Testing the Domino Method of Nonlinear Integer/Continuous/Discrete Programming with a Nonlinear System of 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.

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 32000
    14 RANDOMIZE JJJJ
    16 M = -1D+50

    22 h = 1 / (119 + 1)


    91 FOR KK = 1 TO 119

        93 A(KK) = (h * (KK * h - 1))

    95 NEXT KK



    128 FOR I = 1 TO 500000 STEP 1


        129 FOR K = 1 TO 119

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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 122)

            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 118
            611 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 118
            619 P0 = P0 - ABS(P(j11))
        629 NEXT j11


        622 FOR j33 = 1 TO 119

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


        655 NEXT j33


        677 P119 = 2 * X(119) + .5 * h ^ 2 * (X(119) + h * 119) ^ 3 - X(118)




        999 P = -ABS(P119) + P0



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 119

            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(117), A(118), A(119)

    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.

-2.68801128311508D-11            -3.36663472559238D-11        -1.202984329827381D-10
-1.688823992425807D-05          -1.318892470245795D-05       -2.352394767238498D-05
-8.339331381840622D-12          -32000

-2.687959843339338D-11           -3.366531846040526D-11       -1.202995204456031D-10
-1.688824009365936D-05           -1.318892476496033D-05       -2.35239477694988D-05
-8.327981091731493D-12          -31999

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

Of the 119 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 one minute.

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 an Integer Solution to a Five-Diagonal System of Nonlinear Equations from the Literature, Revised Edition

Jsun Yui Wong

The following computer program seeks an integer solution to the five-diagonal system of nonlinear equations on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 35, Five-diagonal system]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present case has 32035 nonlinear equations and 32035 unknowns.

One notes line 611, which is 611 IF Snew ^ .5 <> Xnew ^ .5 THEN 1670.
.
0 DEFDBL A-Z
3 DEFINT J, K, X
4 DIM X(32768), A(32768), L(32768), K(32768), S(32768)

5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 32035


        94 A(KK) = RND



    95 NEXT KK

    128 FOR I = 1 TO 1000000 STEP 1



        129 FOR K = 1 TO 32035


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 32038)

            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
        433 S(1) = (4 * X(2) ^ 2 - X(2) + X(3) ^ 2) / 4


        455 X(1) = (4 * X(2) ^ 2 - X(2) + X(3) ^ 2) / 4


        456 IF S(1) <> X(1) THEN 1670


        557 IF (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) < 0 THEN 1670


        558 S(4) = (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) ^ .5

        559 X(4) = (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) ^ .5

        588 IF S(4) <> X(4) THEN 1670



        605 FOR J44 = 3 TO 32033



            607 Xnew = 8 * X(J44) * (X(J44) ^ 2 - X(J44 - 1)) - 2 * (1 - X(J44)) + 4 * (X(J44) - X(J44 + 1) ^ 2) + X(J44 - 1) ^ 2 - X(J44 - 2) + X(J44 + 1)

            618 IF Xnew < 0 THEN 1670
            606 Snew = 8 * X(J44) * (X(J44) ^ 2 - X(J44 - 1)) - 2 * (1 - X(J44)) + 4 * (X(J44) - X(J44 + 1) ^ 2) + X(J44 - 1) ^ 2 - X(J44 - 2) + X(J44 + 1)

            611 IF Snew ^ .5 <> Xnew ^ .5 THEN 1670



            621 X(J44 + 2) = Xnew ^ .5



        641 NEXT J44

        651 FOR j47 = 1 TO 32035


            666 IF ABS(X(j47)) > 30 THEN 1670


        688 NEXT j47


        699 P1 = 8 * X(32035) * (X(32035) ^ 2 - X(32034)) - 2 * (1 - X(32035)) + X(32034) ^ 2 - X(32033)


        711 P2 = 8 * X(32034) * (X(32034) ^ 2 - X(32033)) - 2 * (1 - X(32034)) + 4 * (X(32034) - X(32035) ^ 2) + X(32033) ^ 2 - X(32032)


        999 P = -ABS(P1) - ABS(P2)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 32035


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM PRINT A(1), A(2), A(32035), M, JJJJ


        1668 IF M > -.000001 THEN 1912


    1670 NEXT I
    1890 IF M < -1 THEN 1999


    1912 PRINT A(1), A(2), A(3)


    1917 PRINT A(4), A(5), A(6)
    1943 PRINT A(32030), A(32031), A(32032)


    1946 PRINT A(32033), A(32034), A(32035)

    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= -31996 is shown below.

