The second input is optional, and indicates the alternative ways to provide output either using an exact rational interval QQi, a real interval RRi, or by taking a rational or real approximation of the midpoint of the intervals.
i1 : R = QQ[x,y]
o1 = R
o1 : PolynomialRing
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i2 : I = ideal {(x-1)*x, y^2-5}
2 2
o2 = ideal (x - x, y - 5)
o2 : Ideal of R
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i3 : rationalIntervalSols = msolveRealSolutions I
1 1 4801919417 9603838835 4294967295
o3 = {{{- ----------, ----------}, {----------, ----------}}, {{----------,
4294967296 4294967296 2147483648 4294967296 4294967296
------------------------------------------------------------------------
4294967297 4801919417 9603838835 1 1
----------}, {----------, ----------}}, {{- ----------, ----------}, {-
4294967296 2147483648 4294967296 4294967296 4294967296
------------------------------------------------------------------------
9603838835 4801919417 4294967295 4294967297 9603838835
----------, - ----------}}, {{----------, ----------}, {- ----------, -
4294967296 2147483648 4294967296 4294967296 4294967296
------------------------------------------------------------------------
4801919417
----------}}}
2147483648
o3 : List
|
i4 : rationalApproxSols = msolveRealSolutions(I, QQ)
19207677669 19207677669 19207677669
o4 = {{0, - -----------}, {1, - -----------}, {0, -----------}, {1,
8589934592 8589934592 8589934592
------------------------------------------------------------------------
19207677669
-----------}}
8589934592
o4 : List
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i5 : floatIntervalSols = msolveRealSolutions(I, RRi)
o5 = {{[1,1], [-2.23607,-2.23607]}, {[-2.32831e-10,2.32831e-10],
------------------------------------------------------------------------
[-2.23607,-2.23607]}, {[1,1], [2.23607,2.23607]},
------------------------------------------------------------------------
{[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}}
o5 : List
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i6 : floatIntervalSols = msolveRealSolutions(I, RRi_10)
o6 = {{[.999023,1.00098], [-2.23633,-2.23535]}, {[-.000976562,.000976562],
------------------------------------------------------------------------
[-2.23633,-2.23535]}, {[.999023,1.00098], [2.23535,2.23633]},
------------------------------------------------------------------------
{[-.000976562,.000976562], [2.23535,2.23633]}}
o6 : List
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i7 : floatApproxSols = msolveRealSolutions(I, RR)
o7 = {{1, -2.23607}, {0, -2.23607}, {1, 2.23607}, {0, 2.23607}}
o7 : List
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i8 : floatApproxSols = msolveRealSolutions(I, RR_10)
o8 = {{1, 2.23584}, {0, 2.23584}, {1, -2.23584}, {0, -2.23584}}
o8 : List
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i9 : I = ideal {(x-1)*x^3, (y^2-5)^2}
4 3 4 2
o9 = ideal (x - x , y - 10y + 25)
o9 : Ideal of R
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i10 : floatApproxSols = msolveRealSolutions(I, RRi)
o10 = {{[-2.32831e-10,2.32831e-10], [2.23607,2.23607]}, {[1,1],
-----------------------------------------------------------------------
[2.23607,2.23607]}, {[-2.32831e-10,2.32831e-10], [-2.23607,-2.23607]},
-----------------------------------------------------------------------
{[1,1], [-2.23607,-2.23607]}}
o10 : List
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