Seeking an Integer Solution to a System of 20155 Simultaneous Equations from the Literature

Jsun Yui Wong

The following computer program seeks to find an integer solution to Problem 7 in Cao [2, page 9, Problem 7 (penalty I function)]--http://dx.doi.org/10.1155/2014/251587.  See also La Cruz et al. [5, page 25]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present paper considers the case of 20155 equations with 20155 variables.  One notes the initial guess, 94 A(KK) = FIX(RND * 1.9).

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 20155

        94 A(KK) = FIX(RND * 1.9)


    95 NEXT KK

    128 FOR I = 1 TO 12000000 STEP 1


        129 FOR K = 1 TO 20155


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 20158)


            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


        666 sumssqq = 0
        669 FOR J44 = 1 TO 20155

            677 sumssqq = sumssqq + X(J44) ^ 2


        688 NEXT J44


        770 REM


        772 P(20155) = (1 / (4 * 20155)) * (sumssqq) - 1 / 4

        773 FOR J44 = 1 TO 20154


            775 P(J44) = -ABS(((1 / 10 ^ 5) ^ .5 * (X(J44) - 1)))

        777 NEXT J44
        822 P = 0


        833 FOR J44 = 1 TO 20154

            837 P = P + P(J44)



        855 NEXT J44


        999 P = P - ABS(P(20155))


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 20155


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

        1666 PRINT A(20155), M, JJJJ

        1668 IF M > -.00001 THEN 1912


    1670 NEXT I        
    1890 REM  IF M < -.00001 THEN 1999
    1912 PRINT A(1), A(2), A(3)
    1913 PRINT A(4), A(5), A(6)

    1914 PRINT A(7), A(8), A(9)


    1915 PRINT A(3152), A(3153), A(3154)


    1916 PRINT A(4152), A(4153), A(4154)


    1917 PRINT A(5152), A(5153), A(5154)


    1918 PRINT A(6149), A(6150), A(6151)

    1919 PRINT A(20152), A(20153), A(20154)

    1939 PRINT A(20155), 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.


.
.
.

1          -33.8039589923291         -32000
1          -33.8007843108295         -32000
.
.
.
0        -3.18708536961386D-03           -32000
0        -1.240387000744233D-05         -32000
1         0       -32000
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1      1      1
1          0       -32000

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

Of the 20155 unknowns, only the 25 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 two hours and ten 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 of a System of 8265 Simultaneous Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to find an integer solution to Problem 6 in Cao [2, page 9, Problem 6 (exponential problem 2)]--http://dx.doi.org/10.1155/2014/251587.  See also La Cruz et al. [5, page 21]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present paper considers the case of 8265 nonlinear equations with 8265 variables.  One notes the hot starts, 94 A(KK) = FIX(RND * 1.9).  While line 187 of the preceding paper is 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, line 190 here is 190 IF RND < .333 THEN X(B) = A(B) + RND * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1.

0 REM  DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(52768), A(52768), K(52768), P(52222)

5 FOR JJJJ = -32000 TO -32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 8265
        94 A(KK) = FIX(RND * 1.9)



    95 NEXT KK

    128 FOR I = 1 TO 12000000 STEP 1

        129 FOR K = 1 TO 8265
            131 X(K) = A(K)
        132 NEXT K

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 8265)

            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




            189 REM


            190 IF RND < .333 THEN X(B) = A(B) + RND * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1





        191 NEXT IPP
        222 FOR J44 = 1 TO 8265

            227 IF X(J44) > 80 THEN 1670
        229 NEXT J44
        770 X(1) = 0
        771 FOR J44 = 2 TO 8265
            774 P(J44) = -ABS(.1 * J44 * (EXP(X(J44)) + X(J44 - 1) - 1))

        777 NEXT J44
        800 P = 0

        801 FOR J44 = 2 TO 8265

            822 P = P + P(J44)

        888 NEXT J44

        1111 P = P

        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 8265

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

        1666 PRINT A(8265), M, JJJJ

        1668 IF M > -.0001 THEN 1912
    1670 NEXT I
    1890 REM IF M < -500 THEN 1999

    1912 PRINT A(1), A(2), A(3)
    1913 PRINT A(4), A(5), A(6)
    1914 PRINT A(7), A(8), A(9)
    1915 PRINT A(557), A(558), A(559)

    1917 PRINT A(4777), A(4778), A(4779)
    1928 PRINT A(4877), A(4878), A(4879)
    1947 PRINT A(5762), A(5763), A(5764)

    1948 PRINT A(8262), A(8263), A(8264)



    1949 PRINT A(8265), 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.

.
.
.

