Roundoff error is bounded in terms of the machine precision
,
which is the smallest value satisfying
where 
and 
are floating-point numbers ,
is any one of the four operations +, 
, 
and 
,
and 
is the floating-point result of 
.
Machine epsilon, , is the smallest value for which this inequality
is true for all , and for all and such that
is neither too large (magnitude exceeds the overflow
threshold)
nor too small
(is nonzero with magnitude less than the underflow threshold)
to be represented accurately in the machine.
We also assume bounds the relative error in unary
operations like square root:
A precise characterization of depends on the details of the
machine arithmetic and sometimes even of the compiler.
For example, if addition and
subtraction are implemented without a guard digit
we must redefine to be the smallest number
such that
In order to assure portability ,
machine parameters such as machine epsilon, the overflow threshold and
underflow threshold are computed at runtime by the auxiliary
routine xLAMCH. The alternative,
keeping a fixed table of machine parameter values, would degrade portability
because the table would have to be changed when moving from one machine,
or even one compiler, to another.
Actually, most machines, but not yet all, do have the same machine
parameters because they implement IEEE Standard Floating Point Arithmetic
[5][4], which exactly specifies floating-point number
representations and operations. For
these machines, including all modern workstations and
PCs,
the values of these parameters are given in
Table 4.1.
Table 4.1: Values of Machine Parameters in IEEE Floating Point Arithmetic
As stated above, we will ignore overflow and underflow in discussing error
bounds. Reference [18] discusses extending error bounds
to include
underflow, and shows that for many common computations, when underflow occurs it
is less significant than roundoff. Overflow generally causes an error
message and stops execution, so the error bounds do not
apply.
Therefore, most of our error bounds will simply be proportional to machine epsilon.
This means, for example, that if the
same problem in solved in double precision and single precision, the error bound
in double precision will be smaller than the error bound in single precision
by a factor
of 
. In IEEE arithmetic, this
ratio is 
,
meaning that one expects the double precision answer to have approximately nine
more decimal digits correct than the single precision answer.
LAPACK routines are generally insensitive to the details of rounding, like their counterparts in LINPACK and EISPACK. One newer algorithm (xLASV2) can return significantly more accurate results if addition and subtraction have a guard digit (see the end of section 4.9).