Substance

The Domino Method of Solving Nonlinear Systems of Equations with Adjustment of Residuals, Second Edition

Jsun Yui Wong

Using the nonlinear system of seven equations of a chemical equilibrium model from Carnahan, Luther, and Wilkes [1, pp. 321-329], the following computer program is an illustration of the domino algorithm merging the ideas on page 251-256 of Miller [2] and the ideas in the falling domino principle, domino effect, domino phenomenon , and domino theory.  It is important to note that the domino effect is a chain reaction.  X(1) of line 501, X(5) of line 511, X(7) of  line 516, and X(6) of line 521 are like the first domino, the second domino, the third domino, and the fourth domino of the falling domino principle, respectively.  Using the latest number for X(2), the latest number for X(4), and the latest number for X(3) from the lines preceding line 444, the right-hand side of line 501, (2.6058*X(2)*X(4))/X(3), can knock the first domino, X(1), down to its resolution, solution, optimal value.  It is a good thing that the right-hand side of line 511 does not add new uncertainty because the right-hand side of line 511 involves the same variables, X(1), X(2), and X(3) and X(4) of line 501.   After line 511, the right-hand side of 516 does not add new uncertainty.  Similarly, after line 516, the right-hand side of line 521 also does not add new uncertainty.  Immediately after line 521, there is a specific number for each of X(1), X(2), X(3), X(4), X(5), X(6), and X(7).  These seven values are a solution to the nonlinear system of equations if these numbers make P(2) close to zero, P(4) close to zero, and P(6) close to zero.

0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
61 FOR KLQ=1 TO 7
62 A(KLQ)=RND
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=2+FIX(RND*3)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
444 FOR J432=1 TO 7
445 IF X(J432)<.00002 THEN 1670
446 NEXT J432
501 X(1)=(2.6058*X(2)*X(4))/X(3)
511 X(5)=1-X(1)-X(2)-X(3)-X(4)
516 X(7)=(2)/(+X(3)+X(4)+2*X(5))
521 X(6)=.5*X(1)*X(7)+X(2)*X(7)+.5*X(3)*X(7)
622 P(2)=X(1)+X(2)+X(5)-1/X(7)
623 P(4)=-28837*X(1)-139009!*X(2)-78213!*X(3)+18927*X(4)+8427*X(5)+13492/X(7)-10690*X(6)/X(7)
625 P(6)=20*20*X(1)*X(4)^3-178370!*X(3)*X(5)
733 FOR J95=1 TO 7
734 IF X(J95)<.00002 THEN 1670
736 NEXT J95
1450        P=-ABS(P(2))-ABS(P(4))-ABS(P(6))
1451 IF P<=M THEN 1670
1452 M=P
1453 PP(2)=P(2):PP(4)=P(4):PP(6)=P(6)
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1890 IF M>-.11 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),M,JJJJ
1930 PRINT PP(2),PP(4),PP(6)
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The best candidate solution through JJJJ=-29409 is presented below.  What immediately follows is a hand copy from the computer monitor.

.256978   1.209713E-02   7.964728E-02
.6492968   1.980841E-03   .4923137   2.728864
-9.943289E-02   -29409
-9.539694E-02   -3.66211E-04   -3.669739E-03

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-29409 was ten minutes.  

The output of the computer program above suggests the following change: change line 1450 from
1450        P=-ABS(P(2))-ABS(P(4))-ABS(P(6)) of the computer program above
to
1450        P=-ABS(P(2))-RJJJJ*ABS(P(4))-ABS(P(6)) of the computer program below.
                
0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 RJJJJ=RND
61 FOR KLQ=1 TO 7
62 A(KLQ)=RND
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=2+FIX(RND*3)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
444 FOR J432=1 TO 7
445 IF X(J432)<.00002 THEN 1670
446 NEXT J432
501 X(1)=(2.6058*X(2)*X(4))/X(3)
511 X(5)=1-X(1)-X(2)-X(3)-X(4)
516 X(7)=(2)/(+X(3)+X(4)+2*X(5))
521 X(6)=.5*X(1)*X(7)+X(2)*X(7)+.5*X(3)*X(7)
622 P(2)=X(1)+X(2)+X(5)-1/X(7)
623 P(4)=-28837*X(1)-139009!*X(2)-78213!*X(3)+18927*X(4)+8427*X(5)+13492/X(7)-10690*X(6)/X(7)
625 P(6)=20*20*X(1)*X(4)^3-178370!*X(3)*X(5)
733 FOR J95=1 TO 7
734 IF X(J95)<.00002 THEN 1670
736 NEXT J95
1450        P=-ABS(P(2))-RJJJJ*ABS(P(4))-ABS(P(6))
1451 IF P<=M THEN 1670
1452 M=P
1453 PP(2)=P(2):PP(4)=P(4):PP(6)=P(6):PRJJJJ=RJJJJ
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1890 IF M>-.007 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),M,JJJJ,PRJJJJ
1930 PRINT PP(2),PP(4),PP(6)
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-26222 is presented below.  What immediately follows is a hand copy from the computer monitor.

