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.
Friday, June 3, 2016
Sunday, May 29, 2016
Solving the Boundary Value Problem of Gilat and Subramaniam's Example 9.5
Jsun Yui Wong
The following computer program seeks to solve the discrete boundary value problem of Gilat and Subramaniam's Example 9.5 [7]. Also see Gilat and Subramaniam [8, Example 11.5].
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
55 H = 1 / 8
91 FOR KK = 1 TO 9
94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 9
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 9)
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
571 X(1) = 1
580 X(3) = ((4 + H ^ 2) * X(2) - 2 - H ^ 2 * EXP(-.2 * H)) / 2
581 FOR J44 = 4 TO 9 STEP 1
582 X(J44) = (-2 * X(J44 - 2) + (4 + H ^ 2) * X(J44 - 1) - H ^ 2 * EXP(-.2 * (J44 - 2) * H)) / 2
584 NEXT J44
610 PNEW = ABS(2 * H * X(9) + X(7) - 4 * X(8) + 3 * X(9))
1111 P = -PNEW
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 9
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [19]. Copied by hand from the screen, the computer program’s output through JJJJ= -31997 is shown below.
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -32000
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -31999
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -31998
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -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 fifteen seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] 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-19x19-syste.html.
[21] 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 discrete boundary value problem of Gilat and Subramaniam's Example 9.5 [7]. Also see Gilat and Subramaniam [8, Example 11.5].
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
55 H = 1 / 8
91 FOR KK = 1 TO 9
94 IF RND < .5 THEN A(KK) = 1 - RND ELSE A(KK) = 1 + RND
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 9
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 9)
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
571 X(1) = 1
580 X(3) = ((4 + H ^ 2) * X(2) - 2 - H ^ 2 * EXP(-.2 * H)) / 2
581 FOR J44 = 4 TO 9 STEP 1
582 X(J44) = (-2 * X(J44 - 2) + (4 + H ^ 2) * X(J44 - 1) - H ^ 2 * EXP(-.2 * (J44 - 2) * H)) / 2
584 NEXT J44
610 PNEW = ABS(2 * H * X(9) + X(7) - 4 * X(8) + 3 * X(9))
1111 P = -PNEW
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 9
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [19]. Copied by hand from the screen, the computer program’s output through JJJJ= -31997 is shown below.
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -32000
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -31999
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -31998
1 .943153486079235 .8860557500807424
.828448844750844 .7700662000336197 .7106306551757421
.649852405260003 .5874268461945267 .5230323013901858
-1.110223024625157D-16 -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 fifteen seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] 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-19x19-syste.html.
[21] 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.
Monday, May 23, 2016
Solving a Boundary Value Problem, Part 2
Jsun Yui Wong
The following computer program seeks to solve the system of nonlinear equations of the discrete boundary value problem on pp. 402-405 of Gilat and Subramanian [6, Example 9.4]. Also see Gilat and Subramanian [7]. While the preceding paper has five sub-intervals, the present paper has twenty sub-intervals.
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
33 betaa = (40 * .016) / (240 * 1.6D-05)
35 betab = (.4 * 5.67D-08 * .016) / (240 * 1.6D-05)
91 FOR KK = 1 TO 21
94 A(KK) = 293 + RND * 200
95 NEXT KK
128 FOR I = 1 TO 20000000 STEP 1
129 FOR K = 1 TO 21
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 22)
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(1) = 473
577 X(21) = 293
605 FOR J49 = 21 TO 3 STEP -1
610 P(J49) = X(J49 - 2) - (2 + .005 ^ 2 * betaa) * X(J49 - 1) - .005 ^ 2 * betab * X(J49 - 1) ^ 4 + X(J49) + .005 ^ 2 * (betaa * 293 + betab * 293 ^ 4)
612 NEXT J49
660 PS = 0
661 FOR J33 = 21 TO 3 STEP -1
663 PS = PS + ABS(P(J33))
668 NEXT J33
1111 P = -PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 21
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT A(6), A(7), A(8), A(9), A(10)