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

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

1      1      1
1      1      1
1      1      1
1      1      1
0      -31997

1      1      1
1      1      1
1      1      1
1      1      1
0      -31996

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

Of the 32035 unknowns, only the 12 A's of line 1912 through 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= -31996 was 16 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.

The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking an Integer Solution to a Five-Diagonal System of Nonlinear Equations from the Literature

Jsun Yui Wong

The following computer program seeks an integer solution to the five-diagonal system of nonlinear equations on page 27 of La Cruz, Martinez, and Raydan [7, p. 27, Test function 35, Five-diagonal system]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present case has 32035 nonlinear equations and 32035 unknowns.

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

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

5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 32035


        94 A(KK) = RND



    95 NEXT KK

    128 FOR I = 1 TO 1000000 STEP 1



        129 FOR K = 1 TO 32035


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 32038)

            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(1) = (4 * X(2) ^ 2 - X(2) + X(3) ^ 2) / 4
        557 IF (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) < 0 THEN 1670


        559 X(4) = (8 * X(2) * (X(2) ^ 2 - X(1)) - 2 * (1 - X(2)) + 4 * (X(2) - X(3) ^ 2) + X(3) ^ 2) ^ .5



        605 FOR J44 = 3 TO 32033


            607 Xnew = 8 * X(J44) * (X(J44) ^ 2 - X(J44 - 1)) - 2 * (1 - X(J44)) + 4 * (X(J44) - X(J44 + 1) ^ 2) + X(J44 - 1) ^ 2 - X(J44 - 2) + X(J44 + 1)

            618 IF Xnew < 0 THEN 1670

            621 X(J44 + 2) = Xnew ^ .5



        641 NEXT J44

        651 FOR j47 = 1 TO 32035


            666 IF ABS(X(j47)) > 30 THEN 1670


        688 NEXT j47


        699 P1 = 8 * X(32035) * (X(32035) ^ 2 - X(32034)) - 2 * (1 - X(32035)) + X(32034) ^ 2 - X(32033)


        711 P2 = 8 * X(32034) * (X(32034) ^ 2 - X(32033)) - 2 * (1 - X(32034)) + 4 * (X(32034) - X(32035) ^ 2) + X(32033) ^ 2 - X(32032)


        999 P = -ABS(P1) - ABS(P2)



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 32035


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM PRINT A(1), A(2), A(32035), M, JJJJ

        1668 IF M > -.000001 THEN 1912


    1670 NEXT I
    1890 IF M < -1 THEN 1999


    1912 PRINT A(1), A(2), A(3)


    1917 PRINT A(4), A(5), A(6)
    1943 PRINT A(32030), A(32031), A(32032)


    1946 PRINT A(32033), A(32034), A(32035)

    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= -31996 is shown below.

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

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

1      1      1
1      1      1
1      1      1
1      1      1
0      -31997

1      1      1
1      1      1
1      1      1
1      1      1
0      -31996

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

Of the 32035 unknowns, only the 12 A's of line 1912 through 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= -31996 was 16 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

The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To Solve a Case of Broyden's Tridiagonal Simultaneous Equations

Jsun Yui Wong

The following computer program seeks to solve the Broyden case on 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.  See also Broyden [1, p. 587], More, Garbow, Hillstrom [10, page 28], and Cao [3, p. 7]--http://dx.doi.org/10.1155/2014/251587.  The present case has 40 nonlinear equations and 40 unknowns.

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


    16 M = -1D+50

    91 FOR KK = 1 TO 40


        94 A(KK) = -RND


    95 NEXT KK

    128 FOR I = 1 TO 10000000 STEP 1


        129 FOR K = 1 TO 40


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 40)

            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 - .5 * X(1)) * X(1) + 1) / 2


        605 FOR J44 = 2 TO 39


            609 X(J44 + 1) = (-X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1) / 2


        611 NEXT J44
        651 FOR j47 = 1 TO 40


            666 IF ABS(X(j47)) > 40 THEN 1670


        688 NEXT j47


        699 P1 = -X(39) + (3 - .5 * X(40)) * X(40) + 1


        999 P = -ABS(P1)


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 40


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM  PRINT A(1), A(2), A(40), M, JJJJ
        1668 IF M > -.000001 THEN 1912


