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added 1788 characters in body

edited Dec 29, 2015 at 13:24

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Edit1 As it was correctly noted in the comment below, I try to select all solutions which can be given probability interpretation.

Note that in the code below I removed (command Most[]) one pair of GroebnerBase, which actually cases the problem for NSolve. (The coefficients of Groebner base seems are very large, may be somebody will comment on that)

positiveSols = Select[Flatten[ NSolve[#, {wa, wb}] & /@ Most[(Part[gb1, #] & /@ Subsets[Range[4], {2}])], 1], MatchQ[({wa, wb} /. #), {_?Positive, _?Positive}] &]

{{wb -> 0.436467, wa -> 0.436467}, {wb -> 0.436467, wa -> 2.25209}, {wb -> 0.5, wa -> 0.6}, {wb -> 0.5, wa -> 2.9374}, {wb -> 1.42126, wa -> 0.646862}, {wb -> 1.42126, wa -> 0.98405}, {wb -> 1.42126, wa -> 1.41911}, {wb -> 1.45037, wa -> 0.63115}, {wb -> 1.45037, wa -> 0.980516}, {wb -> 1.45037, wa -> 1.45037}, {wb -> 0.5, wa -> 0.194035}, {wb -> 0.5, wa -> 2.26841}, {wb -> 1.45037, wa -> 0.63115}, {wb -> 1.45037, wa -> 0.980516}, {wb -> 1.45037, wa -> 1.45037}, {wb -> 1.42126, wa -> 0.646862}, {wb -> 1.42126, wa -> 0.98405}, {wb -> 1.42126, wa -> 1.41911}, {wb -> 0.5, wa -> 0.6}, {wb -> 0.5, wa -> 2.9374}, {wb -> 0.436467, wa -> 0.436467}, {wb -> 0.436467, wa -> 2.25209}}

And at last

probabilitySols = Select[positiveSols, MatchQ[({wa, wb} /. #), {_?(0 <= # <= 1 &), _?(0 <= # <= 1 &)}] &]

{{wb -> 0.436467, wa -> 0.436467}, {wb -> 0.436467, wa -> 0.436467}, {wb -> 0.5, wa -> 0.6}, {wb -> 0.5, wa -> 0.194035}, {wb -> 0.5, wa -> 0.6}}

we see that mentioned solution {wb -> 0.5, wa -> 0.6} indeed is in the list. Once again, this list of solutions is still not complete, because one pair of Groebner base equations was removed.

Edit1 As it was correctly noted in the comment below, I try to select all solutions which can be given probability interpretation.

positiveSols = Select[Flatten[ NSolve[#, {wa, wb}] & /@ Most[(Part[gb1, #] & /@ Subsets[Range[4], {2}])], 1], MatchQ[({wa, wb} /. #), {_?Positive, _?Positive}] &]

{{wb -> 0.436467, wa -> 0.436467}, {wb -> 0.436467, wa -> 2.25209}, {wb -> 0.5, wa -> 0.6}, {wb -> 0.5, wa -> 2.9374}, {wb -> 1.42126, wa -> 0.646862}, {wb -> 1.42126, wa -> 0.98405}, {wb -> 1.42126, wa -> 1.41911}, {wb -> 1.45037, wa -> 0.63115}, {wb -> 1.45037, wa -> 0.980516}, {wb -> 1.45037, wa -> 1.45037}, {wb -> 0.5, wa -> 0.194035}, {wb -> 0.5, wa -> 2.26841}, {wb -> 1.45037, wa -> 0.63115}, {wb -> 1.45037, wa -> 0.980516}, {wb -> 1.45037, wa -> 1.45037}, {wb -> 1.42126, wa -> 0.646862}, {wb -> 1.42126, wa -> 0.98405}, {wb -> 1.42126, wa -> 1.41911}, {wb -> 0.5, wa -> 0.6}, {wb -> 0.5, wa -> 2.9374}, {wb -> 0.436467, wa -> 0.436467}, {wb -> 0.436467, wa -> 2.25209}}

And at last

probabilitySols = Select[positiveSols, MatchQ[({wa, wb} /. #), {_?(0 <= # <= 1 &), _?(0 <= # <= 1 &)}] &]

{{wb -> 0.436467, wa -> 0.436467}, {wb -> 0.436467, wa -> 0.436467}, {wb -> 0.5, wa -> 0.6}, {wb -> 0.5, wa -> 0.194035}, {wb -> 0.5, wa -> 0.6}}

we see that mentioned solution {wb -> 0.5, wa -> 0.6} indeed is in the list. Once again, this list of solutions is still not complete, because one pair of Groebner base equations was removed.