1       -4368401         -32000
1       -4367295         -32000
.
.
.
0      -1759.129          -32000
0      -1440.989          -32000
0      -1412.076          -32000
0      -188.7487          -32000
0        0                       -32000
0   0   0
0   0   0
0   0   0
0   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 8265 unknowns, only the 25 A's of line 1912 through line 1949 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

Seeking an Integer Solution with Hot Starts of Another System of 5765 Nonlinear Equations

Jsun Yui Wong

Using qb64v1000-win [9], the following computer program seeks to find an integer solution to Problem 6 in Cao [2, page 9, Problem 6 (exponential problem 2)]--http://dx.doi.org/10.1155/2014/251587.  See also La Cruz et al. [5, page 21]--www.ime.unicamp.br/~martinez/lmrreport.pdf.  The present paper considers the case of 5765 nonlinear equations with 5765 variables.  One notes the hot starts, 94 A(KK) = FIX(RND * 1.9).

0 REM  DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(52768), A(52768), K(52768), P(52222)

5 FOR JJJJ = -32000 TO -32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 5765
        94 A(KK) = FIX(RND * 1.9)

    95 NEXT KK

    128 FOR I = 1 TO 12000000 STEP 1

        129 FOR K = 1 TO 5765
            131 X(K) = A(K)
        132 NEXT K

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 5765)

            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
        222 FOR J44 = 1 TO 5765

            227 IF X(J44) > 80 THEN 1670
        229 NEXT J44
        770 X(1) = 0
        771 FOR J44 = 2 TO 5765
            774 P(J44) = -ABS(.1 * J44 * (EXP(X(J44)) + X(J44 - 1) - 1))

        777 NEXT J44
        800 P = 0

        801 FOR J44 = 2 TO 5765
            822 P = P + P(J44)

        888 NEXT J44

        1111 P = P

        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 5765

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

        1666 PRINT A(5765), M, JJJJ

        1668 IF M > -.0001 THEN 1912
    1670 NEXT I
    1890 REM IF M < -500 THEN 1999

    1912 PRINT A(1), A(2), A(3)
    1913 PRINT A(4), A(5), A(6)
    1914 PRINT A(7), A(8), A(9)
    1915 PRINT A(557), A(558), A(559)

    1917 PRINT A(4777), A(4778), A(4779)
    1928 PRINT A(4877), A(4878), A(4879)
    1947 PRINT A(5762), A(5763), A(5764)

    1948 PRINT A(5762), A(5763), A(5764)

    1949 PRINT A(5765), 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.

.
.
.
1        -2088833        -32000
1        -2088358        -32000
.
.
.
0     -2964.686    -32000
0     -1854.712     -32000
0     -970.8984     -32000
0     -378.7567     -32000
0       0      -32000
0   0   0
0   0   0
0   0   0
0   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 5765 unknowns, only the 25 A's of line 1912 through line 1949 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 13 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.    
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 with Hot Starts of a System of 18760 Nonlinear Equations

Jsun Yui Wong

Using qb64v1000-win [9], the following computer program seeks to find an integer solution to Problem 5 in Cao [2, p. 7, Problem 5 (exponential problem 1)]--see also La Cruz et al. [5].  Here the case of 18760 nonlinear equations with 18760 variables is considered.  One notes the hot starts, 94 A(KK) = FIX(RND * 2.2).