.324529   9.122397E-03   4.521917E-02
.6173423   .0037871   .5789768   2.98447   -6.795566E-03 
-31034   .2459211
2.370685E-03   -9.765625E-04   -4.184723-03

.3257676   9.045929E-03   4.462472E-02
.6167215   3.840268E-03   .5806706   2.989417
-6.365321E-03   -29679   .2269497
4.140407E-03   -2.441406E-03   -1.670837E-03

Division by zero
Overflow

.3247342   9.109785E-03   4.512064E-02
.6172399   3.795505E-03   .5792573   2.985291
-4.121109E-03   -26222   .2971025
2.663672E-03   -9.765625E-04   -1.167297E-03

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-26222 was twenty minutes.  

Continuing the run through JJJJ=32000 produced the following candidate solution, which is the best of the whole computer run if one takes the best of the worst among PP(2), PP(4), and PP(6) of each of the candidate solutions.

.3222536   9.260837E-03   4.631469E-02
.6184794   3.691495E-03   .575875   2.975407
-1.588749E-03   10181   .2049015
-8.825958E-04   2.441406E-04   -6.56128E-04

In general, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 126s.  Here is one example: 126 IMAR=10+FIX(RND*1000), 126 IMAR=10+FIX(RND*1200), and 126 IMAR=10+FIX(RND*1500).  This multicomputer approach can considerably reduce the time to obtain one usable solution.

References

[1] Bruce Carnahan, H. A. Luther, and James O. Wilkes, Applied Numerical Methods.  New York: John Wiley and Sons, Inc., 1969.
[2] Ronald E. Miller, Optimization: Foundations and Applications.  New York: John Wiley and Sons, Inc., 2000.
[3] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

The Domino Method Applied to Solving Nonlinear Systems of Equations with Adjustment of Residuals

Jsun Yui Wong

Using the nonlinear system of seven equations of a chemical equilibrium model from Carnahan, Luther, and Wilkes [1, pp. 321-329], the following computer program is an illustration of the domino algorithm merging the ideas on page 251-256 of Miller [2] and the ideas in the falling domino principle, domino effect, domino phenomenon , and domino theory.  It is important to note that the domino effect is a chain reaction.  X(1) of line 501, X(5) of line 511, X(7) of  line 516, and X(6) of line 521 are like the first domino, the second domino, the third domino, and the fourth domino of the falling domino principle, respectively.  Using the latest number for X(2), the latest number for X(4), and the latest number for X(3) from the lines preceding line 444, the right-hand side of line 501, (2.6058*X(2)*X(4))/X(3), can knock the first domino, X(1), down to its resolution, solution, optimal value.  It is a good thing that the right-hand side of line 511 does not add new uncertainty because the right-hand side of line 511 involves the same variables, X(1), X(2), and X(3) and X(4) of line 501.   After line 511, the right-hand side of 516 does not add new uncertainty.  Similarly, after line 516, the right-hand side of line 521 also does not add new uncertainty.  Immediately after line 521, there is a specific number for each of X(1), X(2), X(3), X(4), X(5), X(6), and X(7).  These seven values are a solution to the nonlinear system of equations if these numbers make P(2) close to zero, P(4) close to zero, and P(6) close to zero.

0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
61 FOR KLQ=1 TO 7
62 A(KLQ)=RND
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=2+FIX(RND*3)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
444 FOR J432=1 TO 7
445 IF X(J432)<.00002 THEN 1670
446 NEXT J432
501 X(1)=(2.6058*X(2)*X(4))/X(3)
511 X(5)=1-X(1)-X(2)-X(3)-X(4)
516 X(7)=(2)/(+X(3)+X(4)+2*X(5))
521 X(6)=.5*X(1)*X(7)+X(2)*X(7)+.5*X(3)*X(7)
622 P(2)=X(1)+X(2)+X(5)-1/X(7)
623 P(4)=-28837*X(1)-139009!*X(2)-78213!*X(3)+18927*X(4)+8427*X(5)+13492/X(7)-10690*X(6)/X(7)
625 P(6)=20*20*X(1)*X(4)^3-178370!*X(3)*X(5)
733 FOR J95=1 TO 7
734 IF X(J95)<.00002 THEN 1670
736 NEXT J95
1450        P=-ABS(P(2))-ABS(P(4))-ABS(P(6))
1451 IF P<=M THEN 1670
1452 M=P
1453 PP(2)=P(2):PP(4)=P(4):PP(6)=P(6)
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1890 IF M>-.11 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),M,JJJJ
1930 PRINT PP(2),PP(4),PP(6)
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The best candidate solution through JJJJ=-29409 is presented below.  What immediately follows is a hand copy from the computer monitor.

.256978   1.209713E-02   7.964728E-02
.6492968   1.980841E-03   .4923137   2.728864
-9.943289E-02   -29409
-9.539694E-02   -3.66211E-04   -3.669739E-03

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-29409 was ten minutes.  

The output of the computer program above suggests the following change: change line 1450 from
1450        P=-ABS(P(2))-ABS(P(4))-ABS(P(6)) of the computer program above
to
1450        P=-ABS(P(2))-RJJJJ*ABS(P(4))-ABS(P(6)) of the computer program below.
                
