Strict Estimations

The assertion of proposition 8.2.1 is of a local kind. Namely, this proposition is proved for relatively small deviations. Next we will give the evaluation of the deviation of the solution in general case. First we will prove one necessary auxiliary result.

Proposition 8.3.1. If the matrix is a regular matrix and

then A+E is regular and

Proof. A regular matrix can be represented in the form

where F=-A^-1E. Since then, in virtue of proposition 7.1.1, the matrix I-F is regular and

Hence

and

From the equality

it follows that

and

Proposition 8.3.2. Let

while and If then is a regular matrix, and

and

Proof.Since then, in virtue of proposition 8.3.1, is a regular matrix. Applying proposition 7.1.1 and the equality

we find



In addition, we find that Hence

and that means that the first part of the assertion is true. Since the relation

holds, then

and therefore,

Example 8.3.1.^* Let

We will estimate the relative error of the solution of the system using the Euclidean norm. Since

and the eigenvalues A^TA are and then

Let us find the Euclidean norm of the vector :

If one takes then the conditions and are satisfied. From the singular value decomposition of the matrix A

we find that

Hence

and, in virtue of proposition 8.3.2, we get

Problem 8.3.1.^* Let

Find the relative error of the solution of the system . Use the Euklidean norm.