0 REM DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(52768), A(52768), K(52768), P(52222)

5 FOR JJJJ = -32000 TO -32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 18760

        94 A(KK) = FIX(RND * 2.2)


    95 NEXT KK

    128 FOR I = 1 TO 12000000 STEP 1

        129 FOR K = 1 TO 18760


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 18763)


            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


        222 FOR J44 = 1 TO 18760


            227 IF X(J44) > 80 THEN 1670
        229 NEXT J44


        770 X(1) = 1


        771 FOR J44 = 2 TO 18760


            774 P(J44) = -ABS(J44 * (EXP(X(J44) - 1) - X(J44)))



        777 NEXT J44
        800 P = 0

        801 FOR J44 = 2 TO 18760


            822 P = P + P(J44)


        888 NEXT J44


        1111 P = P


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 18760


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 PRINT A(18760), M, JJJJ


        1668 IF M > -.0001 THEN 1912


    1670 NEXT I
    1890 REM IF M < -500 THEN 1999


    1912 PRINT A(1), A(2), A(3)
    1915 PRINT A(4), A(5), A(6)


    1917 PRINT A(7), A(8), A(9)


    1927 PRINT A(557), A(558), A(559)

    1937 PRINT A(1333), A(1334), A(1335)

    1938 PRINT A(1337), A(1338), A(1339)


    1947 PRINT A(15444), A(15445), A(15446)


    1948 PRINT A(18757), A(18758), A(18759)


    1949 PRINT A(18760), 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.

.
.
.
1     -18900.87                      -32000
1     -16979.96                      -32000
1     -14963.25                      -32000
1     -5720.396                      -32000
1     -1886.926                      -32000
1      0                                    -32000
1   1   1
1   1   1
1   1   1
1   1   1
1   1   1
1   1   1
1   1   1
1   1   1
1       0                                  -32000

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

Of the 18760 unknowns, only the 25 A's of line 1912 through line 1949 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 47 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.    
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

Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with a Discrete Boundary Value Problem of 32760 Nonlinear Equations

Jsun Yui Wong

"Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations," Rice [10, 1993, p. 355].

"We make an extreme, but wholly defensible, statement: There are no good, general methods for solving systems of more than one nonlinear equation. Furthermore, it is not hard to see why (very likely) there never will be any good, general methods," Press, Teukolsky, Vetterling, and Flannery [9, 2007, p. 473].

"Solving a system of nonlinear equations is a problem that is avoided where possible, customarily by approximating the nonlinear system by a system of linear equations.  When this is unsatisfactory, the problem must be tackled directly," Burden, Faires, and Burden [1, 2016, page 642].

Using qb64v1000-win [9], the following computer program seeks to solve Problem 4 in Cao [2, p. 7, Problem 4 (discrete boundary value problem)]--see also La Cruz et al. [5] and Han and Han [4].  Here the case of 32760 nonlinear equations with 32760 variables is considered.  Line 94 below incorporates the initial guess in La Cruz [5, p. 29] and in      Cao [2, p. 7].

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 / (32761)


    91 FOR KK = 1 TO 32760


        94 A(KK) = 2 * RND * (h * (KK * h - 1))


    95 NEXT KK

    128 FOR I = 1 TO 12000000 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) + .5 * h ^ 2 * (X(1) + h) ^ 3




        566 X(32759) = 2 * X(32760) + .5 * h ^ 2 * (X(32760) + h * (32760)) ^ 3







        605 FOR J49 = 2 TO 32759



            609 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * (J49)) ^ 3 - X(J49 - 1) + X(J49 + 1)




        611 NEXT J49


        711 P = 0
        714 FOR J44 = 2 TO 32759

            722 P = P - ABS(P(J44))
        733 NEXT J44





        999 P = P





        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(32760), M, JJJJ

        1668 IF M > -.0001 THEN 1912



    1670 NEXT I
    1890 IF M < -.5 THEN 1999

    1912 PRINT A(1), A(2), A(3)

    1915 PRINT A(4), A(5), A(6)




    1917 PRINT A(32707), A(32708), A(32709)


    1919 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 output through JJJJ= -32000 is summarized below.


.
.
.
0      -1.110802806467152D-04    -32000
0      -1.11080274448885D-04    -32000
0      -1.08491489236126D-04    -32000
0      -9.672295211402018D-05    -32000
0       1.324903525236217D-23                                 0
0   0   0
0   0   0
-2.242643852215177D-10                      0                 4.658176444474268D-10  
0      -9.672295211402018D-05                  -32000

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

Of the 32760 unknowns, only the 13 A's of line 1912 through line 1939 are shown above; these thirteen values suggest thirteen integers and an occasional way to produce an integer solution.

