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Update (2011)

Introduction

SEA is an experimental software package which allows easy modification of Fortran 90/95 programs to compute error bounds for all results. The main aim of SEA is however to explore new ideas and techniques in this practical field.

SEA has developed from the old "err.f90" module set. The old modules were yet another simple minded implementation of Interval Arithmetic (IA). Since IA is not a satisfactory practical solution SEA looks now in other directions.

IA deals with bounds on results but we would like to deal with measures of dispersion. It doesn't mean we don't suffer still from the same diseases as IA: correlated terms, coffins and domain splitting at singularities.

What we do

The Fortran 95 modules included do the following:

• Declare new data types, e.g. a number with an associated error bound.

  Example: "|1.0,0.1|", is the number "1.0" with error bound "0.1". This symbol represents the range of real numbers between "0.9" and "1.1".

• Perform an "overloading" of all operators and intrinsic functions, so that ordinary Fortran 90 code will perform the required error analytical operations, instead of the ordinary operations.

  Example: the code line "z = x / y", given "x" and "y" with error bounds, will now automatically perform error analysis and find the error bounds of the result.

  The results of such a computation may look like:

   x   =  |1.00000E+000,1.00000E-001|
   y   =  |1.00000E+000,1.00000E-001|
   x/y =  |1.02020E+000,2.02020E-001|
  

• Provide convenient input/output facilities. The IO library does its own input buffering and scanning in order to implement non-advancing IO.
• Uses interesting new algorithms, e.g. to compute minima and maxima of sin() inside a given interval.
• Provides a testbed for new ideas, for example:
  • Precision loss warnings (ok, this is an old idea but our implementation is new).
  • Rounding error decreasing by representing numbers as a sum of two floating points, the first the result of the last operation and the second its "residue". This is a generalization of compensated summation. If possible we transparently compute intermediaries in twice the precision.
  • Workarounds the correlated terms problem, e.g. direct computation of low order polynomials.
  • 3-value arithmetic with BSA. In development.
  • Analyzing input to find the error range (the decimal spacing).
  See the TODO file for an updated list of ideas tested.

Useful links

William Kahan

How Futile are Mindless Assessments of Roundoff in Floating-Point Computation ? (PDF)

  See section 13 on page 49/56 on the bloated coffins of interval arithmetics

Ferson, Kreinovich, Hajagos, Oberkampf and Ginzburg

Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty (PDF)

Wikipedia

Propagation of uncertainty (HTML)

Wikipedia

List of uncertainty propagation software (HTML)

Wikipedia

Taylor expansions for the moments of functions of random variables (HTML)

Wikipedia

Automatic differentiation (HTML)