Substance

The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To Solve Case 5 with n=60 of Broyden's Tridiagonal Simultaneous Equations

Jsun Yui Wong

The following computer program seeks to solve Broyden's Case 5 [1, p. 587--Case 5; p. 590--Table 5].  See also page 23 of La Cruz, Martinez, and Raydan [7, p. 23, Test function 11, Broyden Tridiagonal function]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf --and page 7 of Cao [3, p. 7]--http://dx.doi.org/10.1155/2014/251587.  Here alpha=-0.1, beta=1, and n=60.

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), L(32768), K(32768), P(999)

5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+50

91 FOR KK = 1 TO 60

94 A(KK) = -RND

95 NEXT KK

128 FOR I = 1 TO 20000000 STEP 1

129 FOR K = 1 TO 60

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 60)

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) = ((3 - .1 * X(1)) * X(1) + 1) / 2

605 FOR J44 = 2 TO 59

609 X(J44 + 1) = (-X(J44 - 1) + (3 - .1 * X(J44)) * X(J44) + 1) / 2

611 NEXT J44

651 FOR J47 = 1 TO 60

666 IF ABS(X(J47)) > 10 THEN 1670

688 NEXT J47

699 PZ = -X(59) + (3 - .1 * X(60)) * X(60) + 1

999 P = -ABS(PZ)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 60

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(2), A(50), M, JJJJ
1668 IF M > -.000001 THEN 1912

1670 NEXT I
1890 REM IF M < -.1 THEN 1999

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

1946 PRINT A(58), A(59), A(60)

1947 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.

-2.046688482096033       -2.779479410281278       -3.032150163982778
-1.93335468679104       -1.436533614136897       -.7913045190370864
-3.772840786009368D-06           -32000

-2.046688482096033       -2.779479410281278       -3.032150163982778
-1.93335468679104       -1.436533614136897       -.7913045190370864
-3.772840786009368D-06           -31999

-2.046688482096033       -2.779479410281278       -3.032150163982778
-1.93335468679104       -1.436533614136897       -.7913045190370864
-3.772840786009368D-06           -31998

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

Of the 50 unknowns, only the six A's of line 1912 and line 1946 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 [15], the wall-clock time for obtaining the output through JJJJ= -31998 was seven minutes.

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

[5] 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.

[6] 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.

[7] 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.

[8] 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.

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

[11] 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.

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

[13] 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.

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

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

[16] 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.

Testing the Domino Method of Nonlinear Integer/Continuous/Discrete Programming with a System of 1119 Nonlinear Equations from a Discrete Boundary Value Problem

Jsun Yui Wong

The following computer program seeks a solution to the system of nonlinear equations of a discrete boundary value problem on page 29 of La Cruz, Martinez, and Raydan [7, p. 29, Test function 41, Discrete boundary value problem]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present case has 1119 nonlinear equations and 1119 continuous variables.

One notes line 93, which comes from La Cruz, Martinez, and Raydan [7, p. 29] and is 93 A(KK) = (h * (KK * h - 1)).

0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 31111

14 RANDOMIZE JJJJ
16 M = -1D+50

22 h = 1 / (1119 + 1)

91 FOR KK = 1 TO 1119

93 A(KK) = (h * (KK * h - 1))

95 NEXT KK

128 FOR I = 1 TO 1000000 STEP 1

129 FOR K = 1 TO 1119

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 1122)

183 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 < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 7 * R

199 NEXT IPP

566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3

605 FOR J49 = 2 TO 1118
610 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * J49) ^ 3 - X(J49 - 1) + X(J49 + 1)

613 NEXT J49
615 PS = 0

617 FOR J49 = 2 TO 1118
619 PS = PS - ABS(P(J49))
629 NEXT J49

622 FOR J33 = 1 TO 1119

633 IF ABS(X(J33)) > 3 THEN 1670

655 NEXT J33

677 PZ = 2 * X(1119) + .5 * h ^ 2 * (X(1119) + h * 1119) ^ 3 - X(1118)

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 1119

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

1670 NEXT I
1890 REM IF M < -9 THEN 1999

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

1917 PRINT A(1117), A(1118), A(1119)

1939 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= -32000 is shown below.

-1.8805044822603D-21          2.837096668503351D-16          -2.923150630666618D-15
-2.112964279412783D-07          -1.635987264543785D-07          -2.805641930142923D-07
-1.249427790689917D-09          -32000

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

Of the 1119 unknowns, only the six A's of line 1912 and line 1917 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 [16], the wall-clock time for obtaining the output through JJJJ= -32000 was five minutes.

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

[5] 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.

[6] 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.

[7] 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.

[8] 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.

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

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

[12] 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.