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 two hours and twenty 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.    
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 1010 Nonlinear Equations from the Literature, Second Edition

Jsun Yui Wong

Suppose that the problem is to find an integer solution to Problem 4 in Cao [2, p. 7]--see also La Cruz et al. [5] and Han and Han [4].  Here the case of 1010 nonlinear equations with 1010 integer variables is considered.  The following computer program and its output illustrate.  Noteworthy is line 1668, which is 1668 IF M > -.0005 THEN 1912; this line saves computer time.

0 REM 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

    22 h = 1 / (1011)


    91 FOR KK = 1 TO 1010
        93 IF RND < .5 THEN A(KK) = 0 ELSE A(KK) = 1


        94 REM  A(KK) = 2 * RND * (h * (KK * h - 1))


    95 NEXT KK

    128 FOR I = 1 TO 1000000 STEP 1


        129 FOR K = 1 TO 1010


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 1013)


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


        566 X(1009) = 2 * X(1010) + .5 * h ^ 2 * (X(1010) + h * (1010)) ^ 3


        605 FOR J49 = 2 TO 1009


            609 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * (J49)) ^ 3 - X(J49 - 1) + X(J49 + 1)



        611 NEXT J49

        711 P = 0
        714 FOR J44 = 2 TO 1009
            722 P = P - ABS(P(J44))
        733 NEXT J44


        999 P = P


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 1010


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

        1668 IF M > -.0005 THEN 1912


    1670 NEXT I
    1890 IF M < -.5 THEN 1999


    1912 PRINT A(1), A(2), A(3)

    1913 PRINT A(4), A(5), A(6)
    1914 PRINT A(7), A(8), A(9)


    1915 PRINT A(10), A(11), A(12)


    1917 PRINT A(1001), A(1002), A(1003)


    1919 PRINT A(1004), A(1005), A(1006)


    1920 PRINT A(1007), A(1008), A(1009)


    1939 PRINT A(1010), 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.

0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0      -1.229078E-04      -32000

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

Of the 1010 unknowns, only the 22 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 72 seconds.

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.    
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 1010 Nonlinear Equations from the Literature

Jsun Yui Wong

Suppose the problem is to find an integer solution to Problem 4 in Cao [2, p. 7]--see also La Cruz et al. [5] and Han and Han [4].  Here the case of 1010 nonlinear equations with 1010 integer variables is considered.  The following computer program and its output illustrate.

0 REM 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

    22 h = 1 / (1011)


    91 FOR KK = 1 TO 1010
        93 IF RND < .5 THEN A(KK) = 0 ELSE A(KK) = 1


        94 REM  A(KK) = 2 * RND * (h * (KK * h - 1))


    95 NEXT KK

    128 FOR I = 1 TO 1000000 STEP 1


        129 FOR K = 1 TO 1010


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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 1013)


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


        566 X(1009) = 2 * X(1010) + .5 * h ^ 2 * (X(1010) + h * (1010)) ^ 3


        605 FOR J49 = 2 TO 1009


            609 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * (J49)) ^ 3 - X(J49 - 1) + X(J49 + 1)



        611 NEXT J49

        711 P = 0
        714 FOR J44 = 2 TO 1009
            722 P = P - ABS(P(J44))
        733 NEXT J44


        999 P = P


        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 1010


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


    1670 NEXT I
    1890 IF M < -.5 THEN 1999


    1912 PRINT A(1), A(2), A(3)

    1913 PRINT A(4), A(5), A(6)
    1914 PRINT A(7), A(8), A(9)


    1915 PRINT A(10), A(11), A(12)


    1917 PRINT A(1001), A(1002), A(1003)


    1919 PRINT A(1004), A(1005), A(1006)


    1920 PRINT A(1007), A(1008), A(1009)


    1939 PRINT A(1010), 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.

0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0   0   0
0      -1.229078E-04      -32000

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

Of the 1010 unknowns, only the 22 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 6 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.    
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