1913 PRINT A(11), A(12), A(13), A(14), A(15)
1914 PRINT A(16), A(17), A(18), A(19), A(20)
1919 PRINT A(21), 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= -32000 is shown below.
473 459.455895068255 446.6932249241943
434.6475924043841 423.2590645622598
412.4716933656588 402.2330921869252 392.4940590698486
383.2082393784908 374.331821732474
365.8232621978106 357.6430325441322 349.7533890177581
342.1181586515432 334.7025405843482
327.4729202056817 320.3966942320405 313.442105080995
306.5780831026808 299.7740953874252
293 -2.877462553516411D-08 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [18], the wall-clock time for obtaining the output through JJJJ= -32000 was four minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[8] 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.
[9] 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.
[10] 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.
[11] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[12] 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.
[13 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[14] 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.
[15] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[16] 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.
[17] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[18] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[19] 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-19x19-syste.html.
[20] 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 system of nonlinear equations of the discrete boundary value problem on pp. 402-405 of Gilat and Subramanian [6, Example 9.4]. Also see Gilat and Subramanian [7]. While the preceding paper has five sub-intervals, the present paper has twenty sub-intervals.
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
33 betaa = (40 * .016) / (240 * 1.6D-05)
35 betab = (.4 * 5.67D-08 * .016) / (240 * 1.6D-05)
91 FOR KK = 1 TO 21
94 A(KK) = 293 + RND * 200
95 NEXT KK
128 FOR I = 1 TO 20000000 STEP 1
129 FOR K = 1 TO 21
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 22)
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(1) = 473
577 X(21) = 293
605 FOR J49 = 21 TO 3 STEP -1
610 P(J49) = X(J49 - 2) - (2 + .005 ^ 2 * betaa) * X(J49 - 1) - .005 ^ 2 * betab * X(J49 - 1) ^ 4 + X(J49) + .005 ^ 2 * (betaa * 293 + betab * 293 ^ 4)
612 NEXT J49
660 PS = 0
661 FOR J33 = 21 TO 3 STEP -1
663 PS = PS + ABS(P(J33))
668 NEXT J33
1111 P = -PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 21
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT A(6), A(7), A(8), A(9), A(10)
1913 PRINT A(11), A(12), A(13), A(14), A(15)
1914 PRINT A(16), A(17), A(18), A(19), A(20)
1919 PRINT A(21), 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= -32000 is shown below.
473 459.455895068255 446.6932249241943
434.6475924043841 423.2590645622598
412.4716933656588 402.2330921869252 392.4940590698486
383.2082393784908 374.331821732474
365.8232621978106 357.6430325441322 349.7533890177581
342.1181586515432 334.7025405843482
327.4729202056817 320.3966942320405 313.442105080995
306.5780831026808 299.7740953874252
293 -2.877462553516411D-08 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [18], the wall-clock time for obtaining the output through JJJJ= -32000 was four minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[8] 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.
[9] 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.
[10] 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.
[11] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[12] 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.
[13 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[14] 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.
[15] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[16] 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.
[17] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[18] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[19] 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-19x19-syste.html.
[20] 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.
Sunday, May 22, 2016
Solving a Boundary Value Problem
Jsun Yui Wong
The following computer program seeks to solve the system of nonlinear equations of the discrete boundary value problem on pages 402-405 of Gilat and Subramanian [6, Example 9.4]. Also see Gilat and Subramanian [7].
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
33 betaa = (40 * .016) / (240 * 1.6D-05)
35 betab = (.4 * 5.67D-08 * .016) / (240 * 1.6D-05)
91 FOR KK = 1 TO 6
94 A(KK) = 293 + RND * 200
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 6
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 6)
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(1) = 473
577 X(6) = 293
605 FOR J49 = 6 TO 3 STEP -1
610 P(J49) = X(J49 - 2) - (2 + .02 ^ 2 * betaa) * X(J49 - 1) - .02 ^ 2 * betab * X(J49 - 1) ^ 4 + X(J49) + .02 ^ 2 * (betaa * 293 + betab * 293 ^ 4)
612 NEXT J49
660 PS = 0
661 FOR J33 = 6 TO 3 STEP -1
663 PS = PS + ABS(P(J33))
668 NEXT J33
1111 P = -PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 6
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1919 PRINT A(6), 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= -31999 is shown below.
