Jsun Yui Wong
The computer program listed below seeks to solve a system of nonlinear equations based on the following system from Kuri-Morales [6, Problem 3]:
x ^ 2 + 3 * y - z^ 3 + 10=0
x ^ 2 + y ^ 2 + z^ 2 - 6=0
x^ 3 - y ^ 2 +z -2=0.
Here one is interested in integer solution/s only.
0 DEFDBL A-Z
1 DEFINT J, K, X
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
111 FOR J44 = 1 TO 3
112 A(J44) = -10 + FIX(RND * 20)
115 NEXT J44
128 FOR I = 1 TO 100
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 3)
150 R = (1 - RND * 2) * A(B)
160 X(B) = (A(B) + RND ^ 3 * R)
168 NEXT IPP
198 X(3) = 2 - X(1) ^ 3 + X(2) ^ 2
216 N(7) = -ABS(X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 - 6)
217 N(8) = -ABS(X(1) ^ 2 + 3 * X(2) - X(3) ^ 3 + 10)
322 PD1 = N(7) + N(8)
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128
1670 NEXT I
1889 IF M < -5 THEN 1999
1904 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [18]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31940 is shown below:
0 0 2 -4 -32000
0 0 2 -4 -31997
0 0 2 -4 -31995
0 0 2 -4 -31989
0 0 2 -4 -31987
0 0 2 -4 -31983
0 0 2 -4 -31978
0 0 2 -4 -31973
0 0 2 -4 -31968
0 0 2 -4 -31963
1 -1 2 0 -31960
0 0 2 -4 -31956
0 0 2 -4 -31955
0 0 2 -4 -31953
0 0 2 -4 -31948
1 -1 2 0 -31945
0 0 2 -4 -31940
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 [20], the wall-clock time for obtaining the output through JJJJ= -31940 was two seconds, not including "Creating .EXE file..." time--the total time was 9 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.
[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.
[3] I.M.M. El-Emary, M.M.Abd El-Kareem, Towards Using Genetic Algorithmfor Solving Nonlinear Equation System. World Applied Sciences Journal 5 (3):282-289, 2008.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.388.158&rep=rep1&type=pdf
[4] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.
[5] Martin Gardner (1983), Diophantine Analysis and Fermat’s Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). http://www.labeee.ufsc.br/~luis/ga/Gardner.pdf
[6] Angel Fernando Kuri-Morales, Solution of Simultaneous Non-Linear Equations Using Genetic Algorithm.
www.wseas.us/e-library/conferences/brazil2002/papers/449-158.pdf
[7] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler’s Conjecture on Sums of Like Powers. The Bulletin of American Mathematical Society, Vol. 72, 1966, page 1079.
[8] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler’s Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.
[9] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.
[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[11] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation , Volume 225, 1 December 2013, Pages 737-746.
[12] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021
[13] Anton Schigur, Solving a System of Diophantine Equations. Mathematics Stack Exchange. http://math.stackexchange.com/questions/398364/solving-a-system-of-diophantine-equations.
[14] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.
[15] E.K. Virtanen (2008-05-26). “Interview With Galleon”.
http://www.basicprogramming.org/PCOPY!issue70/#galleoninterview.
[16] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.
[17] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.
[18] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[19] Jsun Yui Wong (2014, February 8), Testing the Nonlinear Integer Programming Solver with Lander and Parkin’s Counterexample to Euler’s Conjecture on Sums of Like Powers. https://computerprogramsandresults.wordpress.com/2014/02/08/testing.
Wednesday, June 29, 2016
Wednesday, June 22, 2016
The Nonlinear Integer/Continuous Programming Solver Applied to a Nonlinear Diophantine Equation with 500 General Integer Variables
Jsun Yui Wong
The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [18, p. 55; www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf]; also see Abraham, Sanyal, and Sanglikar [1; https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf]:
X(1)^2+ X(2)^2+X(3)^2+...+X(500)^2=500000.