0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 RJJJJ=RND
61 FOR KLQ=1 TO 7
62 A(KLQ)=RND
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=2+FIX(RND*3)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
444 FOR J432=1 TO 7
445 IF X(J432)<.00002 THEN 1670
446 NEXT J432
501 X(1)=(2.6058*X(2)*X(4))/X(3)
511 X(5)=1-X(1)-X(2)-X(3)-X(4)
516 X(7)=(2)/(+X(3)+X(4)+2*X(5))
521 X(6)=.5*X(1)*X(7)+X(2)*X(7)+.5*X(3)*X(7)
622 P(2)=X(1)+X(2)+X(5)-1/X(7)
623 P(4)=-28837*X(1)-139009!*X(2)-78213!*X(3)+18927*X(4)+8427*X(5)+13492/X(7)-10690*X(6)/X(7)
625 P(6)=20*20*X(1)*X(4)^3-178370!*X(3)*X(5)
733 FOR J95=1 TO 7
734 IF X(J95)<.00002 THEN 1670
736 NEXT J95
1450        P=-ABS(P(2))-RJJJJ*ABS(P(4))-ABS(P(6))
1451 IF P<=M THEN 1670
1452 M=P
1453 PP(2)=P(2):PP(4)=P(4):PP(6)=P(6):PRJJJJ=RJJJJ
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1890 IF M>-.007 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),M,JJJJ,PRJJJJ
1930 PRINT PP(2),PP(4),PP(6)
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-26222 is presented below.  What immediately follows is a hand copy from the computer monitor.

.324529   9.122397E-03   4.521917E-02
.6173423   .0037871   .5789768   2.98447   -6.795566E-03 
-31034   .2459211
2.370685E-03   -9.765625E-04   -4.184723-03

.3257676   9.045929E-03   4.462472E-02
.6167215   3.840268E-03   .5806706   2.989417
-6.365321E-03   -29679   .2269497
4.140407E-03   -2.441406E-03   -1.670837E-03

Division by zero
Overflow

.3247342   9.109785E-03   4.512064E-02
.6172399   3.795505E-03   .5792573   2.985291
-4.121109E-03   -26222   .2971025
2.663672E-03   -9.765625E-04   -1.167297E-03

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-26222 was twenty minutes.  

In general, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 126s.  Here is one example: 126 IMAR=10+FIX(RND*1000), 126 IMAR=10+FIX(RND*1200), and 126 IMAR=10+FIX(RND*1500).  This multicomputer approach can considerably reduce the time to obtain one usable solution.

References

[1] Bruce Carnahan, H. A. Luther, and James O. Wilkes, Applied Numerical Methods.  New York: John Wiley and Sons, Inc., 1969.
[2] Ronald E. Miller, Optimization: Foundations and Applications.  New York: John Wiley and Sons, Inc., 2000.
[3] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

An Illustration of the Domino Method of Solving Nonlinear Systems of Equations, Second Edition

Jsun Yui Wong

Using the nonlinear system of four nonlinear equations from Jacques and Judd [1, p. 68], the following computer program is an illustration of the algorithm to merge the ideas on page 251-256 of Miller [2] and the ideas in the falling domino principle, domino effect, domino phenomenon , and domino theory.  One notes that the domino effect is a reaction effect.  X(2),  X(3), and X(4) of line 501, line 525, and line 533 are like the first domino, the second domino, and the third domino, respectively, of the falling domino principle.  Using the latest number for X(1) from the lines preceding line 501, the right-hand side of line 501, ( 2.0625*X(1)+.0625*X(1)^3 -.5 )/2, can knock the first domino, X(2), down to its resolution, solution, optimal value.  Similarly, using the latest number for X(2) from line 501 and the latest number for X(1) from the lines before line 501, the right-hand side of 525 can knock down the second domino, X(3).  Similarly, using the latest number for X(2) from line 501 and the latest number for X(3) from line 525, the right-hand side of 533 can knock down the third domino, X(4).  Then there is a specific value for each of X(1), X(2), X(3), and X(4).  These four values are a solution to the nonlinear system if they make P(4) of line 628 close to zero.

- 2.0625*X(1)+2*x(2)-.0625*X(1)^3  +.5 =0
X(3)-2*X(2)+X(1)-.0625*X(2)^3+.125*X(2)*(X(3)-X(1))=0
X(4)-2*X(3)+X(2)-.0625*X(3)^3+.125*X(3)*(X(4)-X(2))=0
-1.9375*X(4)+X(3)-.0625*X(4)^3-.125*X(3)*X(4)+.5=0

These equations come from Jacques and Judd [1, p. 68] and are used in line 501, line 525, line 533, and line 628, respectively.

0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
61 FOR KLQ=1 TO 4
62 A(KLQ)=(RND*10)
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*4)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
461 FOR J95=1 TO 4
463 IF X(J95)<0 THEN 1670
466 NEXT J95
501 X(2)=(2.0625*X(1)+.0625*X(1)^3-.5 )/2
525 X(3)=(2*X(2)-X(1)+.0625*X(2)^3+.125*X(2)*X(1))/(1+.125*X(2))
533 X(4)=(2*X(3)-X(2)+.0625*X(3)^3+.125*X(3)*X(2))/(.125*X(3)+1)
628 P(4)=-1.9375*X(4)+X(3)-.0625*X(4)^3-.125*X(3)*X(4)+.5
1450 P=-ABS(P(4))
1451 IF P<=M THEN 1670
1452 M=P
1453 PPP(2)=PP(2):PPP(4)=PP(4):PPP(6)=PP(6)
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1890 IF M>-.0001 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31997 is presented below.  What immediately follows is a hand copy from the computer monitor.