473 423.3441267021215 383.3134062263356
349.8410248260544 320.4456637140555
293 -3.890045567611633D-07 -32000
473 423.3441269631637 383.313406594297
349.8410252159446 320.4456639230223
293 -1.725536257490834D-07 -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 [18], the wall-clock time for obtaining the output through JJJJ= -31999 was five seconds, not including "Creating .EXEC 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] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[8] 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.
[9] 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.
[10] 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.
[11] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[12] 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.
[13 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[14] 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.
[15] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[16] 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.
[17] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[18] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[19] 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-19x19-syste.html.
[20] 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 system of nonlinear equations of the discrete boundary value problem on pages 402-405 of Gilat and Subramanian [6, Example 9.4]. Also see Gilat and Subramanian [7].
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
33 betaa = (40 * .016) / (240 * 1.6D-05)
35 betab = (.4 * 5.67D-08 * .016) / (240 * 1.6D-05)
91 FOR KK = 1 TO 6
94 A(KK) = 293 + RND * 200
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 6
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 6)
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(1) = 473
577 X(6) = 293
605 FOR J49 = 6 TO 3 STEP -1
610 P(J49) = X(J49 - 2) - (2 + .02 ^ 2 * betaa) * X(J49 - 1) - .02 ^ 2 * betab * X(J49 - 1) ^ 4 + X(J49) + .02 ^ 2 * (betaa * 293 + betab * 293 ^ 4)
612 NEXT J49
660 PS = 0
661 FOR J33 = 6 TO 3 STEP -1
663 PS = PS + ABS(P(J33))
668 NEXT J33
1111 P = -PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 6
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 REM IF M < -.1 THEN 1999
1911 PRINT A(1), A(2), A(3), A(4), A(5)
1919 PRINT A(6), 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= -31999 is shown below.
473 423.3441267021215 383.3134062263356
349.8410248260544 320.4456637140555
293 -3.890045567611633D-07 -32000
473 423.3441269631637 383.313406594297
349.8410252159446 320.4456639230223
293 -1.725536257490834D-07 -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 [18], the wall-clock time for obtaining the output through JJJJ= -31999 was five seconds, not including "Creating .EXEC 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] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Wiley, 2008.
[7] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB, 3rd Edition. Wiley, 2014.
[8] 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.
[9] 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.
[10] 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.
[11] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207.
[12] 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.
[13 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[14] 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.
[15] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[16] 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.
[17] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[18] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[19] 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-19x19-syste.html.
[20] 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, April 23, 2016
The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To Solve a Case of Broyden's Tridiagonal Simultaneous Equations, Second Edition
Jsun Yui Wong
The following computer program seeks to solve the Broyden case on page 23 of La Cruz, Martinez, and Raydan [7, p. 23, Test function 11, Broyden Tridiagonal function]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Broyden [1, p. 587], More, Garbow, Hillstrom [10, page 28], and Cao [3, p. 7]--http://dx.doi.org/10.1155/2014/251587. The present case has 40 nonlinear equations and 40 unknowns.
