Substance

The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To Solve a 19X19 System of Nonlinear Equations, Fourth Edition

Jsun Yui Wong

The following computer program seeks to solve the nonlinear system of equations on page 25 of Remani [12, page 25],
https://www.lakeheadu.ca/sites/default/files/updates/77/docs/RemaniFinal.pdf. This system comes from
the boundary value problem of nonlinear ordinary differential equation on page 23 of Remani [12, page 23] and on page 710 of Burden and Faires [1, page 710]. The present problem has 19 nonlinear equations with 19 unknowns.

Whereas line 94 of the third edition is 94 A(KK) = 0 + RND * 50, here line 94 is 94 A(KK) = -50 + RND * 100.

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) = -50 + RND * 100

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= -31994 is shown below.

16.76053994774042       16.51343842892427       16.25888121933391
15.99737797455906       15.72983919734564       15.4576793405858
15.18292149322616       14.90831519116772       14.63746489615808
14.37496941298835       14.12657364130707       13.89933619678212
13.7018200458575       13.54431900988234       13.43914173080704
13.40098790803036       13.44747177815207       13.59987931435824
13.8842966338043
-2.566166300609421D-07          -31996

16.76053994604014       16.5134384251316       16.25888121298047
15.99737796508836       15.72983918409988       15.45767932279133
15.18292146997742       14.90831516140924       14.63746485866395
14.37496936633824       14.12657358385704       13.89933612662772
13.70181996078543       13.54431890730475       13.43914160767007
13.40098776068398       13.44747160216182       13.59987910422862
13.88429638256556
-1.86485233271208D-08          -31994

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= -31994 was ten 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] Courtney Remani. Numerical Methods for Solving Systems of Nonlinear Equations. https://www.lakeheadu.ca/sites/default/files/updates/77/docs/RemaniFinal.pdf.

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

[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.

The Domino Method Applied to Solving a System of 15719 Nonlinear Equations from a Discrete Boundary Value Problem, Third Edition

Jsun Yui Wong

The following computer program seeks to solve the system of nonlinear equations of the 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. Also see More, Garbow, and Hillstrom [11]. The present case has 15719 nonlinear equations and 15719 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)). Also one notes that whereas line 128 of the earlier edition is 128 FOR I = 1 TO 6000000 STEP 1, here line 128 is 128 FOR I = 1 TO 3000000 STEP 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 / (15719 + 1)
91 FOR KK = 1 TO 15719
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1

129 FOR K = 1 TO 15719

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15722)
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 < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 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

199 NEXT IPP
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 15718
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 15718
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15719

633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * h - 1))

655 NEXT J33
677 PZ = 2 * X(15719) + .5 * h ^ 2 * (X(15719) + h * 15719) ^ 3 - X(15718)

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15719

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(5555), A(15718), M, JJJJ

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

1911 PRINT A(1), A(2), A(3), A(5555), A(15718), 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.

-3.043274626032177D-15       -6.086548731220798D-15       -2.317457533498823D-15
-2.543587536870807D-11       -8.38100480481299D-10       -5.675305127032603D-07
-32000

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

Of the 15719 unknowns, only the five A’s of line 1911 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 three hours and fifteen 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 Applied to Solving a System of 15719 Nonlinear Equations from a Discrete Boundary Value Problem, Third Edition

Jsun Yui Wong

The following computer program seeks to solve the system of nonlinear equations of the 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. Also see More, Garbow, and Hillstrom [11]. The present case has 15719 nonlinear equations and 15719 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)). Also one notes that whereas line 128 of the earlier edition is 128 FOR I = 1 TO 6000000 STEP 1, here line 128 is 128 FOR I = 1 TO 3000000 STEP 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 / (15719 + 1)
91 FOR KK = 1 TO 15719
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1

129 FOR K = 1 TO 15719

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15722)
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 < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 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

199 NEXT IPP
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 15718
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 15718
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15719

633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * h - 1))

655 NEXT J33
677 PZ = 2 * X(15719) + .5 * h ^ 2 * (X(15719) + h * 15719) ^ 3 - X(15718)

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15719

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(5555), A(15718), M, JJJJ

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

1911 PRINT A(1), A(2), A(3), A(5555), A(15718), 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.

-3.043274626032177D-15       -6.086548731220798D-15       -2.317457533498823D-15
-2.543587536870807D-11       -8.38100480481299D-10       -5.675305127032603D-07
-32000

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

Of the 15719 unknowns, only the five A’s of line 1911 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 three hours and fifteen 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 Applied to Solving a System of 15719 Nonlinear Equations from a Discrete Boundary Value Problem, Second Edition

Jsun Yui Wong

The following computer program seeks to solve the system of nonlinear equations of the 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. Also see More, Garbow, and Hillstrom [11]. The present case has 15719 nonlinear equations and 15719 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)). Also one notes that whereas line 633 of the earlier edition is 633 IF ABS(X(J33)) > 3 THEN 1670, here line 633 is 633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * 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 / (15719 + 1)
91 FOR KK = 1 TO 15719
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK

128 FOR I = 1 TO 6000000 STEP 1

129 FOR K = 1 TO 15719

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15722)
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 < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 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

199 NEXT IPP
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 15718
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 15718
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15719

633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * h - 1))

655 NEXT J33
677 PZ = 2 * X(15719) + .5 * h ^ 2 * (X(15719) + h * 15719) ^ 3 - X(15718)

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15719

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(5555), A(15718), M, JJJJ

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

1911 PRINT A(1), A(5555), A(15718), 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.

-1.800391222568857D-18          -4.41487878395593374D-11          -8.3767776697362D-10
-2.73774500989827D-08           -32000

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

Of the 15719 unknowns, only the three A’s of line 1911 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 six hours and a half.

