Finding by Computer Integer Solutions of Nonlinear Systems of Equations

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system:

         - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2=-106,

         13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)=127,

         - 2 * X(1) * X(2) * X(3) - X(4) - X(5) - X(6)=-70,

         1 * X(1) + 1 * X(2) + 2 * X(3) - 7 * X(4) - 1 * X(5) + 1 * X(6)=-29,

         1 * X(1) + 1 * X(2) + 1 * X(3) + 2 * X(4) - 9 * X(5) - 3 * X(6)=21,

        1 * X(1) + 1 * X(2) + 1 * X(3) + 1 * X(4) + 2 * X(5) - 7 * X(6)=-13.

The first three of these equations are based on page 177 of Conley [4].  The last three come from

Greenspan and Casulli [5, page 41].

One notes line 94, which is 94 A(KK) = -30 + FIX(RND * 61).


0 DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333)

12 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+317

    77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91


    88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128


    91 FOR KK = 1 TO 6

        94 A(KK) = -30 + FIX(RND * 61)



    95 NEXT KK


    128 FOR I = 1 TO 200000

        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)


            182 REM IF RND < -.1 THEN 183 ELSE GOTO 189

            183 r = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * r

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP


        996 N83 = 106 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2


        998 N82 = -127 + 13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)


        1000 N81 = 70 - 2 * X(1) * X(2) * X(3) - X(4) - X(5) - X(6)
        1004 N84 = 29 + 1 * X(1) + 1 * X(2) + 2 * X(3) - 7 * X(4) - 1 * X(5) + 1 * X(6)
        1005 N85 = -21 + 1 * X(1) + 1 * X(2) + 1 * X(3) + 2 * X(4) - 9 * X(5) - 3 * X(6)

        1006 N86 = 13 + 1 * X(1) + 1 * X(2) + 1 * X(3) + 1 * X(4) + 2 * X(5) - 7 * X(6)


        1335 P = -ABS(N81) - ABS(N82) - ABS(N83) - ABS(N84) - ABS(N85) - ABS(N86)

        1499 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 6


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1664 NN81 = N81: NN82 = N82

        1665 NN83 = N83: NN84 = N84

        1666 NN85 = N85: NN86 = N86

    1670 NEXT I
    1888 IF M < 0 THEN 1999


    1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, NN81, NN82, NN83, NN84, NN85, NN86

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. Copied by hand from the screen, the computer program’s complete output through
JJJJ=-24354 is shown below:
 
2      3      5      7      -1  
4      0      -31947      0      0
0      0      0      0

2      3      5      7      -1  
4      0      -31946      0      0
0      0      0      0

2      3      5      7      -1
4      0      -31945      0      0
0      0      0      0

2      3      5      7      -1
4      0      -31944      0      0
0      0      0      0

2      3      5      7      -1
4      0      -31943      0      0
0      0      0      0

2      3      5      7      -1
4      0      -31942      0      0
0      0      0      0

2      3      5      7      -1
4      0      -28653      0      0
0      0      0      0

2      3      5      7      -1
4      0      -28652      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25332      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25331      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25330      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25329      0      0
0      0      0      0
2      3      5      7      -1
4      0      -25328      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25327      0      0
0      0      0      0
2      3      5      7      -1
4      0      -25326      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25325      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25324      0      0
0      0      0      0
2      3      5      7      -1
4      0      -25323      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25322      0      0
0      0      0      0
2      3      5      7      -1
4      0      -25321      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25320      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25319      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25318      0      0
0      0      0      0

2      3      5      7      -1
4      0      -25317      0      0
0      0      0      0

2      3      5      7      -1
4      0      -24355      0      0
0      0      0      0

2      3      5      7      -1
4      0      -24354      0      0
0      0      0      0

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 [11], the wall-clock time through
JJJJ=-24354 was 32 minutes.

Incidentally, one can stop the computer run as soon as the first 0 for M appears on the screen.  For the present problem, that happens at JJJJ=-31947.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] W. Conley, Computer Optimization Techniques, Revised Edition.  Petrocelli Books, Inc., NY/Princeton, 1984.
[5] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[6] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[7] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[8] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[9]  Thomas L. Saaty, Optimization in Integers and Related Extremal Problems.  McGraw-Hill, 1970.
[10] Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

ERRATA: How To Find Integer Solutions (If Any) Of A Nonlinear System Of Equations, Improved Edition      

Jsun Yui Wong

Each 750 of the computer output should read 75.

