fulfils.
3. Condition number of systems of linear equations and matrices. We will write linear equations as (3.1) where A is a non-singular n × n matrix, b ∈
is given vector, and ∈
the exact solution have to be found. Throughout this point will denote a potential solution and
the sequence of iterates. We
will denote the i-th component of a vector by and the i-th component of rarely need to refer to individual components of vectors.
by
. We will
3.1. DEFINITION. Residual vector The vector is called the residual vector for linear equations Most iterative methods terminate when the residual is sufficiently small. One termination criterion is
3.2. DEFINITION. Error vector The difference vector is called the error vector for linear equations solution
Try to estimate relative changes
between approximate solution
of the residual with the relative changes
and exact
of the error!
Since we have =
.
On the other hand Hence
The upper bound between the relative norms
is called the condition number of 14