Contents

# Error Analysis

In practice, most computations are performed with rounding errors. Besides, you often need to solve a system
Ax
=
b
, where the data (the elements of
A
and
b
) are not known exactly. Therefore, it is important to understand how the data errors and rounding errors can affect the solution
x
.
Data perturbations.
If
x
is the exact solution of
Ax
=
b
, and
x
+
δ
x
is the exact solution of a perturbed problem
(
A
+
δ
A
)(
x
+
δ
x
) = (
b
+
δ
b
)
, then this estimate, given up to linear terms of perturbations, holds:
where
A
+
δ
A
is nonsingular and
In other words, relative errors in
A
or
b
may be amplified in the solution vector
x
by a factor
κ
(
A
) = ||
A
|| ||
A
-1
||
called the condition number of
A
.
Rounding errors
have the same effect as relative perturbations
c
(
n
)
ε
in the original data. Here
ε
is the machine precision, defined as the smallest positive number
x
such that 1 +
x
> 1; and
c
(
n
)
is a modest function of the matrix order
n
. The corresponding solution error is
||
δ
x
||/||
x
||
c
(
n
)
κ
(
A
)
ε
. (The value of
c
(
n
)
is seldom greater than 10
n
.)
Machine precision depends on the data type used.
For example, it is usually defined in the
float.h
file as
FLT_EPSILON
the
float
datatype and
DBL_EPSILON
for the
double
datatype.
A
is ill-conditioned (that is, its condition number
κ
(
A
)
is very large), then the error in the solution
x
can also be large; you might even encounter a complete loss of precision. LAPACK provides routines that allow you to estimate
κ
(
A
)
(see Routines for Estimating the Condition Number) and also give you a more precise estimate for the actual solution error (see Refining the Solution and Estimating Its Error).

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804