Interval Arithmetic

Interval arithmetic is a technique used to determine upper bounds for the absolute error in an algorithm, properly considering all roundoff errors in the calculation. It is based on the fact that the real number system modelled by a computer is effectively viewed as an interval with machine representable endpoints in which the exact result lies. All real numbers that enter into a numerical calculation, initial, intermediate, and final, are most often unknown. At best, an interval is known that contains the exact answer. Extending the arithmetic operations used in a numerical algorithm to operate on intervals produces intervals that are guaranteed to contain the exact solution. This type of analysis can be readily implemented in Fortran 90 with its support for derived types and generic functions and operators. It also illustrates several advanced numerical manipulation functions new to Fortran 90 that simplify the implementation and increase portability.

In section 1.4, the following module was used to illustrate generic functions applied to a derived type.

module IntervalArithmetic
   type interval
      real a  ! Left endpoint
      real b  ! Right endpoint
   end type interval

   interface operator (+)
      module procedure addIntervals
   end interface

   contains

   function addIntervals( first, second )
      type( interval ) addIntervals
      type( interval ), intent( in ) :: first, second

      ! Numerically, the left and right endpoints of the interval
      !  sum should be rounded down and up, respectively, to 
      !  ensure that numbers in the two intervals are also in the 
      !  sum.  This has been omitted to simplify the example.

      addIntervals = interval( first%a + second%a, &
                               first%b + second%b )
    end function addIntervals
end module IntervalArithmetic

As pointed out in the comments preceding the calculation of the interval sum, this implementation is simplistic and could give incorrect results if used in a rounding error analysis. An accurate approach is explained below.

Let M be the set of all machine representable reals and let denote interval addition. Then, the interval sum, , must be the smallest interval containing the exact sums, and , with machine representable endpoints, and c₂. More precisely,

,

where and the `+' operator represents exact addition without roundoff or constraints of finite precision.