One notes line 605 through line 611 and line 611 through line 990; the 2016 January 11 edition does not have these lines 690, 693, and 694, which are 690 FOR J44 = 2 TO 39, 693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1), and 694 NEXT J44, respectively.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), P(99)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 40
94 A(KK) = -RND * 5
95 NEXT KK
128 FOR I = 1 TO 5000000 STEP 1
129 FOR K = 1 TO 40
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 40)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
191 NEXT IPP
555 X(2) = ((3 - .5 * X(1)) * X(1) + 1) / 2
605 REM FOR J44 = 2 TO 39
609 REM X(J44 + 1) = (-X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1) / 2
611 REM NEXT J44
651 FOR j47 = 1 TO 40
666 IF ABS(X(j47)) > 40 THEN 1670
688 NEXT j47
690 FOR J44 = 2 TO 39
693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1)
694 NEXT J44
695 PS = 0
696 FOR J55 = 2 TO 39
697 PS = PS + ABS(P(J55))
698 NEXT J55
699 P1 = -X(39) + (3 - .5 * X(40)) * X(40) + 1
999 P = -ABS(P1) - PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 40
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(2), A(40), M, JJJJ
1668 REM IF M > -.000001 THEN 1912
1670 NEXT I
1890 IF M < -1 THEN 1999
1912 PRINT A(1), A(2), A(3)
1914 GOTO 1950
1917 PRINT A(4), A(5), A(6)
1939 PRINT A(7), A(8), A(9)
1940 PRINT A(10), A(11), A(12)
1941 PRINT A(13), A(14), A(15)
1942 PRINT A(16), A(17), A(18)
1943 PRINT A(19), A(20), A(21)
1944 PRINT A(22), A(23), A(24)
1945 PRINT A(25), A(26), A(27)
1946 PRINT A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33)
1949 PRINT A(34), A(35), A(36)
1950 PRINT A(37), A(38), A(39)
1952 PRINT A(40), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.
-1.03239202604849 -1.31504636293486 -1.388710265545094
-1.2907819914299 -1.17751196878769 -.9675105666261581
-.5965290396787168 -1.846300889951635D-12 -32000
-1.032392026048484 -1.31504636293485 -1.388710265545085
-1.290781991429954 -1.177511968787726 -.9675105666261871
-.5965290396787257 -2.658651077069862D-12 -31999
-1.032392026048488 -1.315046362934857 -1.388710265545091
-1.290781991429892 -1.177511968787691 -.9675105666261593
-.5965290396787167 -3.23663318368972D-12 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 40 unknowns, only the seven A's of line 1912, line 1950, and line 1952 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ= -31998 was three minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps
[6] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf
[8] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[9] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[10] J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[11] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf
[12] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
[13] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[14] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[16] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf
The following computer program seeks to solve the Broyden case on page 23 of La Cruz, Martinez, and Raydan [7, p. 23, Test function 11, Broyden Tridiagonal function]--http://www.ime.unicamp.br/~martinez/lmrreport.pdf. See also Broyden [1, p. 587], More, Garbow, Hillstrom [10, page 28], and Cao [3, p. 7]--http://dx.doi.org/10.1155/2014/251587. The present case has 40 nonlinear equations and 40 unknowns.
One notes line 605 through line 611 and line 611 through line 990; the 2016 January 11 edition does not have these lines 690, 693, and 694, which are 690 FOR J44 = 2 TO 39, 693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1), and 694 NEXT J44, respectively.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), P(99)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
91 FOR KK = 1 TO 40
94 A(KK) = -RND * 5
95 NEXT KK
128 FOR I = 1 TO 5000000 STEP 1
129 FOR K = 1 TO 40
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 40)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 5 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE X(B) = A(B) + RND ^ 9 * R
191 NEXT IPP
555 X(2) = ((3 - .5 * X(1)) * X(1) + 1) / 2
605 REM FOR J44 = 2 TO 39
609 REM X(J44 + 1) = (-X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1) / 2
611 REM NEXT J44
651 FOR j47 = 1 TO 40
666 IF ABS(X(j47)) > 40 THEN 1670
688 NEXT j47
690 FOR J44 = 2 TO 39
693 P(J44) = -X(J44 - 1) + (3 - .5 * X(J44)) * X(J44) + 1 - 2 * X(J44 + 1)
694 NEXT J44
695 PS = 0
696 FOR J55 = 2 TO 39
697 PS = PS + ABS(P(J55))
698 NEXT J55
699 P1 = -X(39) + (3 - .5 * X(40)) * X(40) + 1
999 P = -ABS(P1) - PS
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 40
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1666 REM PRINT A(1), A(2), A(40), M, JJJJ
1668 REM IF M > -.000001 THEN 1912
1670 NEXT I
1890 IF M < -1 THEN 1999
1912 PRINT A(1), A(2), A(3)
1914 GOTO 1950
1917 PRINT A(4), A(5), A(6)
1939 PRINT A(7), A(8), A(9)
1940 PRINT A(10), A(11), A(12)
1941 PRINT A(13), A(14), A(15)
1942 PRINT A(16), A(17), A(18)
1943 PRINT A(19), A(20), A(21)
1944 PRINT A(22), A(23), A(24)
1945 PRINT A(25), A(26), A(27)
1946 PRINT A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33)
1949 PRINT A(34), A(35), A(36)
1950 PRINT A(37), A(38), A(39)
1952 PRINT A(40), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [15]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -31998 is shown below.