How To Find Integer Solutions (If Any) Of A Nonlinear System Of Equations, Improved Edition

 
Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the first problem on page 177 of Conley [4], which is to solve 

         X(1)^2 + 3 * X(2) + 5 * X(4) + 6 * X(3) + 7 * X(6)) =5492,

         2 * X(1) * X(2) * X(3) + X(4) +  X(5)  + X(6)= 1114 X(5) = 638213,

         X(1) + X(2) + X(3) + 9 * X(4) + 11 * X(5) + X(6)=2787,

         3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6)=1768,

        13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)=844252,

where 0<=X(i)>=200 and X(i)'s are whole numbers.


0 DEFDBL A-Z

3 DEFINT J, K, X


4 DIM X(342), A(342), L(333), K(333)


12 FOR JJJJ = -32000 TO 32000


    14 RANDOMIZE JJJJ

    16 M = -1D+317

    77 IF JJJJ > -32000 THEN GOTO 88 ELSE GOTO 91


    88 IF RND < .05 THEN GOTO 91 ELSE GOTO 128


    91 FOR KK = 1 TO 6

        94 A(KK) = 100


    95 NEXT KK


    128 FOR I = 1 TO 1250000


        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)


            182 REM IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)



        191 NEXT IPP


        1005 N88 = 5492 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6) - X(1) ^ 2

        1006 N89 = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6) - X(5)


        1009 N91 = -1768 + 3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6)


        1113 REM


        1117 N93 = -2787 + X(1) + X(2) + X(3) + 9 * X(4) + 11 * X(5) + X(6)

        1119 N94 = -844252 + 13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)



        1335 P = -ABS(N91) - ABS(N93) - ABS(N94) - ABS(N88) - ABS(N89)


        1499 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 6


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1664 NN88 = N88: NN89 = N89: NN91 = N91: NN93 = N93: NN94 = N94


    1670 NEXT I
    1888 IF M < 0 THEN 1999


    1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, NN88, NN89, NN91, NN93, NN94

1999 NEXT JJJJ


This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-27462 is shown below:

58      47      117      152     102
750      0      -31268      0      0
0      0      0    

58      47      117      152     102
750      0      -31267      0      0
0      0      0    

58      47      117      152     102
750      0      -31266      0      0
0      0      0    

58      47      117      152     102
750      0      -30638      0      0
0      0      0    

58      47      117      152     102
750      0      -30441      0      0
0      0      0    

58      47      117      152     102
750      0      -30440      0      0
0      0      0    

58      47      117      152     102
750      0      -27462      0      0
0      0      0    

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 [10], the wall-clock time through
JJJJ=-27462 was one hour and forty minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] W. Conley, Computer Optimization Techniques, Revised Edition.  Petrocelli Books, Inc., NY/Princeton, 1984.
[5] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[6] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[7] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[8] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[9] Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[10] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

How To Find Integer Solutions (If Any) Of A Nonlinear System Of Equations--Yet Another System

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the first problem on page 177 of Conley [4], which is to solve 

         X(1)^2 + 3 * X(2) + 5 * X(4) + 6 * X(3) + 7 * X(6)) =5492,

         2 * X(1) * X(2) * X(3) + X(4) +  X(5)  + X(6)= 1114 X(5) = 638213,

         X(1) + X(2) + X(3) + 9 * X(4) + 11 * X(5) + X(6)=2787,

         3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6)=1768,

        13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)=844252,

where 0<=X(i)>=200 and X(i)'s are whole numbers.


0 DEFDBL A-Z

3 DEFINT J, K, X

4 DIM X(342), A(342), L(333), K(333)


12 FOR JJJJ = -32000 TO 32000


    14 RANDOMIZE JJJJ

    16 M = -1D+317

    91 FOR KK = 1 TO 6

        94 A(KK) = FIX(RND * 201)


    95 NEXT KK
    128 FOR I = 1 TO 2000000


        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)


            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP




        1001 REM



        1004 X(1) = (5492 - 3 * X(2) - 5 * X(4) - 6 * X(3) - 7 * X(6)) ^ .5

        1114 X(5) = 638213 - 2 * X(1) * X(2) * X(3) - X(4) - X(6)


        1009 N91 = -1768 + 3 * X(1) + 4 * X(2) + X(3) + X(4) + 6 * X(5) + 7 * X(6)




        1113 REM

        1117 N93 = -2787 + X(1) + X(2) + X(3) + 9 * X(4) + 11 * X(5) + X(6)

        1119 N94 = -844252 + 13 * X(1) + X(2) * X(3) * X(4) + X(5) * X(6)



        1335 P = -ABS(N91) - ABS(N92) - ABS(N93) - ABS(N94)



        1499 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 6


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1664 NN91 = N91: NN92 = N92: NN93 = N93: NN94 = N94

