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Interval arithmetic calculates with closed intervals that contain the possible values of a quantity. If all we know is $x\in X=[a,b]$, an interval enclosure for a function $f$ must contain $f(x)$ for every allowed $x$. The result says what is guaranteed, including uncertainty in the input and, with outward rounding, roundoff error in the calculation. A wide answer can still be correct; a narrow answer is useful only if it contains every possible value.

The attraction is that a finite calculation can make a statement about infinitely many real numbers at once. An interval enclosure can certify that a function stays positive throughout a whole closed interval, or bound its maximum for mathematical optimization. For example, on $X=[0,1]$, the actual image of a function for $f(x)=x(1-x)$ is $[0,1/4]$, but direct interval arithmetic gives the wider $[0,1]$. Interval subdivision and the interval square range improve this answer in different ways.

Historically, Ramon E. Moore's 1962 Stanford dissertation studied interval arithmetic and automatic error analysis in digital computing; his book Interval Analysis followed in 1966. Eldon Hansen's account documents earlier reports as well: these dates describe important steps in developing and disseminating the subject, rather than a claim of sole invention in a particular year.^[ref] Moore, R. Baker Kearfott and Michael J. Cloud later wrote Introduction to Interval Analysis (2009).^[ref]

Interval root exclusion certifies that a function has no zero of a function on $X$ when a valid interval enclosure $F(X)$ excludes zero. For $g(x)=x(1-x)-1/3$ on $[0,1]$, the interval square range yields $G(X)=[-1/3,-1/12]$, which excludes zero. Direct interval arithmetic yields $[-1/3,2/3]$ and is inconclusive, even though the function has no zero of a function there.

Knowing only that a calculated interval enclosure contains zero does not certify a zero of a function: it may contain unattainable values. Even the positive constant function $f(x)=1$ has a valid, loose interval enclosure $[-1,2]$. A separate existence argument, such as reliable opposite endpoint signs together with continuity in the intermediate value theorem, is required; see root bracket and bisection method. Interval subdivision can resolve some inconclusive pieces, but remaining pieces must be reported as unresolved unless an existence argument succeeds. This makes interval root exclusion useful for systematically discarding regions in a search.

// Target: analysis.bigb

The interval square range treats squaring as one function of one input. For $X=[a,b]$ it is exactlyIt retains the relationship that ordinary interval multiplication loses: for $Z=[-1/2,1/2]$, the interval square range is $[0,1/4]$, whereas the independent-input product $ZZ$ is $[-1/4,1/4]$.

Consequently, completing the square for $x(1-x)$ and using the interval square range givesThis is the exact image of a function on $X=[0,1]$. Merely rewriting the expression and evaluating $[1/4,1/4]-ZZ$ gives $[0,1/2]$, so it does not achieve the same sharpening. A mathematical $Z^2$ must be interpreted explicitly here: a square range operation and an independent-input interval product have different meanings. Choosing an operation that preserves known structure can save many steps of interval subdivision.

Interval subdivision covers $X$ by smaller closed intervals $X_j$, computes a valid interval enclosure $F(X_j)$ on each, and takes their union. Every allowed input belongs to a piece, so its output belongs to that union. If one closed interval is wanted, take the smallest one containing the union; gaps between separate output pieces can then be lost. Splitting narrows the freedom given to repeated inputs, while increasing the number of evaluations.

For $f(x)=x(1-x)$, split $[0,1]$ at $1/2$. The natural interval extensions areTheir union is $[0,1/2]$. Four equal pieces sharpen the upper bound again:The union is now $[0,3/8]=[0,0.375]$, still wider than the actual image of a function $[0,1/4]$. These are exact rational number bounds: the excess width comes from the interval dependency problem, rather than roundoff error.