HP-UX Floating-Point Guide

94 Chapter 3

Factors that Affect the Results of Floating-Point Computations

Floating-Point Coding Practices that Affect Application Results

Ill-Conditioned Computations

If relatively small changes to the input of a program or to the

intermediate results generated by a program cause relatively large

changes in the ﬁnal output, the program is said to be ill-conditioned or

numerically unstable.

The following example illustrates an ill-conditioned program:

Sample Program: sloppy_tangent.f

C The following program is an example of how small

C perturbations in the argument to a function near the

C function’s singularity can cause large variations in the

C function’s result. A program may be ill-conditioned because

C of a case like this, where minor inaccuracies created during

C preliminary calculations become major inaccuracies when they

C are passed through a function at a point where the function

C has a very steep slope.

PROGRAM SLOPPY_TANGENT

DOUBLE PRECISION X

X = 1.570796D0

WRITE(*,*) 'TAN( X-1.0D-5 ):', TAN( X-1.0D-5 )

WRITE(*,*) 'TAN( X ): ', TAN( X )

WRITE(*,*) 'TAN( X+1.0D-5 ):', TAN( X+1.0D-5 )

END

The output from this program shows how small changes in the argument

to the TAN function lead to wildly varying results:

TAN( X-1.0D-5 ): 96835.46637430933

TAN( X ): 3060023.306952844

TAN( X+1.0D-5 ): -103378.351773411

Ill-conditioned computations cause trouble not because their input

values may change, but because the seemingly innocuous rounding

errors and loss of signiﬁcance can have large effects on the ﬁnal results.

One way to establish that rounding errors and loss of signiﬁcance are

causing a program to produce incorrect results is to run the program in

various rounding modes. (See “Rounding Mode: fegetround and

fesetround” on page 130 for information on how to change rounding

modes using the fesetround function.) However, this technique does

not always work. In the preceding example, for instance, changing the

rounding mode has little effect on the results. However, if you do observe

that simply changing the rounding mode causes large changes in the

application results, then your application is most likely ill-conditioned.