.9789119   .7888173   .660769   .568915   -2.682209E-06
-32000

.9789092   .7888143   .6607655   .5689107   -3.129244E-06
-31999

.9789092   .7888143   .6607655   .5689107   -3.129244E-06
-31998

.9789104   .7888158   .6607671   .5689128   -2.384186E-07
-31997

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31993 was three seconds.

In general, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 126s.  Here is one example: 126 IMAR=10+FIX(RND*1000), 126 IMAR=10+FIX(RND*5000), and 126 IMAR=10+FIX(RND*10000).  This multicomputer approach can considerably reduce the time to obtain one usable solution.

References

[1] Ian Jacques and Colin Judd, Numerical Analysis.  London: Chapman and Hall, 1987.
[2] Ronald E. Miller, Optimization: Foundations and Applications.  New York: John Wiley and Sons, Inc., 2000.
[3] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

The Domino Method Applied to Solving a Nonlinear System of Four Equations, Second Edition

Jsun Yui Wong

Using the following nonlinear system based on a system in Shoup [2, pp. 40-41], the following computer program is an illustration of the algorithm merging the ideas on page 251-256 of Miller [1] and the ideas in the falling domino principle, domino effect, domino phenomenon , and domino theory.  One notes that the domino effect is a chain reaction.  X(4) of line 683 is like the first domino of the falling domino principle.   X(4) has been chosen as the first domino because it appears that X(4) of line 683 can be knocked down as easily as any other choice.  Using the latest number for X(1), the latest number for X(2), and the latest number for X(3) from the lines preceding line 683, the right-hand side of line 683, 15-5*X(1)-3*X(2)-X(3),can knock the first domino, X(4), down to its resolution, solution, optimal value.  After line 683, there is a specific number for each of X(1), X(2), X(3), and X(4).  These four values are a solution to the nonlinear system of equations if these numbers make
P(2) of line 684 close to zero, P(3) of line 685 close to zero, and P(4) of line 686 close to zero; line 701 specifies the tolerances.

5*X(1)+3*X(2)+X(3)+X(4)=15
X(1)*X(2)+X(2)*X(3)+X(3)*X(4)=17
X(1)^2+X(2)^2+X(3)^2-X(4)^2=9
X(1)*X(3)+X(2)*X(4)+X(1)^3=8

The nonlinear system above is based on that of Shoup [2,  pp. 40-41].  These four equations are used in line 683 through line 686.

0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
61 FOR KLQ=1 TO 4
62 A(KLQ)=FIX(RND*51)
63 NEXT KLQ
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
181 J=1+FIX(RND*4)
183 R=(1-RND*2)*A(J)
191 IF RND<.14 THEN X(J)=A(J)+RND^5*R ELSE IF RND<.17 THEN X(J)=A(J)+R ELSE IF RND<.2 THEN X(J)=A(J)+RND*R ELSE IF RND<.25 THEN X(J)=A(J)+RND^2*R ELSE IF RND<.33 THEN X(J)=A(J)+RND^3*R ELSE IF RND<.5 THEN X(J)=A(J)+RND^4*R ELSE X(J)=INT(X(J))
683 X(4)=15-5*X(1)-3*X(2)-X(3)
684 P(2)=X(1)*X(2)+X(2)*X(3)+X(3)*X(4) -17
685 P(3)=X(1)^2+X(2)^2+X(3)^2  -X(4)^2   -9
686 P(4)=X(1)*X(3)+X(2)*X(4)+X(1)^3 -8
700 FOR J99=2 TO 4
701 IF ABS(P(J99))>.00001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
702 NEXT J99
1450 P=+P(2)+P(3)+P(4)
1451 IF P<=M THEN 1670
1452 M=P
1453 PPP(2)=PP(2):PPP(4)=PP(4):PPP(6)=PP(6)
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1890 IF M>-100000! THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31148 is presented below.  What immediately follows is a hand copy from the computer monitor.

.9994086   .9866962   4.020352   3.022516   -58915.82
-31931        

1   1   4   3   0
-31501

1   1   4   3   0
-31394

1.003966   1.008434   3.978781   2.976089   -79573.98
-31148        

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31148 was two minutes.  

In general, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 126s.  Here is one example: 126 IMAR=10+FIX(RND*1000),
126 IMAR=10+FIX(RND*5000), and 126 IMAR=10+FIX(RND*10000).  This multicomputer approach can considerably reduce the length of time to obtain one usable solution.

References

[1] Ronald E. Miller, Optimization: Foundations and Applications.  New York: John Wiley and Sons, Inc., 2000.
[2] Terry Shoup, A Practical Guide to Computer Methods for Engineers.  Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1979.
[3] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

The Domino Method Applied to Solving a Nonlinear System of Four Equations

Jsun Yui Wong

Using the following nonlinear system based on a system in Shoup [2, pp. 40-41], the following computer program is an illustration of the algorithm merging the ideas on page 251-256 of Miller [1] and the ideas in the falling domino principle, domino effect, domino phenomenon , and domino theory.  X(4) of line 503 is like the first domino of the falling domino principle.  X(4) has been chosen as the first domino because it appears that X(4) of line 503 can be knocked down as easily as any other choice.  Using the latest number for X(1), the latest number for X(2), and the latest number for X(3) from the lines preceding line 503, the right-hand side of line 503, 15-5*X(1)-3*X(2)-X(3), can knock the first domino, X(4), down to its resolution, solution, optimal value.  After line 503, there is a specific number for each of X(1), X(2), X(3), and X(4).  These four values are a solution to the nonlinear system of equations if these numbers make P(2) of line 622 close to zero, P(3) of line 623 close to zero, and P(4) of line 624 close to zero; line 701 specifies the tolerances of 0.0000001s.

