Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with a System of 155 Simultaneous Nonlinear Equations and 155 Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 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 [7, 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].

The present system of 155 nonlinear equations with 155 unknowns is essentially described by line 211 through line 855–it is based on the system in La Cruz et al. [5, p. 27] and the system in Cao [2, p. 9]. These three systems are different; see La Cruz [5, p. 27, Test Function 32] and Cao [2, p. 9, Problem 9].

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), P(2222), K(32768)

5 FOR JJJJ = -32000 TO -31999
14 RANDOMIZE JJJJ
16 M = -1D+50

22 h = 1 / (156)

91 FOR KK = 1 TO 155

94 A(KK) = RND * (3)
95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1
129 FOR K = 1 TO 155

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

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

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
193 FOR J44 = 1 TO 155

195 IF X(J44) < .000000001 THEN 1670

197 IF X(J44) > 80 THEN 1670
199 NEXT J44

211 FOR J44 = 1 TO 155

222 une = LOG(X(J44)) + EXP(X(J44))
266 deux = (LOG(X(J44) + EXP(X(J44))) ^ 2 + (1 / 10 ^ 10)) ^ .5
277 trois = (une – deux) / 2

775 P(J44) = -ABS(trois)

777 NEXT J44

822 P = 0
833 FOR J44 = 1 TO 155
837 P = P + P(J44)

855 NEXT J44
999 P = P

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 155
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)
1917 PRINT A(152), A(153), A(154)

1939 PRINT A(155), M, JJJJ

1999 NEXT

This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.

.4235549195433431    .4235549195341951    .4235549195430381
.4235549195419544   .4235549195404165    .4235549195419311
.4235549195421712    -2.58868593228101D-10    -32000

.4235549195419993    .4235549195240327    .4235549195419812
.4235549195419818    .4235549195392904    .423554919541849
.4235549195428133    -2.99430313877025D-10    -31999

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

Of the 155 unknowns, only the 7 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 [9], the wall-clock time for obtaining the output through JJJJ= -31999 was eight 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

Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with a Tridiagonal Exponential Problem of 155 Simultaneous Nonlinear Equations

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 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 [7, 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 the tridiagonal exponential problem in Cao [2, p. 7, Problem 3]; here the case of 155 simultaneous equations/unknowns is considered.

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), P(2222), K(32768)

5 FOR JJJJ = -32000 TO 32000

14 RANDOMIZE JJJJ
16 M = -1D+50

22 h = 1 / (156)

91 FOR KK = 1 TO 155

94 A(KK) = RND * (3)
95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1
129 FOR K = 1 TO 155

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

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

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

770 P(1) = X(1) – EXP(COS(h * (X(1) + X(2))))

772 P(155) = X(155) – EXP(COS(h * (X(154) + X(155))))
773 FOR J44 = 2 TO 154
775 P(J44) = -ABS(X(J44) – EXP(COS(h * (X(J44 – 1) + X(J44) + X(J44 + 1)))))
777 NEXT J44

822 P = 0
833 FOR J44 = 2 TO 154
837 P = P + P(J44)

855 NEXT J44
999 P = -ABS(P(1)) + P – ABS(P(155))
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 155
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)
1917 PRINT A(152), A(153), A(154)

1939 PRINT A(155), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.

2.716635055455467       2.714579391239117       2.714581256299916
2.714581254608846       2.714581256299776       2.714579391238663
2.716635055455428       -1.887380030041186D-09       -32000

2.71663505544694       2.714579391238138       2.714581256315673
2.714581254610846       2.714581256301268       2.714579391234607
2.716635055457201       -1.343718913915382D-09       -31999

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

Of the 155 unknowns, only the 7 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 [9], the wall-clock time for obtaining the output through JJJJ= -31999 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] 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.

[6] 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

[7] 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.

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

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

[10] 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 Logarithmic Problem of 55 Simultaneous Nonlinear Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 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 [7, 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 the logarithmic problem in Cao [2, p. 7, Problem 11]; here the case of 55 simultaneous equations/unknowns is considered. One notes line 187, which 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.