    1670 NEXT I
    1888 IF M < -1400 THEN 1999


    1917 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ, JJJJ, JJJJ, NN91, NN92, NN93, NN94

1999 NEXT JJJJ


This computer program was run with qb64v1000-win [10]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-29108 is shown below:

53      82      96      105      86
198      -1314      -30920      -30920      -30920
822      0      -467      25

55      139      46      129      111
169      -1029      -29865      -29865      -29865
977      0      4      48

58      47      117      152      102
75      0      -29108      -29108      -29108
0      0      0      0

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 [10], the wall-clock time through
JJJJ=-29108 was two hours.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] W. Conley, Computer Optimization Techniques, Revised Edition.  Petrocelli Books, Inc., NY/Princeton, 1984.
[5] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[6] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[7] Microsoft Corp. BASIC, second edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[8] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[9]  Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[10] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

How to Find Integer Solutions (if any) of a Nonlinear Systems of Equations

Jsun Yui Wong

The computer program listed below seeks to find integer solutions (if any) of the following nonlinear system, which is based on the nonlinear system of five variables on page 238 of Shoup [8].

         X(1) + X(2) + X(3) + X(4) = 31,
      
         X(1) * X(2) + X(2) * X(3) + X(4) * X(5) = 52,

         X(1) ^ 2 + X(3) * X(4) - X(2) ^ 2 + X(1) * X(5) = 86,

         X(1) - X(2) * X(4) + X(3) ^ 2 + X(5) ^ 3 = 19,

         X(1) * X(3) - X(2) ^ 3 * X(5) - X(5) * X(2) + X(3) ^ 2 * X(4) = 240.


0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(342), A(342), L(333), K(333)


12 FOR JJJJ = -32000 TO 32000


    14 RANDOMIZE JJJJ

    16 M = -1D+317

    91 FOR KK = 1 TO 5

        94 A(KK) = -10 + FIX(RND * 20)


    95 NEXT KK
    128 FOR I = 1 TO 3000000


        129 FOR K = 1 TO 5


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)

            181 B = 1 + FIX(RND * 5)


            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP



        1001 X(4) = 31 - X(1) - X(2) - X(3)



        1004 REM

        1111 N96 = -52 + X(1) * X(2) + X(2) * X(3) + X(4) * X(5)


        1116 N97 = -86 + X(1) ^ 2 + X(3) * X(4) - X(2) ^ 2 + X(1) * X(5)
        1118 N98 = -19 + X(1) - X(2) * X(4) + X(3) ^ 2 + X(5) ^ 3
        1119 N99 = -240 + X(1) * X(3) - X(2) ^ 3 * X(5) - X(5) * X(2) + X(3) ^ 2 * X(4)
        1335 P = -ABS(N96) - ABS(N97) - ABS(N98) - ABS(N99)


        1499 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 5


            1658 A(KEW) = X(KEW)
        1659 NEXT KEW
        1661 M = P
        1666 NN96 = N96: NN97 = N97: NN98 = N98: NN99 = N99

    1670 NEXT I
    1888 IF M < -29 THEN 1999

    1917 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ, NN96, NN97, NN98, NN99

1999 NEXT JJJJ


This computer program was run with qb64v1000-win [9]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31997 is shown below:

2      0      3        26       2
0      -32000      0      0      0
0

2      0      3        26       2
0      -31999      0      0      0
0

2      0      3        26       2
0      -31997      0      0      0
0

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [9], the wall-clock time through
JJJJ=-31997 was 25 seconds, not including "Creating .EXE file..." time.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] 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.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[9]  Terry E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, 1984.
[9] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

Computer Solutions/Integer Solutions of Nonlinear Systems of Equations

Jsun Yui Wong

The two computer programs listed below seek to solve a continuous case and an integer case of two nonlinear systems of equations, respectively.  The first case comes from page 44 of Greenspan and Casulli [4] and is as follows:  

         -10 * X(1) + 5 * X(2) - EXP(X(1)),

         5 * X(1) - 10 * X(2) + 5 * X(3) - EXP(X(2)),

         5 * X(2) - 10 * X(3) + 5 * X(4) - EXP(X(3)),

         5*X(3) - 10 * X(4) - EXP(X(4))=0.


