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

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.

One notes line 605 through line 611 and line 611 through line 990; the 2016 January 11 edition does not have these lines 690, 693, and 694, which are 690 FOR J44 = 2 TO 39, 693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1), and 694 NEXT J44, respectively.

0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), P(99)
5 FOR JJJJ = -32000 TO 32000
    14 RANDOMIZE JJJJ

    16 M = -1D+50

    91 FOR KK = 1 TO 40


        94 A(KK) = -RND * 5


    95 NEXT KK

    128 FOR I = 1 TO 5000000 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 ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R


        191 NEXT IPP


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


        605 REM  FOR J44 = 2 TO 39


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


        611 REM NEXT J44

        651 FOR j47 = 1 TO 40


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


        688 NEXT j47
        690 FOR J44 = 2 TO 39


            693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1)


        694 NEXT J44
        695 PS = 0
        696 FOR J55 = 2 TO 39
            697 PS = PS + ABS(P(J55))
        698 NEXT J55


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


        999 P = -ABS(P1) - PS


        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 REM  IF M > -.000001 THEN 1912



    1670 NEXT I
    1890 IF M < -1 THEN 1999


    1912 PRINT A(1), A(2), A(3)

    1914 GOTO 1950

    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= -31998 is shown below.

-1.03239202604849              -1.31504636293486                   -1.388710265545094
-1.2907819914299                -1.17751196878769                   -.9675105666261581
-.5965290396787168            -1.846300889951635D-12           -32000  

-1.032392026048484             -1.31504636293485                  -1.388710265545085
-1.290781991429954             -1.177511968787726                -.9675105666261871
-.5965290396787257             -2.658651077069862D-12         -31999

-1.032392026048488              -1.315046362934857               -1.388710265545091
-1.290781991429892              -1.177511968787691               -.9675105666261593
-.5965290396787167              -3.23663318368972D-12          -31998

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

Of the 40 unknowns, only the seven A's of line 1912, line 1950, and line 1952 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 three 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 19X19 System of Nonlinear Equations

Jsun Yui Wong

The following computer program seeks to solve the nonlinear system of equations on page 25 of Remani [13, page 25].  This system comes from the boundary value problem of nonlinear ordinary differential equation on page 23 of Remani [13, page 23] and on page 710 of Burden and Faires [1, page 710].  The present problem has 19 nonlinear equations with 19 unknowns.

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), L(32768), K(32768), C(22), P(22)


5 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+50

    31 C(1) = .432: C(2) = .5495: C(3) = .686: C(4) = .84375: C(5) = 1.024: C(6) = 1.22825: C(7) = 1.458: C(8) = 1.71475: C(9) = 2


    35 C(10) = 2.31525: C(11) = 2.662: C(12) = 3.04175: C(13) = 3.456: C(14) = 3.90625: C(15) = 4.394: C(16) = 4.92075: C(17) = 5.488


    91 FOR KK = 1 TO 19

        94 A(KK) = 10 + RND * 10

    95 NEXT KK

    128 FOR I = 1 TO 100000 STEP 1



        129 FOR K = 1 TO 19

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

        155 FOR IPP = 1 TO FIX(1 + RND * 3)
            181 B = 1 + FIX(RND * 19)

            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 ^ 2 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 5 * R


        191 NEXT IPP

        222 X(1) = (17 + X(2) - .0433275) / (2 + (.01 * (X(2) - 17)) / 1.6)

        605 FOR J44 = 1 TO 17

            608 X(J44 + 2) = (X(J44) - 2 * X(J44 + 1) - .01 * (4 + C(J44)) + (.01 * X(J44 + 1) * (X(J44)) / (1.6))) / (-1 + (.01 * X(J44 + 1)) / 1.6)


        611 NEXT J44
        615 FOR J46 = 1 TO 19
            617 IF X(J46) < 0 THEN 1670
        619 NEXT J46



        624 PNEW = ABS(-X(18) + 2 * X(19) + .01 * (4 + 6.09725 + X(19) * (14.333333 - X(18)) / (1.6)) - 14.333333)


        1111 P = -PNEW



        1451 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 19

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

    1670 NEXT I
    1890 IF M < -.00001 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), A(19)

    1946 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.

