Solving a system of inequalities for a range of α (alpha)

Solving a system of inequalities for a range of α (alpha)

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following system of inequalities for a range of alpha (α) from Whis [103]:

p1+p4+p3 = α, alpha

p1+p2+p5 = α, alpha

1−p1−p4−p5 = α, alpha

0 ≤  p1 ≤ 1

0 ≤  p2 ≤ 1

0 ≤  p3 ≤ 1

0 ≤  p4 ≤ 1

0 ≤  p5 ≤ 1

                5

0 <=  1− ∑ pi  <= 1, for i=1 to 5

                i=1

alpha, α   epsilon  [1/3,2/3] .    

The key lines of the following computer program are    333 X(1) = -X(4) - X(3) + X(6)   and           

599 PD1 =  - ABS(X(1) + X(2) + X(5) - X(6)) - ABS(1 - X(1) - X(4) - X(5) - X(6)). 

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

    87 RANDOMIZE JJJJ

    88 M = -3D+30

    117 FOR J44 = 1 TO 5

        121 A(J44) = 0 + RND * 1

    122 NEXT J44

    124 A(6) = .33333333 + RND * .3333333

    127 FOR i = 1 TO 30000

        129 FOR KKQQ = 1 TO 6

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 6)

            140 B = 1 + FIX(RND * 6)

            156 IF RND < .5 THEN 160 ELSE GOTO 167

            160 r = (1 - RND * 2) * A(B)

            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            166 GOTO 168

            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

        168 NEXT IPP

        211 FOR J44 = 1 TO 5

            213 X(J44) = INT(X(J44))

        219 NEXT J44

        333 X(1) = -X(4) - X(3) + X(6)

        422 FOR J44 = 1 TO 5

            433 IF X(J44) < 0 THEN 1670

            435 IF X(J44) > 1 THEN 1670

        455 NEXT J44

        464 IF 1 - (X(1) + X(2) + X(3) + X(4) + X(5)) < 0 THEN 1670

        466 IF 1 - (X(1) + X(2) + X(3) + X(4) + X(5)) > 1 THEN 1670

        599 PD1 =  - ABS(X(1) + X(2) + X(5) - X(6)) - ABS(1 - X(1) - X(4) - X(5) - X(6))

        669 p = PD1

        1111 IF p <= M THEN 1670

        1452 M = p

        1454 FOR KLX = 1 TO 6

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT i

    1778 IF M < 0 THEN 1999

    1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through

JJJJ= -31997 is shown below. GW-BASIC also can handle this computer program.

.5      0      0       0      0

.5      0       -32000 

.5      0      0       0      0

.5      0       -31997

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31997 was 2 seconds, counting from “Starting program…”. The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

Acknowledgement

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

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