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answered Dec 29, 2015 at 12:26

Acus

3.9k
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First, simplify you input:

GS65new = Simplify[GS65]

-(((-1 + wb) ((-1 + wb)^3 wb (1 + 2 wb + 4 wb^2 + 8 wb^3) + 15 wa^2 (-1 + wb)^4 wb^5 (-15 - 41 wb + 142 wb^2 - 132 wb^3 + 40 wb^4) - 5 wa^3 (-1 + wb)^3 wb^5 (165 - 389 wb - 708 wb^2 + 2264 wb^3 - 1952 wb^4 + 560 wb^5) + 5 wa^4 (-1 + wb)^2 wb^5 (-255 + 1613 wb - 2402 wb^2 - 984 wb^3 + 4992 wb^4 - 4144 wb^5 + 1120 wb^6) - 5 wa^7 wb^5 (-15 + 244 wb - 1344 wb^2 + 3564 wb^3 - 5166 wb^4 + 4228 wb^5 - 1848 wb^6 + 336 wb^7) - 3 wa^5 wb^5 (-345 + 3812 wb - 14607 wb^2 + 25702 wb^3 - 19648 wb^4 - 878 wb^5 + 12180 wb^6 - 7896 wb^7 + 1680 wb^8) + wa^6 wb^5 (-435 + 5951 wb - 28026 wb^2 + 63316 wb^3 - 75574 wb^4 + 45430 wb^5 - 8148 wb^6 - 4200 wb^7 + 1680 wb^8) - wa (-1 + wb)^3 (-1 + wb^4 - 11 wb^5 + 209 wb^6 - 457 wb^7 + 446 wb^8 - 212 wb^9 + 40 wb^10)))/((1 - 2 wb + 2 wb^2) (-wb + wa (-1 + 2 wb))))

GS56new = Simplify[GS56]

-((wa (-1 + wb)^5 (1 + 2 wb) (1 + 4 wb^2) ((-1 + wb)^5 (1 + 5 wb) - 15 wa (-1 + wb)^4 wb (1 + 5 wb) + 5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) - 5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) + 5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 3 wa^4 wb (23 - 202 wb + 473 wb^2 - 252 wb^3 - 252 wb^4 + 210 wb^5) - wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 210 wb^5)))/((1 - 2 wb + 2 wb^2) (-wb + wa (-1 + 2 wb))))

Then calculate GroebnerBase

(gb = GroebnerBasis[{GS56new - 25/100, GS65new - 75/100}, {wa, wb}]) // Length

4

Taking gb elements which includes both variables we get

NSolve[{gb[[1]] == 0, gb[[3]] == 0}, {wa, wb}]