0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(342), A(342), L(333), K(333)


12 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+317

    91 FOR KK = 1 TO 4


        94 A(KK) = -10 + FIX(RND * 21)



    95 NEXT KK
    128 FOR I = 1 TO 500000



        129 FOR K = 1 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)


            182 REM IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP



        291 FOR j44 = 1 TO 4
            293 IF X(j44) > 30 THEN 1670



        299 NEXT j44



        333 X(3) = (10 * X(4) + EXP(X(4))) / 5

        334 IF X(3) > 30 THEN 1670


        336 N96 = 5 * X(1) - 10 * X(2) + 5 * X(3) - EXP(X(2))
        338 N97 = 5 * X(2) - 10 * X(3) + 5 * X(4) - EXP(X(3))


        344 N98 = -10 * X(1) + 5 * X(2) - EXP(X(1))

        1335 P = -ABS(N96) - ABS(N97) - ABS(N98)



        1499 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 4


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

    1670 NEXT I
    1888 IF M < -.000001 THEN 1999

    1917 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31982 is shown below:

-.282813848003122        -.4148956819788955      -.4148956819702162
-.2828138479984057      -3.60628082951564D-10      -31985

-.2828138480373464        -.414895681997618      -.4148956820101716
-.2828138480169833        -7.031838999971285D-10      -31983

-.2828138480379652        -.4148956820071726      -.4148956819847773
-.282813848005176         -6.283482067992452D-10      -31982

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time through JJJJ=-31982 was 15 seconds, not including "Creating .EXE file..." time.


The second case--shown below--comes from page 47 of Greenspan and Casulli [4]:

        X(1) +X(2) + X(3) + X(4) =0,

        LOG(X(1)) + X(2) + LOG(X(3)) + X(4) = -2,

        X(1) * X(2) * X(3) - X(4) =0,

        X(1) - EXP(X(2)) + X(3) + EXP(X(4)) =2.
 

0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(342), A(342), L(333), K(333)


12 FOR JJJJ = -32000 TO 32000


    14 RANDOMIZE JJJJ

    16 M = -1D+317

    91 FOR KK = 1 TO 4

        94 A(KK) = -50 + FIX(RND * 101)


    95 NEXT KK
    128 FOR I = 1 TO 1000



        129 FOR K = 1 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)


            182 IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP



        1001 X(1) = -X(2) - X(3) - X(4)

        1002 IF X(1) < .01 THEN 1670
        1003 IF X(3) < .01 THEN 1670



        1004 N96 = LOG(X(1)) + X(2) + LOG(X(3)) + X(4) + 2

        1007 N97 = X(1) * X(2) * X(3) - X(4)

        1009 N98 = X(1) - EXP(X(2)) + X(3) + EXP(X(4)) - 2


        1335 P = -ABS(N96) - ABS(N97) - ABS(N98)



        1499 IF P <= M THEN 1670
        1657 FOR KEW = 1 TO 4


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

    1670 NEXT I
    1888 IF M < -1 THEN 1999

    1917 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-31911 is shown below:

1      -1      1      -1      0
-31982

1      -1      1      -1      0
-31947

1      -1      1      -1      0
-31936

1      -1      1      -1      0
-31926

1      -1      1      -1      0
-31923

1      -1      1      -1      0
-31917

1      -1      1      -1      0
-31911

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time through JJJJ=-31911 was 2 seconds, not including "Creating .EXE file..." time.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] 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.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

Solving a Nonlinear System of Equations

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following nonlinear system of equations from page 665 of Burden, Faires, and Burden [1, p. 665, Exercise 6]:

         4 * X(1) -X( 2)   +X(3) = X(1) * X(4),

        -X(1) + 3 * X(2) - 2 * X(3) = X(2) * X(4),

         X(1) - 2 * X(2) + 3 * X(3) = X(3) * X(4),
       
         X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2 = 1.


0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+317


    91 FOR KK = 96 TO 99

        94 A(KK) = -20 + FIX(RND * 150)


    95 NEXT KK
    128 FOR I = 1 TO 8000



        129 FOR K = 96 TO 99


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)

            181 B = 96 + FIX(RND * 9)


            182 REM IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP

        331 X(98) = X(96) * X(99) + X(97) - 4 * X(96)



        335 REM N98 = X(96) ^ 2 - 625 * X(97) ^ 2 - 1 / 4

        339 REM N99 = EXP(-X(96) * X(97)) + 20 * X(98) + (10 * 3.141592654 - 3) / 3

        351 N97 = -X(97) * X(99) - X(96) + 3 * X(97) - 2 * X(98)

        353 N98 = -X(98) * X(99) + X(96) - 2 * X(97) + 3 * X(98)

        355 REM N98 = -X(98) * X(99) + X(97) - 4 * X(96)
        359 N99 = -1 + X(96) ^ 2 + X(97) ^ 2 + X(98) ^ 2

        1335 P = -ABS(N98) - ABS(N99) - ABS(N97)


        1499 IF P <= M THEN 1670
        1657 FOR KEW = 96 TO 99


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

    1670 NEXT I
    1888 IF M < -.0000001 THEN 1999

    1917 PRINT A(96), A(97), A(98), A(99), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-30366 is shown below:

-2.493654401671402D-15      .7071068014058323      .7071068014058398
 .9999999980999506            -5.98758606780458D-08      -31892

