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, Problem 4]; here the case of 1010 nonlinear equations with 1010 variables is condidered.
.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), P(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
22 h = 1 / (1011)
91 FOR KK = 1 TO 1010
94 A(KK) = 2 * RND * (h * (KK * h - 1))
95 NEXT KK
128 FOR I = 1 TO 1000000 STEP 1
129 FOR K = 1 TO 1010
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 1013)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
191 NEXT IPP
555 X(2) = 2 * X(1) + .5 * h ^ 2 * (X(1) + h) ^ 3
566 X(1009) = 2 * X(1010) + .5 * h ^ 2 * (X(1010) + h * (1010)) ^ 3
605 FOR J49 = 2 TO 1009
609 P(J49) = 2 * X(J49) + .5 * h ^ 2 * (X(J49) + h * (J49)) ^ 3 - X(J49 - 1) + X(J49 + 1)
611 NEXT J49
711 P = 0
714 FOR J44 = 2 TO 1009
722 P = P - ABS(P(J44))
733 NEXT J44
999 P = P
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 1010
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.5 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(1007), A(1008), A(1009)
1939 PRINT A(1010), 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 4.733847099716376D-16 0
0 0 4.877287400684883D-07
0 -1.231070701582301D-04 -32000
0 4.733847099716376D-16 0
0 0 4.877287400684883D-07
0 -1.243709501553312D-04 -31999
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the1010 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 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
Tuesday, December 1, 2015
Saturday, November 28, 2015
Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with a System 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 strictly convex problem in Cao [2, p. 7, Problem 12]; here the case of 155 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 155
94 A(KK) = RND * (1)
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
233 REM
771 FOR J44 = 1 TO 155
773 P(J44) = -ABS(EXP(X(J44)) - 1)
777 NEXT J44
800 P = 0
801 FOR J44 = 1 TO 155
822 P = P + P(J44)
888 NEXT J44
999 REM
1111 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 REM IF M < -500 THEN 1999
1912 PRINT A(1), A(2), A(3)
1913 PRINT A(4), A(5), A(6)
1922 PRINT A(149), A(150), A(151)
1924 PRINT A(152), A(153), A(154)
1949 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.
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 155 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 three 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
"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 strictly convex problem in Cao [2, p. 7, Problem 12]; here the case of 155 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 155
94 A(KK) = RND * (1)
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
233 REM
771 FOR J44 = 1 TO 155
773 P(J44) = -ABS(EXP(X(J44)) - 1)
777 NEXT J44
800 P = 0
801 FOR J44 = 1 TO 155
822 P = P + P(J44)
888 NEXT J44
999 REM
1111 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 REM IF M < -500 THEN 1999
1912 PRINT A(1), A(2), A(3)
1913 PRINT A(4), A(5), A(6)
1922 PRINT A(149), A(150), A(151)
1924 PRINT A(152), A(153), A(154)
1949 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.
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 155 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 three 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
Tuesday, November 24, 2015
Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with an Exponential Problem of 2000 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 first exponential problem in Cao [2, p. 7, Problem 5]; here the case of 2000 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 2000
94 A(KK) = RND * (2)
95 NEXT KK
128 FOR I = 1 TO 1000000 STEP 1
129 FOR K = 1 TO 2000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 2003)
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 2000
227 IF X(J44) > 80 THEN 1670
229 NEXT J44
770 P(1) = -ABS(EXP(X(1) - 1) - 1)
771 FOR J44 = 2 TO 2000
774 P(J44) = -ABS(J44 * (EXP(X(J44) - 1) - X(J44)))
777 NEXT J44
800 P = 0
801 FOR J44 = 2 TO 2000
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 2000
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(1333), A(1334), A(1335)
1947 PRINT A(1555), A(1556), A(1557)
1949 PRINT A(2000), 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= -31998 is shown below.
1 1.000000004608297 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1.00000000000034 1
1 0 -32000
1 1 1
1 1 1
1 1 1
1 1 1
1 1.000000000621198 1
1 1 1
1 0 -31999
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 0 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 2000 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= -31998 was 14 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
"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 first exponential problem in Cao [2, p. 7, Problem 5]; here the case of 2000 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 2000
94 A(KK) = RND * (2)
95 NEXT KK
128 FOR I = 1 TO 1000000 STEP 1
129 FOR K = 1 TO 2000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 2003)
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 2000
227 IF X(J44) > 80 THEN 1670
229 NEXT J44
770 P(1) = -ABS(EXP(X(1) - 1) - 1)
771 FOR J44 = 2 TO 2000
774 P(J44) = -ABS(J44 * (EXP(X(J44) - 1) - X(J44)))
777 NEXT J44
800 P = 0
801 FOR J44 = 2 TO 2000
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 2000
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(1333), A(1334), A(1335)