5*X(1)+3*X(2)+X(3)+X(4)=15
X(1)*X(2)+X(2)*X(3)+X(3)*X(4)=17
X(1)^2+X(2)^2+X(3)^2-X(4)^2=9
X(1)*X(3)+X(2)*X(4)+X(1)^3=8

The nonlinear system above is based on that of Shoup [2,  pp. 40-41].  These four equations are used in line 503 through line 624.

0 REM   DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 J34=FIX(200+RND*1000)
27 B(1)=0:B(2)=0:B(3)=0:B(4)=0
39 N(1)=10:N(2)=10:N(3)=10:N(4)=10
61 FOR KLQ=1 TO 4
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 4
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO J34 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 5
111 KKLL=FIX(1+4*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 3
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J<B(K) THEN 250 ELSE 260
250 L(K)=B(K)
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+4*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
503 X(4)=15-5*X(1)-3*X(2)-X(3)
622 P(2)=X(1)*X(2)+X(2)*X(3)+X(3)*X(4) -17
623 P(3)=X(1)^2+X(2)^2+X(3)^2  -X(4)^2   -9
624 P(4)=X(1)*X(3)+X(2)*X(4)+X(1)^3 -8
698 FOR J99=2 TO 4
701 IF ABS(P(J99))>.0000001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
707 NEXT J99
1450 P=P(2)+P(3)+P(4)
1451 IF P<=M THEN 1670
1452 M=P
1453 PPP(2)=PP(2):PPP(4)=PP(4):PPP(6)=PP(6)
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1888 NEXT JJJ
1890 IF M>-99999! THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The top-two candidate solutions through JJJJ=-31962 are presented below.  What immediately follows is a hand copy from the computer monitor.

.9951868   1.013047   3.99235   2.992576    -5355.406
-31996

1.002835   .9922999   4.004514   3.004409   -3157.189
-31962

One notes that these candidate solutions violate the tolerances of 0.0000001s of line 701.

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31962 was 15 minutes.  

In general, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 22s.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 22s.  Here is one example: 22 J34=FIX(200+RND*200), 22 J34=FIX(200+RND*500), 22 J34=FIX(200+RND*1000), 22 J34=FIX(200+RND*1500), and 22 J34=FIX(200+RND*2000).  This multicomputer approach can considerably reduce the time to obtain one usable solution.

References

[1] Ronald E. Miller, Optimization: Foundations and Applications.  New York: John Wiley and Sons, Inc., 2000.
[2] Terry Shoup, A Practical Guide to Computer Methods for Engineers.  Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1979.
[3] W. C. Conley, Optimization: A Simplified Approach.  Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[4] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

An Illustration of the Domino Method of Solving Nonlinear Systems of Equations

Jsun Yui Wong

Using the nonlinear system of four nonlinear equations from Jacques and Judd [1, p. 68], the following computer program is an illustration of the algorithm to merge the ideas on page 251-256 of Miller [2] and the ideas in the falling domino principle, domino effect, domino phenomenon , and domino theory.  X(2),  X(3), and X(4) of line 501, line 525, and line 533 are like the first domino, the second domino, and the third domino of the falling domino principle, respectively.  Using the latest number for X(1) from the lines preceding line 501, the right-hand-side of line 501, ( 2.0625*X(1)+.0625*X(1)^3 -.5 )/2, can knock the first domino, X(2), down to its resolution, solution, its optimal value.  Similarly, using the latest number for X(2) from line 501, the right-hand-side of 525 can knock down the second domino, X(3).  Similarly, using the latest number for X(2) from line 501 and the latest number for X(3) from line 525, the right-hand-side of 533 can knock down the third domino, X(4).  Then there is a specific value for each of X(1), X(2), X(3), and X(4).  These four values are a solution to the nonlinear system if they make P(4) of line 628 close to zero; line 701 below shows that the tolerances used here are .00001.

- 2.0625*X(1)+2*x(2)-.0625*X(1)^3  +.5 =0
X(3)-2*X(2)+X(1)-.0625*X(2)^3+.125*X(2)*(X(3)-X(1))=0
X(4)-2*X(3)+X(2)-.0625*X(3)^3+.125*X(3)*(X(4)-X(2))=0
-1.9375*X(4)+X(3)-.0625*X(4)^3-.125*X(3)*X(4)+.5=0

These equations are from Jacques and Judd [1, p. 68] and are used in line 501, line 525, line 533, and line 628, respectively.

