3.1.7 Advanced Topic: Interval Arithmetic
| Interval[{min, max}] | the interval from min to max |
Interval[{ ,  }, { ,  }, ... ]
| | the union of intervals from  to  ,  to  , ... |
Representations of real intervals. This represents all numbers between  and  . | |
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Interval[{-2, 5}]
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The square of any number between  and  is always between 0 and 25. | |
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Interval[{-2, 5}]^2
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| Taking the reciprocal gives two distinct intervals. | |
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1/Interval[{-2, 5}]
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| Abs folds the intervals back together again. | |
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| You can use intervals in many kinds of functions. | |
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Solve[3 x + 2  Interval[{-2, 5}], x]
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| Some functions automatically generate intervals. | |
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Limit[Sin[1/x], x -> 0]
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IntervalUnion[ ,  , ... ]
| | find the union of several intervals |
IntervalIntersection[ ,  , ... ]
| | find the intersection of several intervals | | IntervalMemberQ[interval, x] | test whether the point x lies within an interval | IntervalMemberQ[ ,  ] | test whether  lies completely within  |
Operations on intervals. | This finds the overlap of the two intervals. | |
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IntervalIntersection[Interval[{3, 7}], Interval[{-2, 5}]]
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| You can use Max and Min to find the end points of intervals. | |
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| This finds out which of a list of intervals contains the point 7. | |
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IntervalMemberQ[
Table[Interval[{i, i+1}], {i, 1, 20, 3}], 7]
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You can use intervals not only with exact quantities but also with approximate numbers. Even with machine-precision numbers, Mathematica always tries to do rounding in such a way as to preserve the validity of results. | This shows explicitly the interval treated by Mathematica as the machine-precision number 0. | |
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| This shows the corresponding interval around 100., shifted back to zero. | |
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Interval[100.] - 100
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| The same kind of thing works with numbers of any precision. | |
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Interval[N[Pi, 50]] - Pi
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| With ordinary machine-precision arithmetic, this computation gives an incorrect result. | |
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| The interval generated here, however, includes the correct value of 0. | |
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Sin[Interval[N[Pi]]]
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