1947 PRINT A(1555), A(1556), A(1557)
1949 PRINT A(2000), 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= -31998 is shown below.
1 1.000000004608297 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1.00000000000034 1
1 0 -32000
1 1 1
1 1 1
1 1 1
1 1 1
1 1.000000000621198 1
1 1 1
1 0 -31999
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 0 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 2000 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= -31998 was 14 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
Saturday, November 21, 2015
Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with Another System of Ten 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 value problem in Cao [2, p. 7].
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 / (11)
91 FOR KK = 1 TO 10
94 A(KK) = RND * (-3)
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 10
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 10)
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(9) = 2 * X(10) + .5 * h ^ 2 * (X(10) + h * (10)) ^ 3
605 FOR J49 = 3 TO 8
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(8) + .5 * h ^ 2 * (X(8) + h * (8)) ^ 3 - X(7) + X(9)
688 P2 = 2 * X(9) + .5 * h ^ 2 * (X(9) + h * (9)) ^ 3 - X(8) + X(10)
999 P = -ABS(P1) - ABS(P2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10
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)
1915 PRINT A(4), A(5), A(6)
1917 PRINT A(7), A(8), A(9)
1939 PRINT A(10), 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.
-4.567659878764215D-06 -6.031181084693952D-06 -1.733967908702962D-05
-5.516021417404762D-05 -1.056236674287388D-04 -2.31718235140485D-04
-3.119279490344034D-04 -6.711772488166832D-04 -5.54735236918666D-04
-1.820364584748423D-03 -1.421024882752361D-16 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [9], the wall-clock time for obtaining the output through JJJJ= -32000 was 3 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] 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
"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 value problem in Cao [2, p. 7].
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 / (11)
91 FOR KK = 1 TO 10
94 A(KK) = RND * (-3)
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 10
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 10)
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(9) = 2 * X(10) + .5 * h ^ 2 * (X(10) + h * (10)) ^ 3
605 FOR J49 = 3 TO 8
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(8) + .5 * h ^ 2 * (X(8) + h * (8)) ^ 3 - X(7) + X(9)
688 P2 = 2 * X(9) + .5 * h ^ 2 * (X(9) + h * (9)) ^ 3 - X(8) + X(10)
999 P = -ABS(P1) - ABS(P2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10
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)
1915 PRINT A(4), A(5), A(6)
1917 PRINT A(7), A(8), A(9)
1939 PRINT A(10), 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.
-4.567659878764215D-06 -6.031181084693952D-06 -1.733967908702962D-05
-5.516021417404762D-05 -1.056236674287388D-04 -2.31718235140485D-04
-3.119279490344034D-04 -6.711772488166832D-04 -5.54735236918666D-04
-1.820364584748423D-03 -1.421024882752361D-16 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [9], the wall-clock time for obtaining the output through JJJJ= -32000 was 3 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] 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 Broyden Tridiagonal System of Nonlinear Equations
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 Broyden's tridiagonal system of ten nonlinear equations; see Ziani and Guyomarc'h [9, page 211].
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 10
94 A(KK) = RND * (-3)
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 10
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 10)
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) = .5 * (1 + (3 - 2 * X(1)) * X(1))
566 X(9) = 1 + (3 - 2 * X(10)) * X(10)
605 FOR J49 = 3 TO 8
608 X(J49) = .5 * ((3 - 2 * X(J49 - 1)) * X(J49 - 1) - X(J49 - 2) + 1)
611 NEXT J49
655 P1 = ((3 - 2 * X(8)) * X(8) - X(7) - 2 * X(9) + 1)
688 P2 = ((3 - 2 * X(9)) * X(9) - X(8) - 2 * X(10) + 1)
999 P = -ABS(P1) - ABS(P2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10
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)
1915 PRINT A(4), A(5), A(6)
1917 PRINT A(7), A(8), A(9)
1939 PRINT A(10), 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= -31955 is shown below.
-.5707221320112302 -.6818069499842895 -.7022100760176984
-.7055106298951845 -.7049061557290275 -.7014966070306247
-.691889322356902 -.6657965144115334 -.5960351090279503
-.416412257529033 -2.906452856166197D-11 -31955
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 [8], the wall-clock time for obtaining the output through JJJJ= -31955 was 10 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] 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
[9] 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
"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 Broyden's tridiagonal system of ten nonlinear equations; see Ziani and Guyomarc'h [9, page 211].
