The condition number of a system of linear equations Ax=b measures the stability to numerical operations. A system of equations is considered to be well-conditioned/ill-conditioned if a slight perturbation in the coefficient matrix A or in the vector b results in a small/great change in the solution vector x.

In mathematical terms, if x + \delta x is the solution to a perturbed problem, then

                (A + \delta A)(x + \delta x) = b + \delta b,

and
                \frac{ \parallel \delta x \parallel}{ \parallel x \parallel}
\leq
\kappa(A)
\left(\frac{ \parallel \delta A \parallel}{ \parallel A \parallel} +
\frac{ \parallel \delta b \parallel}{ \parallel b \parallel} \right) + ...


\kappa(A) is the condition number of a square matrix A, which is defined as:
                \kappa(A) = \parallel A \parallel \parallel A^{-1} \parallel,

where \parallel A \parallel is the norm of matrix A.

Condition numbers near one indicate well-conditioned matrices, whereas condition numbers much greater than one imply ill-conditioned matrices.

The condition estimation routines in LAPACK compute the reciprocal condition number, which is defined as:

                RCOND = 1/\kappa(A).

Computing \parallel A \parallel directly can be expensive. Therefore, it is estimated by methods of low cost. RCOND is zero or a minimal number on the order of the machine epsilon (or machine precision) if A^{-1} does not exist (i.e., A is singular).

Condition number, or its reciprocal RCOND, is an indicator one can examine before computing the solution to a system of linear equations.