{{wb -> -0.421257, wa -> -0.421257}, {wb -> -0.421257, wa -> 0.0231239}, {wb -> -0.421257, wa -> 0.365881}, {wb -> -0.421257, wa -> 0.703403 - 0.315787 I}, {wb -> -0.421257, wa -> 0.703403 + 0.315787 I}, {wb -> -0.421257, wa -> 1.15004 - 0.190458 I}, {wb -> -0.421257, wa -> 1.15004 + 0.190458 I}, {wb -> -0.421257, wa -> -0.421257}, {wb -> -0.421257, wa -> 0.0231239}, {wb -> -0.421257, wa -> 0.365881}, {wb -> -0.421257, wa -> 0.703403 - 0.315787 I}, {wb -> -0.421257, wa -> 0.703403 + 0.315787 I}, {wb -> -0.421257, wa -> 1.15004 - 0.190458 I}, {wb -> -0.421257, wa -> 1.15004 + 0.190458 I}, {wb -> -0.0023513 + 0.420374 I, wa -> -0.051519 - 0.00405429 I}, {wb -> -0.0023513 + 0.420374 I, wa -> -0.0023513 + 0.420374 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.0793953 + 0.468781 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.699276 - 0.192388 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.774672 + 0.677983 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 1.17814 - 0.187992 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 1.27259 + 0.290135 I}, {wb -> -0.0023513 + 0.420374 I, wa -> -0.051519 - 0.00405429 I}, {wb -> -0.0023513 + 0.420374 I, wa -> -0.0023513 + 0.420374 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.0793953 + 0.468781 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.699276 - 0.192388 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.774672 + 0.677983 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 1.17814 - 0.187992 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 1.27259 + 0.290135 I}, {wb -> -0.0023513 - 0.420374 I, wa -> -0.051519 + 0.00405429 I}, {wb -> -0.0023513 - 0.420374 I, wa -> -0.0023513 - 0.420374 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.0793953 - 0.468781 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.699276 + 0.192388 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.774672 - 0.677983 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 1.17814 + 0.187992 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 1.27259 - 0.290135 I}, {wb -> -0.0023513 - 0.420374 I, wa -> -0.051519 + 0.00405429 I}, {wb -> -0.0023513 - 0.420374 I, wa -> -0.0023513 - 0.420374 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.0793953 - 0.468781 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.699276 + 0.192388 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.774672 - 0.677983 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 1.17814 + 0.187992 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 1.27259 - 0.290135 I}, {wb -> 0.5, wa -> -1.9389}, {wb -> 0.5, wa -> -0.400713 - 0.332275 I}, {wb -> 0.5, wa -> -0.400713 + 0.332275 I}, {wb -> 0.5, wa -> 0.5}, {wb -> 0.5, wa -> 1.40071 - 0.332275 I}, {wb -> 0.5, wa -> 1.40071 + 0.332275 I}, {wb -> 0.5, wa -> 2.9389}, {wb -> 0.5, wa -> -1.9389}, {wb -> 0.5, wa -> -0.400713 - 0.332275 I}, {wb -> 0.5, wa -> -0.400713 + 0.332275 I}, {wb -> 0.5, wa -> 0.5}, {wb -> 0.5, wa -> 1.40071 - 0.332275 I}, {wb -> 0.5, wa -> 1.40071 + 0.332275 I}, {wb -> 0.5, wa -> 2.9389}, {wb -> 1.00235 - 0.420374 I, wa -> -0.272591 - 0.290135 I}, {wb -> 1.00235 - 0.420374 I, wa -> -0.178142 + 0.187992 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.225328 - 0.677983 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.300724 + 0.192388 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.920605 - 0.468781 I}, {wb -> 1.00235 - 0.420374 I, wa -> 1.00235 - 0.420374 I}, {wb -> 1.00235 - 0.420374 I, wa -> 1.05152 + 0.00405429 I}, {wb -> 1.00235 - 0.420374 I, wa -> -0.272591 - 0.290135 I}, {wb -> 1.00235 - 0.420374 I, wa -> -0.178142 + 0.187992 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.225328 - 0.677983 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.300724 + 0.192388 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.920605 - 0.468781 I}, {wb -> 1.00235 - 0.420374 I, wa -> 1.00235 - 0.420374 I}, {wb -> 1.00235 - 0.420374 I, wa -> 1.05152 + 0.00405429 I}, {wb -> 1.00235 + 0.420374 I, wa -> -0.272591 + 0.290135 I}, {wb -> 1.00235 + 0.420374 I, wa -> -0.178142 - 0.187992 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.225328 + 0.677983 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.300724 - 0.192388 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.920605 + 0.468781 I}, {wb -> 1.00235 + 0.420374 I, wa -> 1.00235 + 0.420374 I}, {wb -> 1.00235 + 0.420374 I, wa -> 1.05152 - 0.00405429 I}, {wb -> 1.00235 + 0.420374 I, wa -> -0.272591 + 0.290135 I}, {wb -> 1.00235 + 0.420374 I, wa -> -0.178142 - 0.187992 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.225328 + 0.677983 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.300724 - 0.192388 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.920605 + 0.468781 I}, {wb -> 1.00235 + 0.420374 I, wa -> 1.00235 + 0.420374 I}, {wb -> 1.00235 + 0.420374 I, wa -> 1.05152 - 0.00405429 I}, {wb -> 1.42126, wa -> -0.150039 - 0.190458 I}, {wb -> 1.42126, wa -> -0.150039 + 0.190458 I}, {wb -> 1.42126, wa -> 0.296597 - 0.315787 I}, {wb -> 1.42126, wa -> 0.296597 + 0.315787 I}, {wb -> 1.42126, wa -> 0.634119}, {wb -> 1.42126, wa -> 0.976876}, {wb -> 1.42126, wa -> 1.42126}, {wb -> 1.42126, wa -> -0.150039 - 0.190458 I}, {wb -> 1.42126, wa -> -0.150039 + 0.190458 I}, {wb -> 1.42126, wa -> 0.296597 - 0.315787 I}, {wb -> 1.42126, wa -> 0.296597 + 0.315787 I}, {wb -> 1.42126, wa -> 0.634119}, {wb -> 1.42126, wa -> 0.976876}, {wb -> 1.42126, wa -> 1.42126}}

which does not match your claim. Similarly, for other case we have:

