Interval Analysis - Computer Graphics and Geometric Modeling - page 729

() = [

] H A 00

11
01

,, if d < a or b < c ,

= [] ,, if a === b c d ,

= [] ,, otherwise .

Note that in the special case h( x ) = (f( x ) = 0), where we are simply testing for zeros
of f, then

() = [] Œ [ ] [ ] π [ ] HA 01 ,, if and only if 0 ab but ab , , 0 0 , .

We can also define Boolean operators between relational expressions. For
example, if

n

Æ {}

rr ,: R 01 ,

12

are relational expressions with inclusion functions R 1 and R 2 , respectively, then an
inclusion function B for the logical and operator

n

Æ {} () = () ()

b : R 01 , , b x r x and r x ,

1 2

is

() = [

]

( () = [

]

) ( ( ) = [

]

) BA 00 ,, if R A 00 , or

and

R A 00 , ,

1 2

= [] ( () = [] ) ( ( ) = [] ) 11 ,, if R A 11 , R A 11 , ,

1 2

= [] 01 ,, otherwise ,

for all A ΠI( R n ).
If one is going to use interval analysis, then all relevant theorems and their proofs
need to be reformulated in that context. As an example of this, we consider one impor-
tant theorem from calculus.

18.3.1 Theorem. (The Mean Value Theorem in Interval Analysis) Let f : R m Æ R n
be a differentiable function. Let H ij be an inclusion function for ∂f i /∂x j and H the
inclusion function for the Jacobian matrix defined by

= ( .

HH ij

Then there exists an inclusion function F for f satisfying

() = (

[ ]

) + () - ( ) FA F xx , HA A x ,

for all A ΠI( R m ) and x ΠA.

Proof. The theorem follows from basic facts about the Taylor expansion for f and
its error bounds.