10.2.4.2 Numerical Accuracy and Reduction Operations

Floating-point sum or product reduction operations may be inaccurate due to the following conditions:

• The order in which the calculations are performed in parallel is not the same as when performed serially on a single processor.

• The order of calculation affects the sum or product of floating-point numbers. Hardware floating-point addition and multiplication are not associative. Roundoff, overflow, or underflow errors may result depending on how the operands associate. For example, (X*Y)*Z and X*(Y*Z) may not have the same numerical significance.

In some situations, the error may not be acceptable.

Example: Roundoff, get the sum of 100,000 random numbers between– 1 and +1:

demo% cat t4.f
      parameter ( n = 100000 )
      double precision d_lcrans, lb / -1.0 /, s, ub / +1.0 /, v(n)
      s = d_lcrans ( v, n, lb, ub ) ! Get n random nos. between -1 and +1
      s = 0.0
      do i = 1, n
        s = s + v(i)
      end do
      write(*, ’(" s = ", e21.15)’) s
      end
demo% f95 -O4 -autopar -reduction t4.f

Results vary with the number of processors. The following table shows the sum of 100,000 random numbers between– 1 and +1.

In this situation, roundoff error on the order of 10^-14 is acceptable for data that is random to begin with. For more information, see the Sun Numerical Computation Guide.