(gb1 = GroebnerBasis[{GS56new - Rationalize[0.327531, 0], GS65new - Rationalize[0.827531, 0]}, {wa, wb}]) // Length

4

NSolve[{gb1[[1]] == 0, gb[[3]] == 0}, {wa, wb}]

{{wb -> -0.421257, wa -> -0.421257}, {wb -> -0.421257, wa -> 0.0231239}, {wb -> -0.421257, wa -> 0.365881}, {wb -> -0.421257, wa -> 0.703403 - 0.315787 I}, {wb -> -0.421257, wa -> 0.703403 + 0.315787 I}, {wb -> -0.421257, wa -> 1.15004 - 0.190458 I}, {wb -> -0.421257, wa -> 1.15004 + 0.190458 I}, {wb -> -0.384258, wa -> -2.00696}, {wb -> -0.384258, wa -> -0.814158}, {wb -> -0.384258, wa -> -0.0321924 - 0.434251 I}, {wb -> -0.384258, wa -> -0.0321924 + 0.434251 I}, {wb -> -0.384258, wa -> 1.07842 - 0.461507 I}, {wb -> -0.384258, wa -> 1.07842 + 0.461507 I}, {wb -> -0.384258, wa -> 1.62828}, {wb -> -0.00294383 - 0.38345 I, wa -> -0.356037 - 0.0105394 I}, {wb -> -0.00294383 - 0.38345 I, wa -> -0.0682849 - 2.03837 I}, {wb -> -0.00294383 - 0.38345 I, wa -> 0.0271765 - 0.415577 I}, {wb -> -0.00294383 - 0.38345 I, wa -> 0.144308 + 0.378458 I}, {wb -> -0.00294383 - 0.38345 I, wa -> 1.03701 - 0.484021 I}, {wb -> -0.00294383 - 0.38345 I, wa -> 1.0963 + 0.378683 I}, {wb -> -0.00294383 - 0.38345 I, wa -> 1.54464 - 0.0621063 I}, {wb -> -0.00294383 + 0.38345 I, wa -> -0.356037 + 0.0105394 I}, {wb -> -0.00294383 + 0.38345 I, wa -> -0.0682849 + 2.03837 I}, {wb -> -0.00294383 + 0.38345 I, wa -> 0.0271765 + 0.415577 I}, {wb -> -0.00294383 + 0.38345 I, wa -> 0.144308 - 0.378458 I}, {wb -> -0.00294383 + 0.38345 I, wa -> 1.03701 + 0.484021 I}, {wb -> -0.00294383 + 0.38345 I, wa -> 1.0963 - 0.378683 I}, {wb -> -0.00294383 + 0.38345 I, wa -> 1.54464 + 0.0621063 I}, {wb -> -0.0023513 - 0.420374 I, wa -> -0.051519 + 0.00405429 I}, {wb -> -0.0023513 - 0.420374 I, wa -> -0.0023513 - 0.420374 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.0793953 - 0.468781 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.699276 + 0.192388 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 0.774672 - 0.677983 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 1.17814 + 0.187992 I}, {wb -> -0.0023513 - 0.420374 I, wa -> 1.27259 - 0.290135 I}, {wb -> -0.0023513 + 0.420374 I, wa -> -0.051519 - 0.00405429 I}, {wb -> -0.0023513 + 0.420374 I, wa -> -0.0023513 + 0.420374 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.0793953 + 0.468781 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.699276 - 0.192388 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 0.774672 + 0.677983 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 1.17814 - 0.187992 I}, {wb -> -0.0023513 + 0.420374 I, wa -> 1.27259 + 0.290135 I}, {wb -> 0.436467, wa -> -3.19878}, {wb -> 0.436467, wa -> -0.376344 - 0.188877 I}, {wb -> 0.436467, wa -> -0.376344 + 0.188877 I}, {wb -> 