-1.03239202604849 -1.31504636293486 -1.388710265545094
-1.2907819914299 -1.17751196878769 -.9675105666261581
-.5965290396787168 -1.846300889951635D-12 -32000
-1.032392026048484 -1.31504636293485 -1.388710265545085
-1.290781991429954 -1.177511968787726 -.9675105666261871
-.5965290396787257 -2.658651077069862D-12 -31999
-1.032392026048488 -1.315046362934857 -1.388710265545091
-1.290781991429892 -1.177511968787691 -.9675105666261593
-.5965290396787167 -3.23663318368972D-12 -31998
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 40 unknowns, only the seven A's of line 1912, line 1950, and line 1952 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ= -31998 was three minutes.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps
[6] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf
[8] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[9] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[10] J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[11] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf
[12] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
[13] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[14] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[16] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf
Sunday, April 17, 2016
The Domino Method of Nonlinear Integer/Continuous/Discrete Programming Seeking To Solve a 19X19 System of Nonlinear Equations
Jsun Yui Wong
The following computer program seeks to solve the nonlinear system of equations on page 25 of Remani [13, page 25]. This system comes from the boundary value problem of nonlinear ordinary differential equation on page 23 of Remani [13, page 23] and on page 710 of Burden and Faires [1, page 710]. The present problem has 19 nonlinear equations with 19 unknowns.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), C(22), P(22)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
31 C(1) = .432: C(2) = .5495: C(3) = .686: C(4) = .84375: C(5) = 1.024: C(6) = 1.22825: C(7) = 1.458: C(8) = 1.71475: C(9) = 2
35 C(10) = 2.31525: C(11) = 2.662: C(12) = 3.04175: C(13) = 3.456: C(14) = 3.90625: C(15) = 4.394: C(16) = 4.92075: C(17) = 5.488
91 FOR KK = 1 TO 19
94 A(KK) = 10 + RND * 10
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 19
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 19)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 2 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 5 * R
191 NEXT IPP
222 X(1) = (17 + X(2) - .0433275) / (2 + (.01 * (X(2) - 17)) / 1.6)
605 FOR J44 = 1 TO 17
608 X(J44 + 2) = (X(J44) - 2 * X(J44 + 1) - .01 * (4 + C(J44)) + (.01 * X(J44 + 1) * (X(J44)) / (1.6))) / (-1 + (.01 * X(J44 + 1)) / 1.6)
611 NEXT J44
615 FOR J46 = 1 TO 19
617 IF X(J46) < 0 THEN 1670
619 NEXT J46
624 PNEW = ABS(-X(18) + 2 * X(19) + .01 * (4 + 6.09725 + X(19) * (14.333333 - X(18)) / (1.6)) - 14.333333)
1111 P = -PNEW
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 19
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.00001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(4), A(5), A(6)
1939 PRINT A(7), A(8), A(9)
1940 PRINT A(10), A(11), A(12)
1941 PRINT A(13), A(14), A(15)
1942 PRINT A(16), A(17), A(18), A(19)
1946 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.