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
91 FOR KK = 1 TO 500
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK
128 FOR I = 1 TO 20000 STEP 1
129 FOR K = 1 TO 500
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 500)
183 REM R = (1 - RND * 2) * A(B)
188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)
191 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 500
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 500
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44
261 PZ = -ABS(SUML - 500000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 500
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9), A(10)
1975 PRINT A(491), A(492), A(493), A(494), A(495)
1977 PRINT A(496), A(497), A(498), A(499), A(500)
1989 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [21]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.
21 30 35 20 15
33 54 44 29 45
26 41 60 25 48
37 34 17 9 5
0 -32000
38 20 41 22 30
39 41 32 17 24
29 22 26 29 10
32 25 35 39 41
0 -31999
27 41 24 38 24
12 26 46 31 34
45 30 48 41 22
22 11 39 45 7
0 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 500 unknowns, only the 20 A's of line 1911 through line 1977 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 [21], the wall-clock time for obtaining the output through JJJJ= -31998 was seven seconds, not including "Creating .EXE file..." time--the total time was twenty seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] S. Abraham, S. Sanyal, M. Sanglikar, Particle Swarm Optimization Based Diophantine Equation Solver. https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf.
[2] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[3] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[4] 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.
[5] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[6] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[7] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[9] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[10] 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.
[11] 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.
[12] 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.
[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[14] 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.
[15 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[16] 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.
[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[18] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adapttive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014. www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf.
[19] 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.
[20] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[21] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[22] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[23] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [18, p. 55; www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf]; also see Abraham, Sanyal, and Sanglikar [1; https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf]:
X(1)^2+ X(2)^2+X(3)^2+...+X(500)^2=500000.
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
91 FOR KK = 1 TO 500
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK
128 FOR I = 1 TO 20000 STEP 1
129 FOR K = 1 TO 500
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 500)
183 REM R = (1 - RND * 2) * A(B)
188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)
191 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 500
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 500
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44
261 PZ = -ABS(SUML - 500000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 500
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9), A(10)
1975 PRINT A(491), A(492), A(493), A(494), A(495)
1977 PRINT A(496), A(497), A(498), A(499), A(500)
1989 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [21]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.
21 30 35 20 15
33 54 44 29 45
26 41 60 25 48
37 34 17 9 5
0 -32000
38 20 41 22 30
39 41 32 17 24
29 22 26 29 10
32 25 35 39 41
0 -31999
27 41 24 38 24
12 26 46 31 34
45 30 48 41 22
22 11 39 45 7
0 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 500 unknowns, only the 20 A's of line 1911 through line 1977 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 [21], the wall-clock time for obtaining the output through JJJJ= -31998 was seven seconds, not including "Creating .EXE file..." time--the total time was twenty seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] S. Abraham, S. Sanyal, M. Sanglikar, Particle Swarm Optimization Based Diophantine Equation Solver. https://arxiv.org/ftp/arxiv/papers/1003/1003.2724.pdf.
[2] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[3] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[4] 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.
[5] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[6] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[7] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[9] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[10] 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.
[11] 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.
[12] 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.
[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[14] 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.
[15 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[16] 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.
[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[18] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adapttive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014. www.wseas.org/multimedia/journals/mathematics/2014/c125706-213.pdf.
[19] 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.
[20] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[21] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[22] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[23] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] 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.
Friday, June 17, 2016
Testing the Nonlinear Integer/Continuous Programming Solver with a Nonlinear Diophantine Equation from the Literature
Jsun Yui Wong
The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [17, p. 55]:
X(1)^2+ X(2)^2+X(3)^2+...+X(39)^2=39000.
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
91 FOR KK = 1 TO 39
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK
128 FOR I = 1 TO 5000 STEP 1
129 FOR K = 1 TO 39
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 39)
183 R = (1 - RND * 2) * A(B)
188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)
191 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 39
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 39
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44
261 PZ = -ABS(SUML - 39000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 39
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9)
1915 PRINT A(10), A(11), A(12), A(13)
1916 PRINT A(14), A(15), A(16), A(17), A(18)
1917 PRINT A(19), A(20), A(21), A(22), A(23)
1918 PRINT A(24), A(25), A(26), A(27), A(28)
1919 PRINT A(29), A(30), A(31), A(32), A(33)
1920 PRINT A(34), A(35)
1925 PRINT A(36), A(37), A(38), A(39), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [20]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.
