Condition Number Definition/Meaning:
A number that gives a measure of how sensitive the solution of a
problem is to changes in the data. In practice such numbers are
often difficult to compute; even so they can play an important part
in comparing algorithms. They have a particularly important role in
numerical linear algebra. As an example, for the
linear algebraic equations
Ax = b,
if b is changed to b + ∆b
(simulating, for example, errors in the data) then the corresponding
change ∆x in the solution
satisfies
||∆x|| / ||x|| ≤ cond (A) ||∆b|| /
||b||
where cond A = ||A|| ||A-1||
is the condition number of A with respect to solving linear
equations. The expression bounds the relative change in the solution
in terms of the relative change in the data b. The actual quantities
are measured in terms of a vector norm. Similarly the condition
number is expressed in terms of a corresponding matrix norm. It can
be shown that cond(A) ≥ l. If
cond(A) is large the problem is said to be ill-conditioned and it
follows that a small relative change in b can lead to a large
relative change in the solution x; this means that any precision in
the solution is spurious.
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