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3.1.7 Advanced Topic: Interval Arithmetic

Representations of real intervals.

This represents all numbers between and .

In[1]:= Interval[{-2, 5}]

Out[1]=

The square of any number between and is always between 0 and 25.

In[2]:= Interval[{-2, 5}]^2

Out[2]=

Taking the reciprocal gives two distinct intervals.

In[3]:= 1/Interval[{-2, 5}]

Out[3]=

Abs folds the intervals back together again.

In[4]:= Abs[%]

Out[4]=

You can use intervals in many kinds of functions.

In[5]:= Solve[3 x + 2 == Interval[{-2, 5}], x]

Out[5]=

Some functions automatically generate intervals.

In[6]:= Limit[Sin[1/x], x -> 0]

Out[6]=

Operations on intervals.

This finds the overlap of the two intervals.

In[7]:= IntervalIntersection[Interval[{3, 7}], Interval[{-2, 5}]]

Out[7]=

You can use Max and Min to find the end points of intervals.

In[8]:= Max[%]

Out[8]=

This finds out which of a list of intervals contains the point 7.

In[9]:= IntervalMemberQ[
Table[Interval[{i, i+1}], {i, 1, 20, 3}], 7]

Out[9]=

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.

In[10]:= Interval[0.]

Out[10]=

This shows the corresponding interval around 100., shifted back to zero.

In[11]:= Interval[100.] - 100

Out[11]=

The same kind of thing works with numbers of any precision.

In[12]:= Interval[N[Pi, 50]] - Pi

Out[12]=

With ordinary machine-precision arithmetic, this computation gives an incorrect result.

In[13]:= Sin[N[Pi]]

Out[13]=

The interval generated here, however, includes the correct value of 0.

In[14]:= Sin[Interval[N[Pi]]]

Out[14]=