Sensitivity Analysis of the Matrix Equation from Interpolation Problems

This paper studies the sensitivity analysis of a nonlinear matrix equation connected to interpolation problems.The backward error estimates of an approximate solution to the equation are derived. A residual bound of an approximate solution to the equation is obtained. A perturbation bound for the unique solution to the equation is evaluated. This perturbation bound is independent of the exact solution of this equation. The theoretical results are illustrated by numerical examples.

We would like to know how the errors of Ã and Q influence the error in X. Motivated by this, we consider in this paper the sensitivity analysis of (1).
For the equation  −  *  −1  =  and related equations −∑  =1  *       =  (0 < |  | < 1) and −∑  =1  *      =  (−1 ≤  < 0 or 0 <  < 1) there were many contributions in the literature to the solvability and numerical solutions [8][9][10][11][12][13][14].However, these papers have not examined the sensitivity analysis about the above equations.Hasanov et al. [12,15] obtained two perturbation estimates of the solutions to the equations  ±  *  −1  = .Li and Zhang [13] derived two perturbation bounds of the unique solution to the equation  −  *  −1  = .They also obtained the explicit expression of the condition number for the unique positive definite solution.The perturbation analysis about the related equations  +  *  −1  = ,  −  *  −  = , and   ±    −  =   were mentioned in papers [16][17][18][19].Yin and Fang [20] obtained an explicit expression of the condition number for the unique positive definite solution of (1).They also gave two perturbation bounds for the unique positive definite solution, whereas, to our best knowledge, there have been no backward error estimates and any computable residual bound for (1) in the known literature.In this paper, we obtain the backward error estimates and a residual bound of the approximate solution to (1) as well as evaluate a new relative perturbation bound for (1).This bound does not need any knowledge of the exact solution of (1), which is important in many practical calculations.
As a continuation of the previous results, this paper gives some preliminary knowledge that will be needed to develop this work in Section 2. In Section 3, the backward error estimates of an approximate solution for the unique solution to (1) are discussed.In Section 4, we derive a residual bound of an approximate solution for the unique solution to (1).In Section 5, we give a new perturbation bound for the unique solution to (1), which is independent of the exact solution of (1).Finally, several numerical examples are presented in Section 6.

Backward Error
In this section, applying the technique developed in [18], we obtain some estimates for the backward error of the approximate solution of (1).Let X ∈ H × be an approximation to the unique solution  to (1), and let Δ  ∈ C × ( = 1, 2, . . ., ) and Δ ∈ H × be the corresponding perturbations of the coefficient matrices   ( = 1, 2, . . ., ) and  in (1) Note that It follows from ( 6) that Let where Π is the vec-permutation.Then (8) can be written as It follows from  > 0 that 2 2 ×2(+1) 2 matrix  is full row rank.Hence,  † =  2 2 , which implies that every solution to the equation must be a solution to (10).Consequently, for any solution  to (11) we have Then we can state the estimates of the backward error as follows.
then one has that where Proof.Let Obviously,  : Condition (13) ensures that the quadratic equation in  has two positive real roots.The smaller one is Define Ω = { ∈ C 2(+1) The last equality is due to the fact that () is a solution to the quadratic equation ( 18).Thus we have proved that (Ω) ⊂ Ω.
By the Schauder fixed-point theorem, there exists a  * ∈ Ω such that ( * ) =  * , which means that  * is a solution to (11), and hence it follows from (12) that Next we derive a lower bound for ( X).Suppose that (Δ Then we have where Let a singular value decomposition of  be  = (, 0) * , where  and  are unitary matrices and Substituting this decomposition into (23), and letting we get Let Since () is a solution to (18), we have that which implies that Then ( X) ≥ ().

Residual Bound
Residual bound reveals the stability of a numerical method.In this section, in order to derive the residual bound of an approximate solution for the unique solution to (1), we first introduce the following lemma.(32) Proof.According to According to (35), we obtain for Δ ∈ Ψ.That is, (Ψ) ⊆ Ψ.By Brouwer fixed point theorem, there exists a Δ ∈ Ψ such that (Δ) = Δ.Hence X + Δ is a solution of (1).Moreover, by Lemma 1, we know that the solution  of ( 1) is unique.Then

Perturbation Bounds
In this section we develop a relative perturbation bound for the unique solution of (1), which does not need any knowledge of the actual solution  of (1) and is easy to calculate.
The results listed in Table 1 show that the backward error of X decreases as the error ‖ X − ‖  decreases.Example 2. This example considers the residual bound of an approximate solution for the unique solution  to (1) in Theorem 5. We consider with Choose X0 = .Let the approximate solution X of  be given with the iterative method   =  + ∑  =1  *   −1 −1   ,  0 > 0,  = 1, 2, . .., where  is the iteration number.
Some results are listed in Table 2.
The results listed in Table 2 show that the residual bound given by Theorem 5 is fairly sharp.
Example 3. In this example, we consider the corresponding perturbation bound for the solution  in Theorem 6.
We consider the matrix equation and  is a random matrix generated by MATLAB function randn.
By Theorem 6, we can compute the relative perturbation bound  1 .The results averaged as the geometric mean of 20 randomly perturbed runs.Some results are listed in Table 3.
The results listed in Table 3 show that the perturbation bound  1 given by Theorem 6 is fairly sharp.

Concluding Remarks
In this paper, we consider the sensitivity analysis of the nonlinear matrix equation −∑  =1  *   −1   = .Compared with existing literature, the contributions of this paper are as follows.
(i) A backward error and a computable residual bound of an approximate solution for the unique solution to (1) are derived, which do not appear in other known literature works.(ii) Some results in this paper can cover the work of Li and Zhang [13] for the matrix equation  −  *  −1  =  as a special case.(iii) This paper develops a new relative perturbation bound for the solution to (1), which does not need any knowledge of the actual solution  of (1) and could be computed easily.

Table 1 :
Backward error for Example 1 with different values of .

Table 2 :
Residual bounds for Example 2 with different values of .

Table 3 :
Perturbation bounds for Example 3 with different values of .