0.436467, wa -> 0.334027}, {wb -> 0.436467, wa -> 1.36623 - 0.459684 I}, {wb -> 0.436467, wa -> 1.36623 + 0.459684 I}, {wb -> 0.436467, wa -> 2.21998}, {wb -> 0.5, wa -> -1.9389}, {wb -> 0.5, wa -> -0.400713 - 0.332275 I}, {wb -> 0.5, wa -> -0.400713 + 0.332275 I}, {wb -> 0.5, wa -> 0.5}, {wb -> 0.5, wa -> 1.40071 - 0.332275 I}, {wb -> 0.5, wa -> 1.40071 + 0.332275 I}, {wb -> 0.5, wa -> 2.9389}, {wb -> 1.00165 - 0.449612 I, wa -> -0.528678 - 0.0307153 I}, {wb -> 1.00165 - 0.449612 I, wa -> -0.064596 + 0.384441 I}, {wb -> 1.00165 - 0.449612 I, wa -> -0.0492989 - 0.491668 I}, {wb -> 1.00165 - 0.449612 I, wa -> 0.883162 - 2.14683 I}, {wb -> 1.00165 - 0.449612 I, wa -> 0.891719 + 0.347417 I}, {wb -> 1.00165 - 0.449612 I, wa -> 0.986124 - 0.425576 I}, {wb -> 1.00165 - 0.449612 I, wa -> 1.37354 - 0.0410766 I}, {wb -> 1.00165 + 0.449612 I, wa -> -0.528678 + 0.0307153 I}, {wb -> 1.00165 + 0.449612 I, wa -> -0.064596 - 0.384441 I}, {wb -> 1.00165 + 0.449612 I, wa -> -0.0492989 + 0.491668 I}, {wb -> 1.00165 + 0.449612 I, wa -> 0.883162 + 2.14683 I}, {wb -> 1.00165 + 0.449612 I, wa -> 0.891719 - 0.347417 I}, {wb -> 1.00165 + 0.449612 I, wa -> 0.986124 + 0.425576 I}, {wb -> 1.00165 + 0.449612 I, wa -> 1.37354 + 0.0410766 I}, {wb -> 1.00235 - 0.420374 I, wa -> -0.272591 - 0.290135 I}, {wb -> 1.00235 - 0.420374 I, wa -> -0.178142 + 0.187992 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.225328 - 0.677983 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.300724 + 0.192388 I}, {wb -> 1.00235 - 0.420374 I, wa -> 0.920605 - 0.468781 I}, {wb -> 1.00235 - 0.420374 I, wa -> 1.00235 - 0.420374 I}, {wb -> 1.00235 - 0.420374 I, wa -> 1.05152 + 0.00405429 I}, {wb -> 1.00235 + 0.420374 I, wa -> -0.272591 + 0.290135 I}, {wb -> 1.00235 + 0.420374 I, wa -> -0.178142 - 0.187992 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.225328 + 0.677983 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.300724 - 0.192388 I}, {wb -> 1.00235 + 0.420374 I, wa -> 0.920605 + 0.468781 I}, {wb -> 1.00235 + 0.420374 I, wa -> 1.00235 + 0.420374 I}, {wb -> 1.00235 + 0.420374 I, wa -> 1.05152 - 0.00405429 I}, {wb -> 1.42126, wa -> -0.150039 - 0.190458 I}, {wb -> 1.42126, wa -> -0.150039 + 0.190458 I}, {wb -> 1.42126, wa -> 0.296597 - 0.315787 I}, {wb -> 1.42126, wa -> 0.296597 + 0.315787 I}, {wb -> 1.42126, wa -> 0.634119}, {wb -> 1.42126, wa -> 0.976876}, {wb -> 1.42126, wa -> 1.42126}, {wb -> 1.45037, wa -> -0.630092}, {wb -> 1.45037, wa -> -0.0748663 - 0.462553 I}, {wb -> 1.45037, wa -> -0.0748663 + 0.462553 I}, {wb -> 1.45037, wa -> 1.02769 - 0.432315 I}, {wb -> 1.45037, wa -> 1.02769 + 0.432315 I}, {wb -> 1.45037, wa -> 1.71992}, {wb -> 1.45037, wa -> 3.70918}}

Taking other Groebner base elements your will get more/less roots.