0 DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 J34=FIX(20+RND*20)
27 B(1)=0:B(2)=0:B(3)=0:B(4)=0
28 B(5)=0:B(6)=0:B(7)=0:B(8)=0
29 B(9)=0:B(10)=0
39 N(1)=10:N(2)=10:N(3)=10:N(4)=10
42 N(5)=10:N(6)=10:N(7)=10:N(8)=10
44 N(9)=10:N(10)=10
61 FOR KLQ=1 TO 10
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 10
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO J34 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 5
111 KKLL=FIX(1+10*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 1
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J<B(K) THEN 250 ELSE 260
250 L(K)=B(K)
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+10*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
461 FOR J95=1 TO 4
463 IF X(J95)<B(J95) THEN 1670
464 IF X(J95)>N(J95) THEN 1670
466 NEXT J95
501 X(2)=( 2.0625*X(1)+.0625*X(1)^3    -.5 )/2
525 X(3)=(2*X(2) -X(1)+.0625*X(2)^3+.125*X(2)*X(1) )/(1+.125*X(2))
533 X(4)=(  2*X(3)-X(2)+.0625*X(3)^3       +.125*X(3)*X(2)  )/ (.125*X(3)+1)
555 FOR J444=1 TO 10
557 IF X(J444)<0 THEN 1670
559 NEXT J444
628 P(4)=  -1.9375*X(4)+X(3)-.0625*X(4)^3       -.125*X(3)*X(4)+.5
698 FOR J99=4 TO 4
701 IF ABS(P(J99))>.00001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
707 NEXT J99
1450 P=P(4)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1888 NEXT JJJ
1890 IF M>-1 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31993 is presented below.  What immediately follows is a hand copy from the computer monitor.

.9789105922984859   .7888158197595746   .6607671896578949
.5689128740645653   0   -32000

.9789078035582719   .7888126933386815   .6607633409837652
.5689080245917145   0   -31994

.9789106279368102   .7888158597132522   .66076723884153
.5689129360378038   0   -31993

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31993 was seven seconds.

References

[1] Ian Jacques and Colin Judd, Numerical Analysis.  London: Chapman and Hall, 1987.
[2] Ronald E. Miller, Optimization: Foundations and Applications.  New York: John Wiley and Sons, Inc., 2000.
[3] W. C. Conley, Optimization: A Simplified Approach.  Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[4] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Second Edition

  
Jsun Yui Wong

Modeled after the computer program of preceding post, the following computer program tries to solve the nonlinear system of ten equations in Rice [1, pp. 356-357] and in Rice [2, p. 250].  These ten equations are used in line 501 through line 633 below.  Here line 132 is just 132 FOR K=4 TO 9, and line 22 is just 22 J34=FIX(20+RND*200).  As shown in line 701, the tolerances are .00001.

0 DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 J34=FIX(20+RND*200)
27 B(1)=0:B(2)=0:B(3)=0:B(4)=0
28 B(5)=0:B(6)=0:B(7)=0:B(8)=0
29 B(9)=0:B(10)=0
39 N(1)=10:N(2)=10:N(3)=10:N(4)=10
42 N(5)=10:N(6)=10:N(7)=10:N(8)=10
44 N(9)=10:N(10)=10
61 FOR KLQ=1 TO 10
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 10
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO J34 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 5
111 KKLL=FIX(1+10*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=4 TO 9
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J<B(K) THEN 250 ELSE 260
250 L(K)=B(K)
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+10*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
461 FOR J95=1 TO 4
463 IF X(J95)<B(J95) THEN 1670
464 IF X(J95)>N(J95) THEN 1670
466 NEXT J95
501 X(1)=3-X(4)
504 X(3)=(-X(5)+4*4.056734)/2
507 X(2)=(X(1)*X(5))/(.193*X(4))
511 X(7)=8-2*X(2)-2*X(5)-X(6)
515 X(10)=(10+  4.056734-2*X(1)-X(2)-X(4)   -X(7)-X(8)-X(9)   )/2
521 REM      X(6)=.5*X(1)*X(7)+X(2)*X(7)+.5*X(3)*X(7)
555 FOR J444=1 TO 10
557 IF X(J444)<0 THEN 1670
559 NEXT J444
566 TOT=X(1)+X(2)+X(3)+X(4)+X(5)+X(6)+X(7)+X(8)+X(9)+X(10)
621 REM        P(2)= -2+X(3)*X(7)+X(4)*X(7)+2*X(5)*X(7)
623 P(6)=X(6)*X(2)^.5-.002597*( X(2)* X(4)* TOT          )^.5
627 P(7)=X(7)*X(4)^.5-.003448*( X(1)* X(4)* TOT          )^.5
629 P(8)=X(8)*X(4)-1.799E-05*X(2)*TOT
631 P(9)=X(9)*X(4)-.0002155*X(1)*(  X(3)* TOT          )^.5
633 P(10)=X(10)*X(4)^2-3.846E-05*X(4)^2*TOT
698 FOR J99=6 TO 10
701 IF ABS(P(J99))>.00001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
707 NEXT J99
1450 P=P(6)+P(7)+P(8)+P(9)+P(10)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 10
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1888 NEXT JJJ
1890 IF M>-20 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5)
1929 PRINT A(6),A(7),A(8),A(9),A(10),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The top-two       candidate solutions through JJJJ=-31342 are presented below.  What immediately follows is a hand copy from the computer monitor.

