TABLE OF CONTENTS In solving the linear system numerically we have to consider the problem s conditioning, algorithm stability, and cost. In using direct methods for solutions of linear systems we discussed efficient elimination schemes, and these schemes are stable when pivoting is employed. But there are some ill-conditioned systems, which are tough to solve by any method. These types of linear systems are identified here.
Here, we will present a parameter, the condition number, which quantitatively measures the conditioning of a linear system. The condition number is greater and equal to one and as a linear system becomes more ill-conditioned, the condition number increases. After factoring a matrix, the condition number can be estimated in roughly the same time as it takes to solve a few factored systems (LU) x= b. Hence, after factoring a matrix, the extra computer time needed to estimate the condition number is usually insignificant.
Any computed solution of a linear system must, because of round-off and other errors, be considered an approximate solution. Here, we shall consider the most natural method for determining the accuracy of a solution of the linear system. One obvious way of estimating the accuracy of the computed solution x* is to compute A x* and to see how close A x* comes to b. Thus, if x* is an approximate solution of the given system A x= b, we compute a vector
which...