16.76053994626924 16.51343842564263 16.25888121383656
15.9973779663645 15.72983918588467 15.45767932518904
15.18292147311006 14.90831516541902 14.63746486371606
14.37496937262407 14.1265735915981 13.8993361360806
13.70181997224838 13.54431892112646 13.43914162426204
13.40098778053801 13.44747162587545 13.59987913254232
13.88429641641846
-1.844179150882475D-08 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [14], the wall-clock time for obtaining the output through JJJJ= -32000 was thirty seconds, not including “Creating .EXE file” time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587
[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[4] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps
[5] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am
[6] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[8] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[9] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf
[10] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
[11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[12] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[13] Courtney Remani. Numerical Methods for Solving Systems of Nonlinear Equations. https://www.lakeheadu.ca/sites/default/files/updates/77/docs/RemaniFinal.pdf.
[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[15] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf
The following computer program seeks to solve the nonlinear system of equations on page 25 of Remani [13, page 25]. This system comes from the boundary value problem of nonlinear ordinary differential equation on page 23 of Remani [13, page 23] and on page 710 of Burden and Faires [1, page 710]. The present problem has 19 nonlinear equations with 19 unknowns.
0 DEFDBL A-Z
3 DEFINT J, K
4 DIM X(32768), A(32768), L(32768), K(32768), C(22), P(22)
5 FOR JJJJ = -32000 TO 32000
14 RANDOMIZE JJJJ
16 M = -1D+50
31 C(1) = .432: C(2) = .5495: C(3) = .686: C(4) = .84375: C(5) = 1.024: C(6) = 1.22825: C(7) = 1.458: C(8) = 1.71475: C(9) = 2
35 C(10) = 2.31525: C(11) = 2.662: C(12) = 3.04175: C(13) = 3.456: C(14) = 3.90625: C(15) = 4.394: C(16) = 4.92075: C(17) = 5.488
91 FOR KK = 1 TO 19
94 A(KK) = 10 + RND * 10
95 NEXT KK
128 FOR I = 1 TO 100000 STEP 1
129 FOR K = 1 TO 19
131 X(K) = A(K)
132 NEXT K
155 FOR IPP = 1 TO FIX(1 + RND * 3)
181 B = 1 + FIX(RND * 19)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 2 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 3 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 4 * R ELSE X(B) = A(B) + RND ^ 5 * R
191 NEXT IPP
222 X(1) = (17 + X(2) - .0433275) / (2 + (.01 * (X(2) - 17)) / 1.6)
605 FOR J44 = 1 TO 17
608 X(J44 + 2) = (X(J44) - 2 * X(J44 + 1) - .01 * (4 + C(J44)) + (.01 * X(J44 + 1) * (X(J44)) / (1.6))) / (-1 + (.01 * X(J44 + 1)) / 1.6)
611 NEXT J44
615 FOR J46 = 1 TO 19
617 IF X(J46) < 0 THEN 1670
619 NEXT J46
624 PNEW = ABS(-X(18) + 2 * X(19) + .01 * (4 + 6.09725 + X(19) * (14.333333 - X(18)) / (1.6)) - 14.333333)
1111 P = -PNEW
1451 IF P <= M THEN 1670
1657 FOR KEW = 1 TO 19
1658 A(KEW) = X(KEW)
1659 NEXT KEW
1661 M = P
1670 NEXT I
1890 IF M < -.00001 THEN 1999
1912 PRINT A(1), A(2), A(3)
1917 PRINT A(4), A(5), A(6)
1939 PRINT A(7), A(8), A(9)
1940 PRINT A(10), A(11), A(12)
1941 PRINT A(13), A(14), A(15)
1942 PRINT A(16), A(17), A(18), A(19)
1946 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [14]. Copied by hand from the screen, the computer program’s complete output through JJJJ= -32000 is shown below.
16.76053994626924 16.51343842564263 16.25888121383656
15.9973779663645 15.72983918588467 15.45767932518904
15.18292147311006 14.90831516541902 14.63746486371606
14.37496937262407 14.1265735915981 13.8993361360806
13.70181997224838 13.54431892112646 13.43914162426204
13.40098778053801 13.44747162587545 13.59987913254232
13.88429641641846
-1.844179150882475D-08 -32000
Above there is no rounding by hand; it is just straight copying by hand from the screen.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [14], the wall-clock time for obtaining the output through JJJJ= -32000 was thirty seconds, not including “Creating .EXE file” time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587
[3] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[4] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps
[5] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am
[6] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[8] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[9] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf
[10] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf
[11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[12] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[13] Courtney Remani. Numerical Methods for Solving Systems of Nonlinear Equations. https://www.lakeheadu.ca/sites/default/files/updates/77/docs/RemaniFinal.pdf.
