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We explore the condition numbers of the nonlinear matrix equation

We consider the nonlinear matrix equation

To the best of our knowledge, no one has studied the condition numbers of (

The following notations will be used throughout this paper:

In this section, we provide useful definitions and lemmas that will be applied in our proofs in the next sections.

The condition number of a matrix function

The Fréchet derivative of a matrix function

Suppose that

Let

Let

For easy expansion and simplification of matrix polynomials, we need Lemma

Let

In this section, we concentrate on the derivation of the explicit expressions for the normwise, mixed, and componentwise condition numbers. In order to derive the explicit expressions for the normwise, mixed, and componentwise condition numbers of (

Let us define a map

In this subsection, we define the two kinds of normwise condition numbers. According to Rice [

Now, we prove that matrix

Suppose that

Denote

Since

From Theorem

Explicit expressions for the two kinds of normwise condition numbers are derived in Theorem

Suppose that

where

Using the fact that

In this case, we rewrite

In this subsection, we derive the explicit expressions for mixed and componentwise condition numbers of (

Based on the work by Gohberg and Koltracht [

Let

The mixed condition number of

The componentwise condition number of

Assume

if

if

Let

Let

where

Let

Moreover, we define two simple upper bounds for

We first prove (I) using

Likewise, to estimate the upper bound of

In this section, we provide some numerical examples and results. Our tests were carried out in MATLAB mark 22.0 on an Intel(R) Core(TM)i3-4005u CPU@1.7GHz 1.70GHz with 64-bit operating system. Four examples are considered. In Example

In each example a comparison table for the computed condition numbers is provided and a general remark is provided for all results.

For

Here, we propose a fixed point algorithm to compute the solutions

Input an

For

Exit the loop if res

Display the solution

We consider the tridiagonal matrix

The matrix

Condition numbers for Example

| | | | | |
---|---|---|---|---|---|

4 | 0.1991 | 0.5114 | 0.5066 | 11.4838 | 1.0071 |

6 | 0.2096 | 0.5124 | 0.5066 | 23.1583 | 1.0079 |

8 | 0.2140 | 0.5128 | 0.5066 | 52.7581 | 1.0082 |

10 | 0.2162 | 0.5130 | 0.5066 | 134.5177 | 1.0084 |

We consider (

We consider (

Normwise and mixed condition numbers for different

| | | |
---|---|---|---|

2 | 34.6169 | 22.7843 | 1.6223 |

4 | 6.5804 | 4.4013 | 0.8919 |

6 | 3.5024 | 2.3663 | 0.6400 |

9 | 2.0393 | 1.3890 | 0.4498 |

Normwise and mixed condition numbers evaluated at different doubly stochastic input matrix

| | | | |
---|---|---|---|---|

(3,3) | 8.7952 | 5.2523 | 3.4629 | 12.6425 |

(4,4) | 3.4727 | 2.0392 | 1.4709 | 9.7038 |

(5,5) | 2.2008 | 1.2491 | 0.8735 | 8.1702 |

Equation (

From Table

In this example, we use the same matrix

The proposed fixed point algorithm is used to obtain the solutions of (

Condition numbers and relative upper perturbation bounds for Example

| 6 | 8 | 10 | 12 |
---|---|---|---|---|

| 1.2816 | 1.3364 | 1.2573 | 1.2935 |

| 3.0635 | 2.8668 | 2.9382 | 2.7542 |

| 5.6886 | 5.0152 | 5.7033 | 5.9011 |

| 5.6991 | 5.0237 | 5.7129 | 5.9093 |

| 4.8743 | 5.6170 | 4.8345 | 5.2817 |

| 5.5744 | 5.9432 | 5.2362 | 5.7641 |

| 5.5987 | 5.9691 | 5.2590 | 5.7893 |

| 1.0861 | 1.1579 | 1.0202 | 1.1230 |

In this example, we consider (

Condition numbers and relative upper perturbation bounds for Example

| 6 | 8 | 10 | 12 |
---|---|---|---|---|

| 1.6657 | 1.6657 | 1.6656 | 1.6601 |

| 1.8385 | 1.8385 | 1.8385 | 1.8385 |

| 2.9229 | 2.9229 | 2.9228 | 2.9087 |

| 3.0312 | 3.0312 | 3.0310 | 3.0164 |

| 1.8081 | 1.8081 | 1.8083 | 1.8081 |

| 1.9060 | 1.9060 | 1.9060 | 1.9060 |

| 1.9376 | 1.9376 | 1.9376 | 1.9376 |

| 3.3083 | 3.3083 | 3.3083 | 3.3083 |

In Table

In Tables

In Tables

In this paper, we studied the normwise, mixed, and componentwise condition numbers of (

Due to the nature of our subject, all the necessary steps are included in the submitted manuscript. However, if more details will be required we will provide them immediately.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to thank Professor Hyun-Min Kim for his remarkable guidance during this research work. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2017R1A5A1015722).