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

[14] 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.

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

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

[17] 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 Only Integer Solutions to a Tridiagonal System of 1019 Nonlinear Equations

Jsun Yui Wong

The following computer program seeks only integer solutions to the system of nonlinear equations of the tridiagonal system on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 34, Tridiagonal system]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present case has 1019 equations and 1019 general integer variables.

0 DEFDBL A-Z
3 DEFINT J, K, X

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 1019

94 A(KK) = RND * 3

95 NEXT KK

128 FOR I = 1 TO 300000 STEP 1

129 FOR K = 1 TO 1019

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 1022)

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

199 NEXT IPP

577 X(1) = X(2) ^ 2

605 FOR J49 = 2 TO 1018

610 P(J49) = 8 * X(J49) * (X(J49) ^ 2 - X(J49 - 1)) - 2 * (1 - X(J49)) + 4 * (X(J49) - X(J49 + 1) ^ 2)

611 NEXT J49
615 PS = 0

617 FOR J49 = 2 TO 1018
619 PS = PS - ABS(P(J49))
629 NEXT J49

622 FOR J33 = 1 TO 1019

633 IF ABS(X(J33)) > 5 THEN 1670

655 NEXT J33

666 PZ = 8 * X(1019) * (X(1019) ^ 2 - X(1018)) - 2 * (1 - X(1019))

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 1019

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1668 IF M > -.000001 THEN 1890

1670 NEXT I
1890 IF M < -.00001 THEN 1999

1912 PRINT A(1), A(2), A(3), A(4), A(5)

1917 PRINT A(1015), A(1016), A(1017), A(1018), A(1019)

1939 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= -31998 is shown below.

1    1    1    1    1
1    1    1    1    1
0       -32000

1    1    1   1   1
1    1    1   1   1
0       -31999

1    1    1   1   1
1    1    1   1   1
0       -31998

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

Of the 1019 unknowns, only the ten A's of line 1912 and line 1917 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 [16], the wall-clock time for obtaining the output through JJJJ= -31998 was 80 seconds.

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

[5] 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.

[6] 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.

[7] 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.

[8] 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.

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

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

[12] 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.

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

[14] 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.

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

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

[17] 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.

Testing the Domino Method of Nonlinear Integer/Continuous/Discrete Programming with a System of 2119 Nonlinear Equations from a Discrete Boundary Value Problem

Jsun Yui Wong

The following computer program seeks a solution to the system of nonlinear equations of a discrete boundary value problem on page 29 of La Cruz, Martinez, and Raydan [7, p. 29, Test function 41, Discrete boundary value problem]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. The present case has 2119 equations and 2119 variables.

One notes line 93, which comes from La Cruz, Martinez, and Raydan [7, p. 29] and is 93 A(KK) = (h * (KK * h - 1)).

0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 31111

14 RANDOMIZE JJJJ
16 M = -1D+50

22 h = 1 / (2119 + 1)

91 FOR KK = 1 TO 2119

93 A(KK) = (h * (KK * h - 1))

95 NEXT KK

128 FOR I = 1 TO 1000000 STEP 1

129 FOR K = 1 TO 2119

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 2122)

183 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 < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 7 * R

199 NEXT IPP

566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3

605 FOR J49 = 2 TO 2118
610 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * J49) ^ 3 - X(J49 - 1) + X(J49 + 1)

613 NEXT J49
615 P0 = 0

617 FOR j11 = 2 TO 2118
619 P0 = P0 - ABS(P(j11))
629 NEXT j11

622 FOR j33 = 1 TO 2119

633 IF ABS(X(j33)) > 3 THEN 1670

655 NEXT j33

677 P2119 = 2 * X(2119) + .5 * h ^ 2 * (X(2119) + h * 2119) ^ 3 - X(2118)

999 P = -ABS(P2119) + P0

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 2119

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

1670 NEXT I
1890 IF M < -9 THEN 1999

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

1917 PRINT A(2117), A(2118), A(2119)

1939 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= -31999 is shown below.

-7.705519547750754D-18       -3.735130144469935D-18       -2.262544050832445D-17
-5.925820507706342D-08        -4.58567482475182D-08      -7.847318931529344D-08
-1.55636724387727D-08          -32000

-4.223719379446417D-17       -7.279847863789934D-17       -3.690039609107596D-16
-5.925264284304708D-08       -4.58572195745407D-08      -7.847394218806585D-08
-2.371508528785914D-08          -31999

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

Of the 2119 unknowns, only the six A's of line 1912 and line 1917 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 [16], the wall-clock time for obtaining the output through JJJJ= -31999 was 20 minutes.

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

[5] 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.

[6] 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.

[7] 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.

[8] 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.

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

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