0 DEFDBL A-Z

3 DEFINT J, K

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 55

94 A(KK) = RND * (1)

95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1
129 FOR K = 1 TO 55
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 55)

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

771 FOR J44 = 1 TO 55

773 P(J44) = -ABS(LOG(X(J44) + 1) – X(J44) / 55)
777 NEXT J44
800 P = 0

801 FOR J44 = 1 TO 55

822 P = P + P(J44)

888 NEXT J44
999 REM

1111 P = P
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 55
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -500 THEN 1999

1912 PRINT A(1), A(2), A(3)
1914 PRINT A(4), A(5), A(5)

1917 PRINT A(49), A(50), A(51)

1927 PRINT A(52), A(53), A(54)

1949 PRINT A(55), M, JJJJ

1999 NEXT JJJJ
This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -31999 is shown below.

0      0      0
0      0      0
0      0      0
0      0      0
0      0       -32000

0      0      0
0      0      0
0      0      0
0      0      0
0      0        -31999

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

Of the 55 unknowns, only the 13 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 [9], the wall-clock time for obtaining the output through JJJJ= -31999 was 80 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] 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.

[6] 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

[7] 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.

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

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

[10] 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 Trigonometric-Exponential Problem of 55 Nonlinear Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 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 [7, 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 the trigonometric-exponential problem in Cao [2, p. 7, Problem 2]; here the case of 55 equations/unknowns is considered. One notes line 187, which 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.

0 DEFDBL A-Z

3 DEFINT J, K

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 55

94 A(KK) = RND * (-2)
95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1
129 FOR K = 1 TO 55
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 55)

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 55

227 IF X(J44) > 80 THEN 1670
229 NEXT J44
670 P(1) = -ABS(3 * X(1) ^ 2 + 2 * X(2) – 5 + SIN(X(1) – X(2)) * SIN(X(1) + X(2)))
770 P(55) = -ABS(4 * X(55) – X(54) * EXP(X(54) – X(55)) – 3)
771 FOR J44 = 2 TO 54

774 P(J44) = -ABS(3 * X(J44) ^ 2 + 2 * X(J44 + 1) – 5 + SIN(X(J44) – X(J44 + 1)) * SIN(X(J44) + X(J44 + 1)) + 4 * X(J44) – X(J44 – 1) * EXP(X(J44) – 1) – X(J44) – 3)

777 NEXT J44
800 P = 0

801 FOR J44 = 2 TO 54
822 P = P + P(J44)

888 NEXT J44
999 REM

1111 P = P(1) + P + P(55)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 55
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -500 THEN 1999

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

1927 PRINT A(52), A(53), A(54)

1949 PRINT A(55), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -31999 is shown below.

-1.779522017287711    -2.592287972015785    -1.906583815040876
1.0749978076199       1.104172198037183       1.116279391087924
1.048608417481643       -1.06447428454247D-11       -32000

.9843334682928363       1.094770751433046       1.105478573859634
-2.016944516802347       .8121801181243233      1.083468020425818
1.03446959358922       -1.955411530531756D-11       -31999

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

Of the 55 unknowns, only the 7 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 [9], the wall-clock time for obtaining the output through JJJJ= -31999 was 4 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] 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.

[6] 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

[7] 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.

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

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

[10] 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 Large Exponential Problem of 32765 Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 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 [7, 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 the second exponential problem in Cao [2, p. 9, Problem 6]; here the case of 32765 equations/unknowns is considered.

0 DEFDBL A-Z

3 DEFINT J, K

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 32765

94 A(KK) = RND
95 NEXT KK

128 FOR I = 1 TO 3000000 STEP 1

129 FOR K = 1 TO 32765

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 32768)

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 X(B) = INT(A(B))

191 NEXT IPP

222 FOR J44 = 1 TO 32765

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

770 P(1) = -ABS(EXP(X(1)) – 1)

771 FOR J44 = 2 TO 32765

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

777 NEXT J44
800 P = 0

801 FOR J44 = 2 TO 32765

822 P = P + P(J44)

888 NEXT J44
999 REM

1111 P = P(1) + P

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 32765

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
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(7777), A(7778), A(7779)
1947 PRINT A(32762), A(32763), A(32764)
1949 PRINT A(32765), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [9]. 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    -32000

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

Of the 32765 unknowns, only the 19 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 [9], the wall-clock time for obtaining the output through JJJJ= -32000 was three hours.