[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64
[15] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf
Wednesday, April 6, 2016
A General Computer Program Applied to Schittkowski’s Test Problem 395 but with 10000 Unknowns instead of 50 Unknowns
Jsun Yui Wong
The computer program listed below seeks to solve Schittkowski’s last test problem [16, p. 213, Test Problem 395] but with 10000 unknowns instead of 50 unknowns. The source of this Test Problem 395 is given in Schittkowski [16]. Thus, the problem here is to minimize
10000
SIGMA i*(X(i)^2+X(i)^4 )
i=1
subject to
10000
SIGMA X(i)^2 =1.
i=1
0 REM DEFDBL A-Z
1 DEFINT J, K, B
2 DIM A(10000), X(10000)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+300
110 FOR J44 = 1 TO 10000
112 A(J44) = RND * .01
114 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 10000
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10000)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
199 NEXT IPP
400 SONE = 0
401 FOR J44 = 2 TO 10000
403 SONE = SONE + X(J44) ^ 2
404 NEXT J44
405 IF (1 - SONE) < .0000001 THEN 1670
406 X(1) = (1 - SONE) ^ (1 / 2)
410 STWO = 0
411 FOR J44 = 1 TO 10000
413 STWO = STWO + J44 * (X(J44) ^ 2 + X(J44) ^ 4)
415 NEXT J44
457 PD1 = -STWO
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 10000
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M < -999999999# THEN 1999
1935 PRINT A(1), A(2), A(3), A(4), A(5)
1937 PRINT A(9996), A(9997), A(9998), A(9999), A(10000)
1939 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [17]. Copied by hand from the screen, the computer program’s completer output through JJJJ= -31994 is shown below.
1 0 0 0 0
0 0 0 0 0
-2 -32000
.911705 .4108454 0 0 0
0 0 0 0 0
-1.91668 -31999
.9999998 0 6.966768E-04 0
0
0 0 0 0 0
-2 -31998
.9231066 .384544 0 0 0
0 0 0 0 0
-1.917726 -31997
.9122139 .4097142 0 0 0
0 0 0 0 0
-1.916671 -31996
.9127767 .4084589 0 0 0
0 0 0 0 0
-1.916667 -31995
1 0 0 0 0
0 0 0 0 0
-2 -31994
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 10000 unknowns, only the ten A's of line 1935 and line 1937 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [17], the wall-clock time for obtaining the output through JJJJ= -31994 was four hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am.
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[8] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[9] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207. .
[10] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[11 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[12] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.
[13] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[14] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[15] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[18] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
The computer program listed below seeks to solve Schittkowski’s last test problem [16, p. 213, Test Problem 395] but with 10000 unknowns instead of 50 unknowns. The source of this Test Problem 395 is given in Schittkowski [16]. Thus, the problem here is to minimize
10000
SIGMA i*(X(i)^2+X(i)^4 )
i=1
subject to
10000
SIGMA X(i)^2 =1.
