Improvement of Solution Problems of Matrix Equations Calculated by Computers

: Solution of linear algebra systems may come out with “ill-condition” or “well-condition” based on input information and solution methods. The aim of this study is to determine and correction of problems that may come out from the solution of matrix equations by computers and to calculate . A x f  linear algebra.


Introduction
If we solve . A x f  equation system computer, we have to know about numbers subset in the numbering system that called "FORMAT". It is

A. Error in Placing The Real Number into Format
Let's take real numbers set of R and set F define an operator  A . We saw before that is unsuccessful. In addition, let us give another example that is shown det A is unsuccessful.
Let we take following equations system: That is, the computer assume that 1/ 2 N is zero. According to that, Thus, the absence of a well-condition problem's solution is given as a result. We can say that at problem solving with computers some methods given in classical mathematics, like Cramer rule, cause to obtain unsuccessful results.

Condition Number
In computer-aided mathematical education, if it was started to solve a problem by using some methods that no having robust basics, it is clear that we will have important problems. This has been appearing in the last 50 years. Some studies have been done about this subject by Neuman [4] and Turing [3]. After that, for example, studies given by [1] [2] and for the defined problem at above the condition number is The condition number has two important attributes. We will see these at following theorems.
The proof of these theorems could be found at [1]. If we solve the Ax f  problem, we don't need the more accreted system in practice. Because, we can't explain about the computed solution is near the real solution. In this situation, the solution is to use a parameter that accepted the upper limit of practical invertible matrices [5,6]. By using this parameter, we can express the Theorem 1 in other way. . In such problems, it can be thought that, how far or how near the problem is to the bad-defined region. How much we can change the elements of A , so it is still in a well-defined region.
For the problem . Such problems whose solutions exist, are unique and stable in terms of their elements are called well-conditioned (correct). The set of matrices can be divided into two parts which are the subset of well-conditioned ones and the subset of ill-conditioned ones by the condition number () A  , we can illustrate this situation by a Fig.1: There are many studies concerning with the distance of a matrix to the region for well-conditioned matrices.
In most of these studies indefiniteness principle arises. Namely, when we get a matrix A from the set   , , is the given matrix, ( )  It should be defined that practical well-conditioned matrices so that we can obtain the distance of a matrix to the set of the illconditioned matrices. .

A x f
 is a well-conditioned problem practically. Otherwise it is a ill-conditioned one. Here the number *  is a boundary between practically well-conditioned problems and practically ill-conditioned problems. We can illustrate this situation by a Fig.4.

Conclution
So, the most important advantage of this approach is to warn the users about the problem (system) whether it is a well-conditioned or ill-conditioned problem by computing the exact value of () A  by given formula; then if necessary, they change the data and input values which make the problem ill-conditioned or improve the approach partially or fully so that they have a chance to prevent themselves from wasting their time and work.