2.996835774765609   3.985981027349706   8.113062038367777
3.164224234391303D-03   8.122635964778723D-04
6.278568056159382D-04   2.578556130201621D-02   .4188405100411829
2.507390002296335   .5609506046640796   0
-31536

2.996460549376552   3.985876793698319   8.11301383276877
3.539450623447628D-03   9.086747944914114D-04
6.634639434560849D-04   2.576559907092223D-02   .3733257691574096
2.232064025692125   .7216206740438398   0
-31342

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31342 was one hour and half an hour.

Generally speaking, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 22s.  Here is one example: 22 J34=FIX(200+RND*300), 22 J34=FIX(200+RND*500), 22 J34=FIX(200+RND*800), 22 J34=FIX(200+RND*1000), and 22 J34=FIX(200+RND*2000).  This multicomputer approach can considerably reduce the time to obtain one usable solution for a difficult nonlinear system of equations.

References

[1] John R. Rice: Numerical Methods, Software, and Analysis, Second Edition, Academic Press, 1993.
[2] John R. Rice: Numerical Methods, Software, and Analysis, Academic Press, 1983.
[3] W. C. Conley, Optimization: A Simplified Approach.  Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[4] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Solving a Nonlinear System of Seven Equations, Second Edition

 
Jsun Yui Wong

Modeled after the preceding post, the following computer program tries to solve the nonlinear system of equations in Carnahan, Luther, and Wilkes [1, pp. 321-329] and in Morris [2, p. 11].  Developed from a chemical equilibrium analysis, these equations are used in line 501 through line 625 below.

0 DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 J34=FIX(200+RND*1500)
27 B(1)=0:B(2)=0:B(3)=0:B(4)=0
39 N(1)=1:N(2)=1:N(3)=1:N(4)=1
61 FOR KLQ=1 TO 4
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 4
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO J34 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 5
111 KKLL=FIX(1+4*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 4
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J<B(K) THEN 250 ELSE 260
250 L(K)=B(K)
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+4*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
461 FOR J95=1 TO 4
463 IF X(J95)<B(J95) THEN 1670
464 IF X(J95)>N(J95) THEN 1670
466 NEXT J95
501 X(1)=(2.6058*X(2)*X(4))/X(3)
505 REM     X(5)=(20*X(1)*X(4)^3)/(170000!*X(3))
511 X(5)=1-X(1)-X(2)-X(3)-X(4)
515 X(7)=1/(X(1)+X(2)+X(5))
521 X(6)=              .5*X(1)*X(7)+X(2)*X(7)+.5*X(3)*X(7)
612 REM       X(1)=-X(2)^3+5*X(2)^2+2*X(2)+10
614 REM
616 REM     P(2)=X(1)+X(2)^3+X(2)^2-14*X(2)-29
621 P(2)= -2+X(3)*X(7)+X(4)*X(7)+2*X(5)*X(7)
623 P(4)=   -28837*X(1)    -139009!*X(2)-78213!*X(3)+18927*X(4)+8427*X(5)+13492/X(7)-10690*X(6)/X(7)
625 P(6)=              20*20*X(1)*X(4)^3-178370!*X(3)*X(5)
698 FOR J99=1 TO 7
701 IF ABS(P(J99))>.0001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
707 NEXT J99
1450 P=P(1)+P(2)+P(3)+P(4)+P(6)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1778 REM  PRINT A(1),A(2),A(3),A(4),M,JJJJ
1888 NEXT JJJ
1890 IF M>-66666! THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),A(5),A(6),A(7),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  The top-two candidate solutions through JJJJ=-31867 are presented below.  What immediately follows is a hand copy from the computer monitor.

.3223993959851501   9.237589381512722D-03   4.617769111186494D-02        
.6184815019787562   3.703821542716063D-03   .5771028426681097
2.982040895101192   -3476.43801496463   -31996

.3226257131033424   9.230860589283036D-03   4.610058086803693D-02        
.6183325623200472   3.710283119290433D-03   .576916354059011
2.980032085112913   -1579.877219226698   -31867

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31867 was three hours.

Generally speaking, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 22s.  Here is one example: 22 J34=FIX(200+RND*300), 22 J34=FIX(200+RND*500), 22 J34=FIX(200+RND*800), 22 J34=FIX(200+RND*1000), and 22 J34=FIX(200+RND*2000).  This multicomputer approach can considerably reduce the time to obtain one usable solution for a difficult nonlinear system of equations.

References

[1] B. Carnahan, H. A. Luther, and J. O. Wilkes: Applied Numerical Methods, John Wiley, 1969.
[2] J. Ll. Morris: Computational Methods in Elementary Numerical Analysis, John Wiley, 1983.
[3] W. C. Conley, Optimization: A Simplified Approach.  Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[4] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

A Nonlinear Integer/Discrete/Continuous Programming Solver Solving a Nonlinear System of Equations with Three Alternative Solutions from One Computer Run

 
Jsun Yui Wong

Modeled after the computer program of the preceding post, the following computer program tries to produce all alternative solutions for the following nonlinear system of equations from Burden and Faires [1, p. 502]:

2x(1)+x(2)+x(3)-4=0
x(1)+2*x(2)+x(3)-4=0
x(1)*x(2)*x(3)-1=0

These equations are used in line 612, line 614, and line 616.  One notes that line 132 below is just 132 FOR K=1 TO 2.  Line 701 shows that the tolerances are .00001.

