Consider the nonlinear matrix equation
In this paper we consider the Hermitian positive definite solution of the nonlinear matrix equation
When
For the similar equations
As a continuation of the previous results, the rest of the paper is organized as follows. In Section
We denote by
If
For any Hermitian positive definite matrix
In addition, if
The matrix equation
Here the perturbed equation
By Lemma
To prove the next theorem, we first verify the following lemma.
If
By Lemma
The next theorem generalizes [
Let
Let
For every
According to
Next, a sharper perturbation estimate is derived.
Subtracting (
If
It suffices to show that the following equation:
Furthermore, we define operators
Suppose that
Let
From Theorem
Combining this with (
In this section, a backward error of an approximate solution for the unique solution to (
Let
Let
According to (
In this section, we apply the theory of condition number developed by Rice [
Suppose that
By the theory of condition number developed by Rice [
Substituting (
Let
Then we have the following theorem.
If
From (
In this subsection we consider the real case. That is, all the coefficient matrices
Let
In the real case the relative condition number is given by
To illustrate the results of the previous sections, in this section three simple examples are given, which were carried out using MATLAB 7.1. For the stopping criterion we take
We consider the matrix equation
Suppose that the coefficient matrices
We now consider the corresponding perturbation bounds for the solution
The assumptions in Theorem
The assumptions in Theorem
Assumptions check for Example
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The results listed in Table
By Theorems
Perturbation bounds for Example
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The results listed in Table
We consider the matrix equation
The residual
By Theorem
Backward error bound for Example
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The results listed in Table
We study the matrix equation
Table
Relative condition number for Example
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1.0717 | 1.0228 | 1.0225 | 1.0225 | 1.0225 |
The author wishes to express her gratitude to the referees for their fruitful comments. The work was supported in part by National Nature Science Foundation of China (11201263), Natural Science Foundation of Shandong Province (ZR2012AQ004), and Independent Innovation Foundation of Shandong University (IIFSDU), China.