M, JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s output through JJJJ= -32000 is summarized below.

.
.
.
0       0       -42641.96       -32000
0       0       -34927.57       -32000
0       0       -28168.44       -32000
0       0       -20810.77       -32000
0       0       -13471.76       -32000
0       0       -6852.624       -32000
0       0       0       -32000
0       0       0       0       0
0       0       0       0       0
0       0       0       0       0
0       -32000

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

Of the 7000 unknowns, only the 15 A’s of line 1891 through line 1897 are shown above.

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 for obtaining the output through JJJJ= -32000 was 11 minutes.

Acknowledgment

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

References

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

[2] 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

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

[4] 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

[5] 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

[6] 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.

[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] 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

[9] 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.

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

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

[12] 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