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 10
94 A(KK) = RND * (-3)
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 10
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 10)
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) = .5 * (1 + (3 - 2 * X(1)) * X(1))
566 X(9) = 1 + (3 - 2 * X(10)) * X(10)
605 FOR J49 = 3 TO 8
608 X(J49) = .5 * ((3 - 2 * X(J49 - 1)) * X(J49 - 1) - X(J49 - 2) + 1)
611 NEXT J49
655 P1 = ((3 - 2 * X(8)) * X(8) - X(7) - 2 * X(9) + 1)
688 P2 = ((3 - 2 * X(9)) * X(9) - X(8) - 2 * X(10) + 1)
999 P = -ABS(P1) - ABS(P2)
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 10
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)
1915 PRINT A(4), A(5), A(6)
1917 PRINT A(7), A(8), A(9)
1939 PRINT A(10), 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= -31955 is shown below.
-.5707221320112302 -.6818069499842895 -.7022100760176984
-.7055106298951845 -.7049061557290275 -.7014966070306247
-.691889322356902 -.6657965144115334 -.5960351090279503
-.416412257529033 -2.906452856166197D-11 -31955
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 [8], the wall-clock time for obtaining the output through JJJJ= -31955 was 10 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] 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
[9] 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
Wednesday, November 4, 2015
Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with Brown’s Almost Linear System of 5000 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 5000 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. While line 94 and line 128 of the preceding paper are 94 A(KK) = RND * 5 and 128 FOR I = 1 TO 400000 STEP 1, respectively, here line 94 and line 128 are 94 A(KK) = RND * 3 and 128 FOR I = 1 TO 2000000 STEP 1, respectively.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(5003), A(5003), L(5003), K(5003)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 5000
94 A(KK) = RND * 3
95 NEXT KK
128 FOR I = 1 TO 2000000 STEP 1
129 FOR K = 1 TO 5000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 5000)
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 5000
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 5000
511 summ = summ + X(j27)
521 NEXT j27
901 DIFF = 0
905 FOR J77 = 1 TO 4999
911 DIFF = DIFF - ABS(X(J77) + summ - 5001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 5000
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)
1917 PRINT A(7), A(8), A(9)
1930 PRINT A(4991), A(4992), A(4993)
1931 PRINT A(4994), A(4995), A(4996)
1933 PRINT A(4997), A(4998), A(4999)
1939 PRINT A(5000), 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= -31996 is shown below.
1.000000005436172 1 .99999999965176291
1 1 1
1 1 1
.9999999999986794 1 1
1 1 1
1 1 1
1 -1.920321127446556D-06 -32000
.9999999662810266 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 -7.11476723314064D-06 -31997
.999999709188973 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1.000000000000012 -7.247766151685653D-07 -31996
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 5000 values for the 5000 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= -31996 was 80 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
"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 5000 equations; see Morgan [5, page 15], Floudas [2, page 660], and Han and Han [3, page 227, Example 3]. While line 94 and line 128 of the preceding paper are 94 A(KK) = RND * 5 and 128 FOR I = 1 TO 400000 STEP 1, respectively, here line 94 and line 128 are 94 A(KK) = RND * 3 and 128 FOR I = 1 TO 2000000 STEP 1, respectively.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(5003), A(5003), L(5003), K(5003)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 5000
94 A(KK) = RND * 3
95 NEXT KK
128 FOR I = 1 TO 2000000 STEP 1
129 FOR K = 1 TO 5000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 5000)
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 5000
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 5000
511 summ = summ + X(j27)
521 NEXT j27
901 DIFF = 0
905 FOR J77 = 1 TO 4999
911 DIFF = DIFF - ABS(X(J77) + summ - 5001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 5000
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)