16.76053994626924      16.51343842564263      16.25888121383656
15.9973779663645        15.72983918588467      15.45767932518904
15.18292147311006      14.90831516541902      14.63746486371606    
14.37496937262407      14.1265735915981        13.8993361360806      
13.70181997224838      13.54431892112646      13.43914162426204
13.40098778053801      13.44747162587545      13.59987913254232
13.88429641641846
-1.844179150882475D-08      -32000

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 [14], the wall-clock time for obtaining the output through JJJJ= -32000 was thirty seconds, not including “Creating .EXE file” time.

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

[11] 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.

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

[13] Courtney Remani.  Numerical Methods for Solving Systems of Nonlinear Equations.  https://www.lakeheadu.ca/sites/default/files/updates/77/docs/RemaniFinal.pdf.

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

[15] 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

A General Computer Program Applied to Schittkowski’s Test Problem 395 but with 10000 Unknowns instead of 50 Unknowns

Jsun Yui Wong

The computer program listed below seeks to solve Schittkowski’s last test problem [16, p. 213, Test Problem 395] but with 10000 unknowns instead of 50 unknowns. The source of this Test Problem 395 is given in Schittkowski [16]. Thus, the problem here is to minimize

10000
SIGMA i*(X(i)^2+X(i)^4 )
i=1

subject to

10000
SIGMA X(i)^2 =1.
i=1

0 REM DEFDBL A-Z
1 DEFINT J, K, B
2 DIM A(10000), X(10000)
88 FOR JJJJ = -32000 TO 32000
    89 RANDOMIZE JJJJ
    90 M = -3D+300

    110 FOR J44 = 1 TO 10000


        112 A(J44) = RND * .01


    114 NEXT J44
    128 FOR I = 1 TO 100000


        129 FOR KKQQ = 1 TO 10000
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        139 FOR IPP = 1 TO FIX(1 + RND * 3)
            140 B = 1 + FIX(RND * 10000)


            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
        199 NEXT IPP


        400 SONE = 0
        401 FOR J44 = 2 TO 10000
            403 SONE = SONE + X(J44) ^ 2
        404 NEXT J44
        405 IF (1 - SONE) < .0000001 THEN 1670


        406 X(1) = (1 - SONE) ^ (1 / 2)


        410 STWO = 0
        411 FOR J44 = 1 TO 10000
            413 STWO = STWO + J44 * (X(J44) ^ 2 + X(J44) ^ 4)
        415 NEXT J44
        457 PD1 = -STWO


        1111 IF PD1 <= M THEN 1670
        1452 M = PD1
        1454 FOR KLX = 1 TO 10000

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX


        1557 GOTO 128


    1670 NEXT I
    1889 REM  IF M < -999999999# THEN 1999

    1935 PRINT A(1), A(2), A(3), A(4), A(5)

    1937 PRINT A(9996), A(9997), A(9998), A(9999), A(10000)

    1939 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [17]. Copied by hand from the screen, the computer program’s completer output through JJJJ= -31994 is shown below.

1   0   0   0   0
0   0   0   0   0
-2   -32000

.911705   .4108454   0   0   0
0   0   0   0   0
-1.91668   -31999

.9999998   0   6.966768E-04   0  
0
0   0   0   0   0
-2   -31998

.9231066   .384544   0   0   0
0   0   0   0   0
-1.917726   -31997

.9122139   .4097142   0   0   0
0   0   0   0   0
-1.916671   -31996

.9127767   .4084589   0   0   0
0   0   0   0   0
-1.916667   -31995

1   0   0   0   0
0   0   0   0   0
-2   -31994

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

Of the 10000 unknowns, only the ten A's of line 1935 and line 1937 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 [17], the wall-clock time for obtaining the output through JJJJ= -31994 was four hours.    

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]  K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.

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

[18] 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.