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 Applied to Solving a System of 15719 Nonlinear Equations from a Discrete Boundary Value Problem, Second Edition

Jsun Yui Wong

The following computer program seeks to solve the system of nonlinear equations of the 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. Also see More, Garbow, and Hillstrom [11]. The present case has 15719 nonlinear equations and 15719 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)). Also one notes that whereas line 633 of the earlier edition is 633 IF ABS(X(J33)) > 3 THEN 1670, here line 633 is 633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * 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 / (15719 + 1)
91 FOR KK = 1 TO 15719
93 A(KK) = (h * (KK * h - 1))
95 NEXT KK

128 FOR I = 1 TO 6000000 STEP 1

129 FOR K = 1 TO 15719

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 15722)
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 < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 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

199 NEXT IPP
566 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
605 FOR J49 = 2 TO 15718
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 15718
619 PS = PS - ABS(P(J49))
629 NEXT J49
622 FOR J33 = 1 TO 15719

633 IF ABS(X(J33)) > 3 THEN X(J33) = (h * (J33 * h - 1))

655 NEXT J33
677 PZ = 2 * X(15719) + .5 * h ^ 2 * (X(15719) + h * 15719) ^ 3 - X(15718)

999 P = -ABS(PZ) + PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 15719

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(5555), A(15718), M, JJJJ

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

1911 PRINT A(1), A(5555), A(15718), 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.

-1.800391222568857D-18          -4.41487878395593374D-11          -8.3767776697362D-10
-2.73774500989827D-08           -32000

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

Of the 15719 unknowns, only the three A’s of line 1911 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 six hours and a half.

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 Applied to a Case of Broyden's Tridiagonal Simultaneous Nonlinear 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 140 nonlinear equations and 140 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 139, 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(199)
5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ

16 M = -1D+50

91 FOR KK = 1 TO 140

94 A(KK) = -RND * 5

95 NEXT KK

128 FOR I = 1 TO 5000000 STEP 1

129 FOR K = 1 TO 140

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 140)

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 140

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

688 NEXT j47
690 FOR J44 = 2 TO 139

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 139
697 PS = PS + ABS(P(J55))
698 NEXT J55

699 P1 = -X(139) + (3 - .5 * X(140)) * X(140) + 1

999 P = -ABS(P1) - PS

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 140

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(137), A(138), A(139)

1952 PRINT A(140), 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.

-1.032392026035996       -1.315046362909672       -1.38871026547211
-1.290781991584355       -1.177511968994598       -.9675105667118213
-.5965290397083675       -3.211300825256558D-08          -32000

-1.032392026020284       -1.315046362877993       -1.388710265428533
-1.290781991570645        -1.177511968919872       -.967510566708735
-.59652903972024       -3.75924307238762D-08          -31999

-1.032392026098676       -1.315046363036047       -1.38871026564822
-1.290781991438694       -1.177511968811573       -.967510566638756
-.5965290396848243       -3.921834401676705D-08          -31998

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

Of the 140 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 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] 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

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

Jsun Yui Wong

The second paragraph should read as follows:

Whereas line 128 and line 187 of the earlier edition are 128 FOR I = 1 TO 20000000 STEP 1 and 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, respectively, here line 128 and line 187 are 128 FOR I = 1 TO 1000000 STEP 1 and 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) = A(B) + RND ^ 8 * R ELSE X(B) = A(B) + RND ^ 9 * R, respectively.

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

Jsun Yui Wong

The second paragraph should read as follows:

Whereas line 128 and line 187 of the earlier edition are 128 FOR I = 1 TO 20000000 STEP 1 and 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, respectively, here line 128 and line 187 are 128 FOR I = 1 TO 1000000 STEP 1 and 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) = A(B) + RND ^ 8 * R ELSE X(B) = A(B) + RND ^ 9 * R, respectively.

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

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; the present problem has 60 simultaneous nonlinear equations with 60 unknowns.

Whereas line 128 and line 187 are 128 FOR I = 1 TO 20000000 STEP 1 and 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, here line 128 and line 187 are
128 FOR I = 1 TO 1000000 STEP 1 and 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) = A(B) + RND ^ 8 * R ELSE X(B) = A(B) + RND ^ 9 * R, respectively.

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 1000000 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) = A(B) + RND ^ 8 * R ELSE X(B) = A(B) + RND ^ 9 * R

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(60), 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 60 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 fifty 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] 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 Case 5 with n=60 of Broyden's Tridiagonal Simultaneous Equations, Second Edition

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; the present problem has 60 simultaneous nonlinear equations with 60 unknowns.

Whereas line 128 and line 187 are 128 FOR I = 1 TO 20000000 STEP 1 and 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, here line 128 and line 187 are
128 FOR I = 1 TO 1000000 STEP 1 and 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) = A(B) + RND ^ 8 * R ELSE X(B) = A(B) + RND ^ 9 * R, respectively.

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 1000000 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) = A(B) + RND ^ 8 * R ELSE X(B) = A(B) + RND ^ 9 * R

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(60), 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 60 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 fifty 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] 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.