8.33475294629724D-18        .7071067794128311      .7071067794128311
1.000000018454071            -3.111482494474582D-08      -31647

7.720047542022925D-10        .707106789031514       .7071067867154998
1.000000005454302             -2.663362473018802D-08      -31561

3.038262892746562D-09        .7071067999755675      .707106790860779
1.00000002148609               -7063893669385927D-08      -30989

4.231662939811113D-10      .7071067906077655      .7071067893382665
.9999999970098467            -2.908354898334712D-08      -30691

0                     -.7071067723044366           -.7071067723044366
1.00000000499632             -3.218826598465926D-08      -30366

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time through
JJJJ=-30366 was 30 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] 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.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

Simultaneously Solving a Nonlinear System of Transcendental Equations

Jsun Yui Wong

The computer program listed below seeks to solve simultaneously the following nonlinear system of equations from Burden, Faires,and Burden [1, p. 656, Exercise 7d]:

         6*X(1) - 2 * COS(X(2) * X(3)) - 1 = 0,

         9 * X(2) + (X(1) ^ 2 + SIN(X(3)) + 1.06) ^ .5 + .9 = 0,

         60 * X(3)  +  3 * EXP(-X(1) * X(2)) + 10 * 3.141592654 - 3 = 0.


0 DEFDBL A-Z

3 DEFINT J, K

4 DIM X(342), A(342), L(333), K(333)
12 FOR JJJJ = -32000 TO 32000

    14 RANDOMIZE JJJJ

    16 M = -1D+317


    91 FOR KK = 91 TO 98

        94 A(KK) = -20 + FIX(RND * 150)


    95 NEXT KK
    128 FOR I = 1 TO 8000




        129 FOR K = 91 TO 98


            131 X(K) = A(K)
        132 NEXT K
        155 FOR IPP = 1 TO FIX(1 + RND * 3)

            181 B = 91 + FIX(RND * 8)


            182 REM IF RND < -.1 THEN 183 ELSE GOTO 189

            183 R = (1 - RND * 2) * A(B)
            186 X(B) = A(B) + (RND ^ 3) * R

            188 GOTO 191
            189 IF RND < .5 THEN X(B) = A(B) - FIX(1 + RND * 1.99) ELSE X(B) = A(B) + FIX(1 + RND * 1.99)


        191 NEXT IPP
        331 X(96) = (2 * COS(X(97) * X(98)) + 1) / 6


        335 N98 = 9 * X(97) + (X(96) ^ 2 + SIN(X(98)) + 1.06) ^ .5 + .9


        339 N99 = 3 * EXP(-X(96) * X(97)) + 60 * X(98) + (10 * 3.141592654 - 3)
        1335 P = -ABS(N98) - ABS(N99)




        1499 IF P <= M THEN 1670
        1657 FOR KEW = 96 TO 98



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

    1670 NEXT I
    1888 IF M < -.000001 THEN 1999

    1917 PRINT A(96), A(97), A(98), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [8]. Copied by hand from the screen, the computer program’s complete output through JJJJ=-30739 is shown below:

.4981446846090066         -.1996058948931147         -.5288259765133804
-7.301866214139618D-08         -31894

.4981446845080947         -.1996058989003793         -.5288259802917031
-1.850737475616996D-07         -30848

.4981446846603527         -.1996058928890903         -.5288259744982781
-2.096859361336691D-07         -30739

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

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and with qb64v1000-win [8], the wall-clock time through
JJJJ=-30739 was 30 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References
[1] R. Burden, J. Faires, A. Burden, Numerical Analysis, Tenth Edition. Cengage Learning, 2016.
[2] R. Burden, J. Faires, Numerical Analysis, Sixth Edition. Brooks/Cole Publishing Company, 1996.
[3] R. Burden, J. Faires, Numerical Analysis, Third Edition. PWS Publishers, 1985.
[4] D. Greenspan, V. Casulli, Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley Publishing Company, 1988
[5] L. W. Johnson, R. D. Riess, Numerical Analysis, Second Edition. Addison-Wesley Publishing Company, 1982
[6] 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.
[7] William H. Mills, A System of Quadratic Diophantine Equations, Pacific Journal of Mathematics, 3 (1953), pp. 209-220.
[8] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.