ABSTRACT
Multiplying the left-hand and right-hand sides of Eqs. (l.l75) and (1.179) gives IIblll18xli ~ IIAllllxllllA-1111l8bll (I.I80)
Dividing Eqs. (l.l80) by IIbllllxll yields
118xll < IIAIIIIA-11I118b ll = C(A) 118bll IIxII - IIbll lib II
where C(A) is the condition number of matrix A:
(1.181)
Equation (l.l82) determines the sensitivity of the solution, 118xll/llxll, to changes in the vector b, 118bll/llbll. The sensitivity is determined directly by the value of the condition number C(A). Small values of C(A), of the order of unity, show a small sensitivity of the solution to changes in b. Such a problem is well-conditioned. Large values of C(A) show a large sensitivity of the solution to changes in b. Such a problem is ill-conditioned.