23 31 45 22 26
30 35 55 15
51 33 30 44
31 6 27 40 6
7 16 11 47 14
42 25 14 51 48
25 35 43 43 12
43 15
3 22 11 19 0
-32000
40 49 43 51 35
31 27 31 11
14 35 36 24
30 12 15 42 27
18 37 35 9 17
42 26 26 21 25
6 44 56 36 46
17 29
30 26 15 24 0
-31999
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 [20], the wall-clock time for obtaining the output through JJJJ= -31999 was two seconds, not including "Creating .EXE file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[9] 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.
[10] 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.
[11] 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.
[12] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[13] 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.
[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[15] 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.
[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[17] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adaptive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014.
[18] 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.
[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[20] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[22] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
The computer program listed below seeks to solve the following nonlinear Diophantine equation from Oliveira [17, p. 55]:
X(1)^2+ X(2)^2+X(3)^2+...+X(39)^2=39000.
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
91 FOR KK = 1 TO 39
94 A(KK) = FIX(1 + RND * 20)
95 NEXT KK
128 FOR I = 1 TO 5000 STEP 1
129 FOR K = 1 TO 39
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 39)
183 R = (1 - RND * 2) * A(B)
188 REM IF RND < .2 THEN X(B) = FIX(A(B) + RND * R) ELSE IF RND < .25 THEN X(B) = FIX(A(B) + RND ^ 3 * R) ELSE IF RND < .333 THEN X(B) = FIX(A(B) + RND ^ 5 * R) ELSE IF RND < .5 THEN X(B) = FIX(A(B) + RND ^ 7 * R) ELSE X(B) = FIX(A(B) + RND ^ 9 * R)
191 IF RND < .5 THEN X(B) = CINT(A(B) - 1) ELSE X(B) = CINT(A(B) + 1)
199 NEXT IPP
201 FOR J43 = 1 TO 39
203 IF X(J43) < 1 THEN 1670
207 NEXT J43
211 SUML = 0
221 FOR J44 = 1 TO 39
231 SUML = SUML + X(J44) ^ 2
251 NEXT J44
261 PZ = -ABS(SUML - 39000)
1111 P = PZ
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 39
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -99999 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1913 PRINT A(6), A(7), A(8), A(9)
1915 PRINT A(10), A(11), A(12), A(13)
1916 PRINT A(14), A(15), A(16), A(17), A(18)
1917 PRINT A(19), A(20), A(21), A(22), A(23)
1918 PRINT A(24), A(25), A(26), A(27), A(28)
1919 PRINT A(29), A(30), A(31), A(32), A(33)
1920 PRINT A(34), A(35)
1925 PRINT A(36), A(37), A(38), A(39), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [20]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31999 is shown below.
23 31 45 22 26
30 35 55 15
51 33 30 44
31 6 27 40 6
7 16 11 47 14
42 25 14 51 48
25 35 43 43 12
43 15
3 22 11 19 0
-32000
40 49 43 51 35
31 27 31 11
14 35 36 24
30 12 15 42 27
18 37 35 9 17
42 26 26 21 25
6 44 56 36 46
17 29
30 26 15 24 0
-31999
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 [20], the wall-clock time for obtaining the output through JJJJ= -31999 was two seconds, not including "Creating .EXE file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[9] 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.
[10] 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.
[11] 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.
[12] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[13] 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.
[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[15] 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.
[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[17] Hime Aguiar E. Oliveira, Junior, Diophantine Equations and Fuzzy Adaptive Simulated Annealing. WSEAS Transactions on Mathematics, Volume 13, 2014.
[18] 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.
[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[20] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[22] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] 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, June 4, 2016
The Domino Method Applied to Solving an Ordinary Differential Equation
Jsun Yui Wong
The following computer program seeks to solve the boundary value problem in Example 8.2 of Jacques and Judd [10, pp.269-270].