    1670 NEXT I
    1890 IF M < -1 THEN 1999


    1912 PRINT A(1), A(2), A(3)


    1917 PRINT A(4), A(5), A(6)
    1939 PRINT A(7), A(8), A(9)
    1940 PRINT A(10), A(11), A(12)

    1941 PRINT A(13), A(14), A(15)
    1942 PRINT A(16), A(17), A(18)

    1943 PRINT A(19), A(20), A(21)


    1944 PRINT A(22), A(23), A(24)

    1945 PRINT A(25), A(26), A(27)
    1946 PRINT A(28), A(29), A(30)
    1948 PRINT A(31), A(32), A(33)


    1949 PRINT A(34), A(35), A(36)


    1950 PRINT A(37), A(38), A(39)


    1952 PRINT A(40), 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= -31970 is shown below.

-1.032392026048492              -1.315046362934866           -1.388710265545107
-1.40767126725782                -1.412536367279982           -1.413783664513196
-1.414103375640894              -1.414185320454485           -1.41420632300852
-1.414211705294856              -1.414213083286271           -1.414213433516494
-1.414213517516233              -1.414213527797511           -1.41421350848952
-1.414213460734085              -1.414213364986737           -1.414213177539841
-1.414212811698226              -1.414212097970268           -1.414210705617657
-1.414207989412248              -1.414202690638903           -1.41419235380481
-1.414172188777757              -1.41413285114235             -1.414056112494643
-1.413906415491656              -1.413614404932278           -1.413044821110598
-1.41193394581663                -1.409767875006974           -1.405546204952566
-1.397325403489502              -1.381344573567245           -1.350382366337035
-1.290784396550433              -1.177516501251648           -.9675188312822373
-.5965431685189682              -4.255022799226627D-05           -31970    

 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 [15], the wall-clock time for obtaining the output through JJJJ= -31970 was 23 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

The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To Solve a Modified Broyden Tridiagonal Problem

Jsun Yui Wong

The following computer program seeks to solve an enlarged version of Example 3 on page 290 of  Ge, Liu, and Xu [4, page 290, modified Broyden tridiagonal function]–    http://www.scirp.org/journal/ojapps  The present paper considers the case of 4000 equations with 4000 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 REM   h = 1 / (32760 + 1)


    16 M = -1D+50

    91 FOR KK = 1 TO 4000

        94 A(KK) = FIX(RND * 6)


    95 NEXT KK

    128 FOR I = 1 TO 12000 STEP 1


        129 FOR K = 1 TO 4000


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 4003)

            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 - 2 * X(1)) * X(1) + 1) / 2


        605 FOR J44 = 2 TO 3999

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

        611 NEXT J44


        699 P1 = (3 - 2 * X(4000)) * X(4000) - X(3999)



        999 P = -ABS(P1)


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 4000


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 REM PRINT A(1), A(4000), M, JJJJ

        1668 IF M > -.000000000001 THEN 1912

    1670 NEXT I
    1890 REM IF M < -5555 THEN 1999

    1912 PRINT A(1), A(2), A(3)
    1915 PRINT A(3997), A(3998), A(3999)
    1917 PRINT A(4000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31994 is shown below.

3.903018182689294        -8.879023660369297        -93.1071057432964
 -1.#IND        -1.#IND         -1.#IND
-1.#IND         -1.#IND      -32000

0      .5     1.5
1      1      1
1      0      -31999

4.34741353824193          -11.87888416512631        -160.0999220252991
 -1.#IND        -1.#IND         -1.#IND
-1.#IND         -1.#IND      -31998

1.653270079584755        .24660316332695       .482456585035189
1      1      1
1      0      -31997

3.02737948516394          -4.123957319445566         -23.70664969435698
 -1.#IND        -1.#IND         -1.#IND
-1.#IND         -1.#IND      -31996

3.613930553710306        -7.139598216475422         -62.49022529426731
 -1.#IND        -1.#IND         -1.#IND
-1.#IND         -1.#IND      -31995

1      1      1
1      1      1
1      0      -31994

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

Of the 4000 unknowns, only the 7 A’s of line 1912 through 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 [12], the wall-clock time for obtaining the output through JJJJ= -31994 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] 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

[5] 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

[6] 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

[7] 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.

[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] 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

[10] 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.

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

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

[13] 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