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] 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.

[6] 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

[7] 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.

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

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

[10] 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 System of Twenty Nonlinear Equations

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [8, 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 [7, 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 the discrete boundary value problem in Cao [2, p. 7] and in Han and Han [4, p. 228].

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32768), A(32768), L(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50

22 h = 1 / (21)

91 FOR KK = 1 TO 20

94 A(KK) = RND * (-3)
95 NEXT KK

128 FOR I = 1 TO 100000 STEP 1

129 FOR K = 1 TO 20

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 20)

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 X(B) = INT(A(B))
191 NEXT IPP

555 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
566 X(19) = 2 * X(20) + .5 * h ^ 2 * (X(20) + h * (20)) ^ 3
605 FOR J49 = 3 TO 18
609 X(J49) = -2 * X(J49 – 1) – .5 * h ^ 2 * (X(J49 – 1) + h * (J49 – 1)) ^ 3 + X(J49 – 2)

611 NEXT J49

655 P1 = 2 * X(18) + .5 * h ^ 2 * (X(18) + h * (18)) ^ 3 – X(17) + X(19)
688 P2 = 2 * X(19) + .5 * h ^ 2 * (X(19) + h * (19)) ^ 3 – X(18) + X(20)

999 P = -ABS(P1) – ABS(P2)

1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 20

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.0000000005 THEN 1999
1912 PRINT A(1), A(2), A(3)
1915 PRINT A(4), A(5), A(6)
1916 PRINT A(7), A(8), A(9)
1917 PRINT A(10), A(11), A(12)

1918 PRINT A(13), A(14), A(15)

1919 PRINT A(16), A(17), A(18)
1939 PRINT A(19), A(20), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -31783 is shown below.

-1.63788436818447D-07 -2.051521733951551D-07 -7.328854689073154D-07
-2.044831377003442D-06 -4.478232039118847D-06 -8.39074925714514D-06
-1.413841191862338D-05 -2.210068781554012D-05 -3.26082208635711D-05
-4.611240338779846D-05 -6.277381530349364D-05 -8.345515337503525D-05
-1.073228977085603D-04 -1.376393319470796D-04 -1.677730490905885D-04
-2.149897776538251D-04 -2.388258657996598D-04 -3.382846166917472D-04
-2.753998249101887D-04 -6.264380791706676D-04 -1.187929732179757D-12
-31783

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 [9], the wall-clock time for obtaining the output through JJJJ= -31783 was 8 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] 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.

[6] 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

[7] 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.

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

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

[10] 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 Brown’s Barely Nonlinear System of 32500 Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [7, 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 [6, 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 [8], the following computer program seeks to solve simultaneously Brown’s almost linear system of 32500 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. Accordingly the presently problem has 32500 equations, and each of these 32500 equations has 32500 variables. Hence the presently problem is to solve simultaneously these 32500 equations. While line 128 of the preceding paper is 128 FOR I = 1 TO 9000000 STEP 1, here line 128 is 128 FOR I = 1 TO 20000000 STEP 1.

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(32503), A(32503), L(32503), K(32503)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50

91 FOR KK = 1 TO 32500

94 A(KK) = RND * 2.75

95 NEXT KK

128 FOR I = 1 TO 20000000 STEP 1

129 FOR K = 1 TO 32500
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 32503)
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 X(B) = CINT(A(B))