i=1
0 REM DEFDBL A-Z
1 DEFINT J, K, B
2 DIM A(10000), X(10000)
88 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+300
110 FOR J44 = 1 TO 10000
112 A(J44) = RND * .01
114 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 10000
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 10000)
183 R = (1 - RND * 2) * A(B)
187 IF RND < .25 THEN X(B) = A(B) + RND * R ELSE IF RND < .333 THEN X(B) = A(B) + RND ^ 4 * R ELSE IF RND < .5 THEN X(B) = A(B) + RND ^ 7 * R ELSE IF RND < .5 THEN X(B) = FIX(A(B)) ELSE X(B) = FIX(A(B)) + 1
199 NEXT IPP
400 SONE = 0
401 FOR J44 = 2 TO 10000
403 SONE = SONE + X(J44) ^ 2
404 NEXT J44
405 IF (1 - SONE) < .0000001 THEN 1670
406 X(1) = (1 - SONE) ^ (1 / 2)
410 STWO = 0
411 FOR J44 = 1 TO 10000
413 STWO = STWO + J44 * (X(J44) ^ 2 + X(J44) ^ 4)
415 NEXT J44
457 PD1 = -STWO
1111 IF PD1 <= M THEN 1670
1452 M = PD1
1454 FOR KLX = 1 TO 10000
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M < -999999999# THEN 1999
1935 PRINT A(1), A(2), A(3), A(4), A(5)
1937 PRINT A(9996), A(9997), A(9998), A(9999), A(10000)
1939 PRINT M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [17]. Copied by hand from the screen, the computer program’s completer output through JJJJ= -31994 is shown below.
1 0 0 0 0
0 0 0 0 0
-2 -32000
.911705 .4108454 0 0 0
0 0 0 0 0
-1.91668 -31999
.9999998 0 6.966768E-04 0
0
0 0 0 0 0
-2 -31998
.9231066 .384544 0 0 0
0 0 0 0 0
-1.917726 -31997
.9122139 .4097142 0 0 0
0 0 0 0 0
-1.916671 -31996
.9127767 .4084589 0 0 0
0 0 0 0 0
-1.916667 -31995
1 0 0 0 0
0 0 0 0 0
-2 -31994
Above there is no rounding by hand; it is just straight copying by hand from the screen.
Of the 10000 unknowns, only the ten A's of line 1935 and line 1937 are shown above.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [17], the wall-clock time for obtaining the output through JJJJ= -31994 was four hours.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] C. G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Mathematics of Computation, Vol. 19, Number 92, pp. 577-593, 1965.
[2] R. L. Burden, J. D. Faires, Annette M. Burden. Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[3] Huiping Cao, Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations, Abstract and Applied Analysis, Volume 2014, Article ID 251587, 12 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2014/251587.
[4] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[5] Rendong Ge, Lijun Liu, Yi Xu, Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables, Open Journal of Applied Sciences, 2013, 3, 285-292. Published Online July 2013. http://www.scirp.org/journal/ojapps.
[6] Tianmin Han, Yuhuan Han, Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method, Applied Mathematics, 2010, 1, 222-229.
http://www.SciRP.org/journal/am.
[7] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: Theory and experiments. Technical Report RT-04-08, July 2004.
http://www.ime.unicamp.br/~martinez/lmrreport.pdf.
[8] William La Cruz, Jose Mario Martinez, Marcos Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, vol. 75, no. 255, pp.1429-1448, 2006.
[9] Guangye Li (1989) Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, pp. 187-207. .
[10] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[11 J. J. More, B. S. Garbow, K. E. Hillstrom (1981) Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol. 7, Pages 17-41.
[12] Alexander P. Morgan, A Method for Computing All Solutions to Systems of Polynomial Equations, ACM Transactions on Mathematical Software, Vol. 9, No. 1, March 1983, Pages 1-17. https://folk.uib.no/ssu029/pdf_file/Morgan83.pdf.
[13] NAG, NAG Fortran Library Routine Document, C05PDF/C05PDA.
http://www.nag.com/numeric/FL/manual/pdf/C05/c05pdf.pdf.
[14] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical recipes: the art of scientific computing, third ed. Cambridge University Press, 2007.
[15] J. Rice. Numerical Methods, Software, and Analysis, Second Edition. Academic Press, 1993.
[16] K. Schittkowski, More Test Examples for Nonlinear Programming Codes. Springer-Verlag, 1987.
[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[18] M. Ziani, F. Guyomarc’h, An Autoadaptive Limited Memory Broyden’s Method To Solve Systems of Nonlinear Equations, Applied Mathematics and Computation 205 (2008) pp. 202-211. web.info.uvt.ro/~cristiana.drogoescu/MC/broyden.pdf.
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