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
91 FOR KK = 0 TO 3
94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND
95 NEXT KK
128 FOR I = 1 TO 10000 STEP 1
129 FOR K = 0 TO 3
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = FIX(RND * 4)
183 R = (1 - RND * 2) * A(B)
188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
199 NEXT IPP
582 X(1) = (2.0625 * X(0) + .0625 * X(0) ^ 3 - .5) / 2
586 FOR J44 = 2 TO 3
611 X(J44) = (2 * X(J44 - 1) - X(J44 - 2) + .0625 * X(J44 - 1) ^ 3 + .125 * X(J44 - 1) * X(J44 - 2)) / (1 + .125 * X(J44 - 1))
613 NEXT J44
811 PNEW = -1.9375 * X(3) + X(2) - .0625 * X(3) ^ 3 - .125 * X(2) * X(3) + .5
999 P = -ABS(PNEW)
1451 IF P <= M THEN 1670
1657 FOR KEW = 0 TO 3
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.1 THEN 1999
1911 PRINT A(0), A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [20]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.
.9789106623451404 .788815898287999 .6607672863276792
.5689129958721464 -2.363544147725161D-12 -32000
.978910662345106 .7888158982879604 .6607672863276316
.5689129958720863 -2.444260017910732D-12 -31999
.9789106623462763 .7888158982892726 .660767286329247
.5689129958741218 -2.906248375636467D-13 -31998
.9789106623603372 .788815898305036 .660767286348652
.5689129958985727 -3.31427122074804D-12 -31997
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 [20], the wall-clock time for obtaining the output through JJJJ= -31997 was three seconds, not including "Creating .EXE file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[9] 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.
[10] Ian Jacques, Colin Judd. Numerical Analysis. Chapman and Hall Ltd., 1987.
[11] 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.
[12] 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.
[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[14] 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.
[15] J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[16] 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.
[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[18] 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.
[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[20] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[22] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
The following computer program seeks to solve the boundary value problem in Example 8.2 of Jacques and Judd [10, pp.269-270].
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
91 FOR KK = 0 TO 3
94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND
95 NEXT KK
128 FOR I = 1 TO 10000 STEP 1
129 FOR K = 0 TO 3
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = FIX(RND * 4)
183 R = (1 - RND * 2) * A(B)
188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
199 NEXT IPP
582 X(1) = (2.0625 * X(0) + .0625 * X(0) ^ 3 - .5) / 2
586 FOR J44 = 2 TO 3
611 X(J44) = (2 * X(J44 - 1) - X(J44 - 2) + .0625 * X(J44 - 1) ^ 3 + .125 * X(J44 - 1) * X(J44 - 2)) / (1 + .125 * X(J44 - 1))
613 NEXT J44
811 PNEW = -1.9375 * X(3) + X(2) - .0625 * X(3) ^ 3 - .125 * X(2) * X(3) + .5
999 P = -ABS(PNEW)
1451 IF P <= M THEN 1670
1657 FOR KEW = 0 TO 3
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.1 THEN 1999
1911 PRINT A(0), A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [20]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.
.9789106623451404 .788815898287999 .6607672863276792
.5689129958721464 -2.363544147725161D-12 -32000
.978910662345106 .7888158982879604 .6607672863276316
.5689129958720863 -2.444260017910732D-12 -31999
.9789106623462763 .7888158982892726 .660767286329247
.5689129958741218 -2.906248375636467D-13 -31998
.9789106623603372 .788815898305036 .660767286348652
.5689129958985727 -3.31427122074804D-12 -31997
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 [20], the wall-clock time for obtaining the output through JJJJ= -31997 was three seconds, not including "Creating .EXE file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[9] 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.
[10] Ian Jacques, Colin Judd. Numerical Analysis. Chapman and Hall Ltd., 1987.
[11] 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.
[12] 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.
[13] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[14] 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.
[15] J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[16] 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.
[17] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[18] 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.
[19] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[20] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[21] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[22] Xin-She Yang, Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[23] 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.
