# Error Bound

This module graphically illustrates a standard bound on the relative
error in the solution vector *x* to a system of linear equations
*A* *x* = *b*, where
*A* is a nonsingular *n* × *n*
matrix and *b* is a known *n*-vector. In particular,
if the right-hand-side vector is perturbed to become
*b* + Δ*b*, then the
relative change in the solution is bounded by
where the condition number
of the matrix is defined by
A similar bound applies for perturbations in the matrix
*A*, but for this module we will consider only
perturbations in the right-hand-side vector *b*. For this
module we will use the Euclidean 2-norm in
two dimensions.

Such an error bound is important because there is usually some
level of uncertainty in the problem data, for example due to
measurement error or rounding error. Because of this uncertainty, the
right-hand-side vector *b* is perhaps better thought of as
a fuzzy region rather than as a single well-defined point, and thus the
corresponding solution vector *x* is also a fuzzy region
rather than a single point. The error bound cited above gives some
idea of the size of the region of uncertainty in which the solution
vector *x* lies for a given level of relative uncertainty
in *b*. We will see that the bound can be tight or loose,
depending on the particular matrix *A* and vector
*b*.

The user first selects a matrix *A* by entering values
for its entries in the text boxes or using a preset example. Next
the user selects a right-hand-side vector *b* by clicking
on the right graph. The vector *b* selected is shown by an
arrow at the selected location in the right graph and also by its
numerical coordinates below. The vector *b* can be dragged
to a new location at any time, if desired. The corresponding solution
vector *x* is shown by an arrow in the left graph and also
by its numerical coordinates below.

The region of uncertainty in the right-hand-side vector
*b* for a given relative error is shown in the right graph
by a shaded circular disk whose size can be altered by dragging its
perimeter, and the resulting numerical value for the relative error in
*b* is shown below. The lightly shaded circular disk in
the left graph shows the corresponding region of uncertainty in the
solution vector *x* given by the condition number of the
matrix, and the corresponding bound on the relative error in
*x* is shown below. In addition, the actual preimage of
the disk on the right is shown by the darkly shaded ellipse in the left
graph, and the corresponding actual worst case relative error in
*x* is shown below. The circular disk on the left always
contains this ellipse, but the bound may be tight or loose, depending
on the particular choice of matrix *A* and right-hand-side
vector *b*. The vectors can be reset by clicking
*Clear*, and the matrix can be changed by entering new values for
its entries.

**Reference:** Michael T. Heath,
*Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 2.3.4.

**Developers:** Sukolsak Sakshuwong and Michael Heath