0 REM    DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 J34=FIX(20+RND*50)
27 B(1)=-20:B(2)=-20:B(3)=-20:B(4)=-20
39 N(1)=20:N(2)=20:N(3)=20:N(4)=20
61 FOR KLQ=1 TO 4
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 4
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO J34 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 5
111 KKLL=FIX(1+4*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 2
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J<B(K) THEN 250 ELSE 260
250 L(K)=B(K)
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+4*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
461 FOR J95=1 TO 4
463 IF X(J95)<B(J95) THEN 1670
464 IF X(J95)>N(J95) THEN 1670
466 NEXT J95
612 X(3)=-2*X(1)-X(2)+4
614 P(2)=X(1)+2*X(2)+X(3)-4
616 P(3)=X(1)*X(2)*X(3)-1
698 FOR J99=1 TO 4
701 IF ABS(P(J99))>.00001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
707 NEXT J99
1450 P=P(1)+P(2)+P(3)+P(4)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1778 REM  PRINT A(1),A(2),A(3),A(4),M,JJJJ
1888 NEXT JJJ
1890 IF M>-10 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter.  Only distinct solutions are presented below.  What immediately follows is a hand copy from the computer monitor.

.7675994   .767594   1.697207   0   -32000

-.4342565   -.4342637   5.302777   0   -31999

1.00001   1.000012   .999969   0   -31968

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the running time from JJJJ=-32000 through JJJJ=-31968 was thirty seconds.

Generally speaking, a usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 22s.  Here is one example: 22 J34=FIX(200+RND*300), 22 J34=FIX(200+RND*500), 22 J34=FIX(200+RND*800), 22 J34=FIX(200+RND*1000), and 22 J34=FIX(200+RND*2000).  This multicomputer approach can considerably reduce the time to obtain one usable solution for a difficult nonlinear system of equations.

References

[1] Richard L. Burden and J. Douglas Faires, Numerical Analysis, Third Edition.  Boston: Prindle, Weber and Schmidt, 1985.   
[2] W. C. Conley, Optimization: A Simplified Approach.  Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[3] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Solving a Nonlinear System of Equations

   
Jsun Yui Wong

"For F(X)=0, the iterative procedures are much more complex because finding roots to a system of nonlinear equations is, in general, a very difficult task," Miller [1, p. 275].

Modelled after the computer program of the preceding post, the following computer program tries to solve the following nonlinear system of four equations from Miller [1, p. 254]:

2x(1)+x(2)=3
x(1)*x(3)+x(2)-x(4)=0
4x(3)-x(4)^2=20
x(1)+(0.5)x(4)=0

These equations are used in line 561 through line 564, and these lines suggest that the present algorithm does not depend on substitutions.  Line 701 below shows that the four tolerances used are .0001.

0 REM    DEFDBL A-Z
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
22 J34=FIX(11+RND*990)
27 B(1)=-20:B(2)=-20:B(3)=-20:B(4)=-20
39 N(1)=20:N(2)=20:N(3)=20:N(4)=20
61 FOR KLQ=1 TO 4
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 4
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO J34 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 5
111 KKLL=FIX(1+4*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 4
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J<B(K) THEN 250 ELSE 260
250 L(K)=B(K)
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+4*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
461 FOR J95=1 TO 4
463 IF X(J95)<B(J95) THEN 1670
464 IF X(J95)>N(J95) THEN 1670
466 NEXT J95
561 P(1)=2*X(1)+X(2)-3
562 P(2)=X(1)*X(3)+X(2)-X(4)
563 P(3)=4*X(3)-X(4)^2-20
564 P(4)=X(1)+.5*X(4)
698 FOR J99=1 TO 4
701 IF ABS(P(J99))>.0001 THEN P(J99)=-555555!*ABS(P(J99)) ELSE P(J99)=0
707 NEXT J99
1450 P=P(1)+P(2)+P(3)+P(4)
1451 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1778 REM
1888 NEXT JJJ
1890 IF M>-1000 THEN 1928 ELSE 1999
1928 PRINT A(1),A(2),A(3),A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with the IBM basica/D interpreter, and the complete output through JJJJ=-31599 is shown below.  What immediately follows is a hand copy from the computer monitor.

-.5640721   4.128173   5.318301   1.128328   0
-31941

-.5640645   4.128123   5.318286   1.128316   0
-31856

-.5641205   4.128199   5.318132   1.128049   0
-31855

-.5640542   4.128192   5.318293   1.128304   0
-31622

-.5646212   4.129342   5.317459   1.126911   -647.7695
-31599

On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, the running time from JJJJ=-32000 through JJJJ=-31599 was two hours.

A usable solution for another application of this algorithm may come early, late, or worse.  To alleviate this shortcoming, one can simultaneously run independent computers with different line 22s.  Here is one example: 22 J34=FIX(11+RND*190), 22 J34=FIX(11+RND*290), 22 J34=FIX(11+RND*490), 22 J34=FIX(11+RND*990), and 22 J34=FIX(11+RND*1990).  This multicomputer approach can considerably reduce the time to obtain one usable solution.

References

[1] Ronald E. Miller, Optimization: Foundations and Applications.  John Wiley and Sons, Inc. New York, 2000, Chapter 5.
[2] W. C. Conley, Optimization: A Simplified Approach.  Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[3] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.