191 NEXT IPP
301 prodd = 1

305 FOR j55 = 2 TO 32500
311 prodd = prodd * X(j55)
321 NEXT j55

371 IF prodd < .00001 THEN 1670
389 X(1) = (1) / prodd
501 summ = 0
505 FOR j27 = 1 TO 32500
511 summ = summ + X(j27)

521 NEXT j27

901 DIFF = 0
905 FOR J77 = 1 TO 32499

911 DIFF = DIFF – ABS(X(J77) + summ – 32501)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 32500

1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.0005 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)
1919 PRINT A(10), A(11), A(12)

1928 PRINT A(32488), A(32489), A(32490)
1930 PRINT A(32491), A(32492), A(32493)
1931 PRINT A(32494), A(32495), A(32496)
1933 PRINT A(32497), A(32498), A(32499)

1939 PRINT A(32500), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.

1.000000225207885             1             1.000000046947952
1            1           1
1             1.000000511925313             1
1             1.000000001632073             1
1.000000985060877                1                .9999999992260287
1            1            1
1            1            1.000000001638765
1.000000000005344             .999999512253733                   1
1             -.145978558468375             -32000

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

One notes that of the 32500 values for the 32500 unknowns, only the 25 A’s of line 1912 through line 1939 of the computer program above are shown above and that the M of line 1939 is essentially defined in line 911, line 995, and line 1661, which are 911 DIFF = DIFF – ABS(X(J77) + summ – 32501), 995 P = DIFF, and 1661 M = P, respectively. This M includes 32499 absolutes values.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time for obtaining the output through JJJJ= -32000 was seventeen hours.

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

[3] 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

[4] 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.

[5] 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

[6] 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.

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

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

Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with Brown’s Barely Nonlinear System of 20000 Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [7, 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 [6, 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 [8], the following computer program seeks to solve simultaneously Brown’s almost linear system of 20000 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. While line 128 of the preceding paper is 128 FOR I = 1 TO 4000000 STEP 1, here line 128 is 128 FOR I = 1 TO 9000000 STEP 1.

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(20003), A(20003), L(20003), K(20003)

5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 20000
94 A(KK) = RND * 2.75

95 NEXT KK

128 FOR I = 1 TO 9000000 STEP 1

129 FOR K = 1 TO 20000
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 20000)
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 X(B) = CINT(A(B))

191 NEXT IPP
301 prodd = 1

305 FOR j55 = 2 TO 20000
311 prodd = prodd * X(j55)
321 NEXT j55

371 IF prodd < .00001 THEN 1670
389 X(1) = (1) / prodd
501 summ = 0
505 FOR j27 = 1 TO 20000

511 summ = summ + X(j27)

521 NEXT j27

901 DIFF = 0
905 FOR J77 = 1 TO 19999
911 DIFF = DIFF – ABS(X(J77) + summ – 20001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 20000

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

1670 NEXT I
1890 REM IF M < -.0005 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)
1919 PRINT A(10), A(11), A(12)

1928 PRINT A(19988), A(19989), A(19990)

1930 PRINT A(19991), A(19992), A(19993)

1931 PRINT A(19994), A(19995), A(19996)

1933 PRINT A(19997), A(19998), A(19999)

1939 PRINT A(20000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.

1.000000013780924             1             1
1            1            1
1            1            1
1            1            1
1            1            1
1            1            1
1            1            1
1            1            1
.9999958660703356             -4.150825875015585D-06             -32000

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

One notes that of the 20000 values for the 20000 unknowns, only the 25 A’s of line 1912 through line 1939 of the computer program above are shown above and that the M of line 1939 is essentially defined in line 911, line 995, and line 1661, which are 911 DIFF = DIFF – ABS(X(J77) + summ – 20001), 995 P = DIFF, and 1661 M = P, respectively. This M includes 19999 absolutes values.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time for obtaining the output through JJJJ= -32000 was five hours.

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

[3] 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

[4] 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.

[5] 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

[6] 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.