1917 PRINT A(7), A(8), A(9)
1930 PRINT A(4991), A(4992), A(4993)
1931 PRINT A(4994), A(4995), A(4996)
1933 PRINT A(4997), A(4998), A(4999)
1939 PRINT A(5000), 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= -31996 is shown below.
1.000000005436172 1 .99999999965176291
1 1 1
1 1 1
.9999999999986794 1 1
1 1 1
1 1 1
1 -1.920321127446556D-06 -32000
.9999999662810266 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 -7.11476723314064D-06 -31997
.999999709188973 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1.000000000000012 -7.247766151685653D-07 -31996
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 5000 values for the 5000 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= -31996 was 80 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
Friday, October 30, 2015
Testing the Domino Method of General Integer/Continuous/Mixed Nonlinear Programming with Brown’s Almost Linear System of 1000 Equations/Unknowns
Jsun Yui Wong
"Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations," Rice [6, 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 [5, 2007, p. 473].
Using qb64v1000-win [7, 9], the following computer program seeks to solve simultaneously Brown's almost linear system of 1000 equations; see Morgan [4, page 15], Floudas [2, page 660], and the preceding paper. While line 128 of the preceding paper is 128 FOR I = 1 TO 100000 STEP 1, here line 128 is 128 FOR I = 1 TO 400000 STEP 1.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(1003), A(1003), L(1003), K(1003)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 1000
94 A(KK) = RND * 5
95 NEXT KK
128 FOR I = 1 TO 400000 STEP 1
129 FOR K = 1 TO 1000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 1000)
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 1000
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 1000
511 summ = summ + X(j27)
521 NEXT j27
901 DIFF = 0
905 FOR J77 = 1 TO 999
911 DIFF = DIFF - ABS(X(J77) + summ - 1001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 1000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.000001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1933 PRINT A(997), A(998), A(999)
1939 PRINT A(1000), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [7, 8]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -31992 is shown below.
.9999999983488452 1 1
1 1 1
1 -5.462564678371251D-08 -32000
1.000000000028562 1 1
1 1 1
1 -8.066667445127962D-09 -31998
.9999999994479734 1 1
1.000000000296977 1 1
1.000000000005344 -1.147819295166386D-08 -31992
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 1000 values for the 1000 unknowns, only the 7 A's of line 1912, line 1933, and 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 [7, 8], the wall-clock time for obtaining the output through JJJJ= -31992 was ten minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. L. Burden, J. D. Faires. Numerical Analysis, Fifth Edition. PWS Publishing Company, 1993.
[2] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[3] 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.
[4] 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
[5] 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.
[6] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[7] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY! issue70/#galleoninterview
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
"Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations," Rice [6, 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 [5, 2007, p. 473].
Using qb64v1000-win [7, 9], the following computer program seeks to solve simultaneously Brown's almost linear system of 1000 equations; see Morgan [4, page 15], Floudas [2, page 660], and the preceding paper. While line 128 of the preceding paper is 128 FOR I = 1 TO 100000 STEP 1, here line 128 is 128 FOR I = 1 TO 400000 STEP 1.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(1003), A(1003), L(1003), K(1003)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+17
91 FOR KK = 1 TO 1000
94 A(KK) = RND * 5
95 NEXT KK
128 FOR I = 1 TO 400000 STEP 1
129 FOR K = 1 TO 1000
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 1000)
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 1000
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 1000
511 summ = summ + X(j27)
521 NEXT j27
901 DIFF = 0
905 FOR J77 = 1 TO 999
911 DIFF = DIFF - ABS(X(J77) + summ - 1001)
921 NEXT J77
995 P = DIFF
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 1000
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.000001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1933 PRINT A(997), A(998), A(999)
1939 PRINT A(1000), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [7, 8]. Copied by hand from the screen, the computer program’s complete output through
JJJJ= -31992 is shown below.
.9999999983488452 1 1
1 1 1
1 -5.462564678371251D-08 -32000
1.000000000028562 1 1
1 1 1
1 -8.066667445127962D-09 -31998
.9999999994479734 1 1
1.000000000296977 1 1
1.000000000005344 -1.147819295166386D-08 -31992
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 1000 values for the 1000 unknowns, only the 7 A's of line 1912, line 1933, and 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 [7, 8], the wall-clock time for obtaining the output through JJJJ= -31992 was ten minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. L. Burden, J. D. Faires. Numerical Analysis, Fifth Edition. PWS Publishing Company, 1993.
[2] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[3] 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.
[4] 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
[5] 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.
[6] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[7] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY! issue70/#galleoninterview
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
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