Friday, June 3, 2016
The Domino Method Applied to Solving a Partial Differential Equation
Jsun Yui Wong
The following computer program seeks to solve the elliptical equation of Example 10.1 of Yang [21, pp.118-119].
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
18 H = .25
91 FOR KK = 0 TO 4
94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND
95 NEXT KK
128 FOR I = 1 TO 10000 STEP 1
129 FOR K = 0 TO 4
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 4)
183 R = (1 - RND * 2) * A(B)
188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
199 NEXT IPP
566 X(0) = 1
577 X(4) = 0
586 FOR J44 = 2 TO 3
611 X(J44) = .0625 - X(J44 - 2) + 2 * X(J44 - 1)
613 NEXT J44
714 PNEW = X(2) - 2 * X(3) - .0625
999 P = -ABS(PNEW)
1451 IF P <= M THEN 1670
1657 FOR KEW = 0 TO 4
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(0), A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [19]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.
1 .6562499999994446 .3749999999988891
.1562499999983337 0 -2.221778316879863D-12
-32000
1 .6562500000000195 .3750000000000391
.1562500000000586 0 -7.815970093361102D-14
-31999
1 .6562500000000126 .3750000000000251
.1562500000000376 0 -5.018208071305708D-14
-31998
1 .6562500000000527 .3750000000001055
.1562500000001582 0 -2.109423746787797D-13
-31997
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 [19], the wall-clock time for obtaining the output through JJJJ= -31997 was three seconds, not including "Creating .EXE file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[9] 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.
[10] 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.
[11] 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.
[12] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[13] 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.
[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[15] 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.
[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[17] 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.
[18] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[20] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[21] Xin-She Yang. Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[22] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
The following computer program seeks to solve the elliptical equation of Example 10.1 of Yang [21, pp.118-119].
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
18 H = .25
91 FOR KK = 0 TO 4
94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND
95 NEXT KK
128 FOR I = 1 TO 10000 STEP 1
129 FOR K = 0 TO 4
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 4)
183 R = (1 - RND * 2) * A(B)
188 IF RND < .2 THEN X(B) = A(B) + RND * R ELSE IF RND < .25 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
199 NEXT IPP
566 X(0) = 1
577 X(4) = 0
586 FOR J44 = 2 TO 3
611 X(J44) = .0625 - X(J44 - 2) + 2 * X(J44 - 1)
613 NEXT J44
714 PNEW = X(2) - 2 * X(3) - .0625
999 P = -ABS(PNEW)
1451 IF P <= M THEN 1670
1657 FOR KEW = 0 TO 4
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(0), A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [19]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31997 is shown below.
1 .6562499999994446 .3749999999988891
.1562499999983337 0 -2.221778316879863D-12
-32000
1 .6562500000000195 .3750000000000391
.1562500000000586 0 -7.815970093361102D-14
-31999
1 .6562500000000126 .3750000000000251
.1562500000000376 0 -5.018208071305708D-14
-31998
1 .6562500000000527 .3750000000001055
.1562500000001582 0 -2.109423746787797D-13
-31997
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 [19], the wall-clock time for obtaining the output through JJJJ= -31997 was three seconds, not including "Creating .EXE file..." time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, Third Edition. McGraw-Hill, 2012. http://www.learngroup.org/uploads/2014-10-27/Applied_Num_Methods_with_MATLAB_for_Engineers_3ed1.pdf.
[5] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[6] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[8] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[9] 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.
[10] 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.
[11] 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.
[12] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[13] 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.
[14 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[15] 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.
[16] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[17] 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.
[18] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[20] J. Y. Wong. April 27 2016. The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To
Solve a 19X19 System of Nonlinear Equations, Fourth Edition.
http://myblogsubstance.typepad.com/substance/2016/04/-the-domino-method-of-nonlinear-integercontinuousdiscrete-programming-seeking-to-solve-a-19×19-syste.html.
[21] Xin-She Yang. Introduction to Computational Mathematics. World Scientific Publishing Co. Pte. Ltd., 2008.
[22] 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.
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