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

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

Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with Brown’s Almost Linear System of 10000 Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [7, 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 [6, 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 [8], the following computer program seeks to solve simultaneously Brown’s almost linear system of 10000 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. While line 128 of the preceding paper is 128 FOR I = 1 TO 1500000 STEP 1, here line 128 is 128 FOR I = 1 TO 4000000 STEP 1.
0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(10003), A(10003), L(10003), K(10003)

5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 10000
94 A(KK) = RND * 2.75

95 NEXT KK

128 FOR I = 1 TO 4000000 STEP 1

129 FOR K = 1 TO 10000
131 X(K) = A(K)
132 NEXT K

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 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 X(B) = CINT(A(B))

191 NEXT IPP
301 prodd = 1

305 FOR j55 = 2 TO 10000

311 prodd = prodd * X(j55)
321 NEXT j55

371 IF prodd < .00001 THEN 1670
389 X(1) = (1) / prodd
501 summ = 0
505 FOR j27 = 1 TO 10000

511 summ = summ + X(j27)

521 NEXT j27

901 DIFF = 0
905 FOR J77 = 1 TO 9999

911 DIFF = DIFF – ABS(X(J77) + summ – 10001)

921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.00005 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)
1930 PRINT A(9991), A(9992), A(9993)
1931 PRINT A(9994), A(9995), A(9996)

1933 PRINT A(9997), A(9998), A(9999)

1939 PRINT A(10000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.

.9999999324320742             1             1
1            1            1
1            1            1
1            1            1
1            1            1
1            1            1
1             -2.099449557402267D-05             -31999

.9999999997858835             1             1
1            1            1
1            1            1
1            1            1
1            1             .9999999992260287
1            1            1
1             -2.953256606375021D-06             -31998

.9999998158525395             1             1
1            1            1
1            1            1
1            1            1
1             .9999999984966788             1
1            1            1
1             -7.954908678442507D-07             -31997

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

Of the 10000 values for the 10000 unknowns, only the 19 A’s of line 1912 through line 1939 of the computer program above 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 [8], the wall-clock time for obtaining the output through JJJJ= -31997 was four 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] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.

[3] 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

[4] 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.

[5] 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

[6] 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.

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

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

Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with Brown’s Almost Linear System of 4000 Equations/Unknowns

Jsun Yui Wong

“Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations,” Rice [7, 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 [6, 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 [8], the following computer program seeks to solve simultaneously Brown’s almost linear system of 4000 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. While line 128 of the preceding paper is 128 FOR I = 1 TO 1200000 STEP 1, here line 128 is 128 FOR I = 1 TO 1500000 STEP 1.

0 DEFDBL A-Z
3 DEFINT J, K

4 DIM X(4003), A(4003), L(4003), K(4003)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 4000
94 A(KK) = RND * 3
95 NEXT KK

128 FOR I = 1 TO 1500000 STEP 1

129 FOR K = 1 TO 4000

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

155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 4000)

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 X(B) = CINT(A(B))

191 NEXT IPP
301 prodd = 1

305 FOR j55 = 2 TO 4000

311 prodd = prodd * X(j55)
321 NEXT j55

371 IF prodd < .00001 THEN 1670
389 X(1) = (1) / prodd
501 summ = 0
505 FOR j27 = 1 TO 4000
511 summ = summ + X(j27)

521 NEXT j27

901 DIFF = 0
905 FOR J77 = 1 TO 3999
911 DIFF = DIFF – ABS(X(J77) + summ – 4001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 4000
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)
1915 PRINT A(4), A(5), A(6)

1931 PRINT A(3994), A(3995), A(3996)
1933 PRINT A(3997), A(3998), A(3999)
1939 PRINT A(4000), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31995 is shown below.

.9999998781626573             1             1
1            1            1
1            1            1
1            1            1
1             -2.189336641755801D-06             -31999
1.00000002911722             1             1
1            1            1
1            1            1
1            1            1
1.000000041739135             -1.86561538106922D-06             -31996
1.000000023820726             1             1
1            1            1
1            1            1
1            1            1
1            -3.191413370018381D-06             -31995

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

Of the 4000 values for the 4000 unknowns, only the 13 A’s of line 1912 through line 1939 of the computer program above 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 [8], the wall-clock time for obtaining the output through JJJJ= -31995 was one hour.

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

[3] 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

[4] 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.

[5] 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

[6] 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.

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

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