# Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method

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## Abstract

**:**

## 1. Introduction

**Algorithm IN**

**[6]**

- Step 1: Choose initial guess ${x}_{0}$, tolerance value $tol$; Set $k=0$
- Step 2: While $F\left({x}_{k}\right)>tol\times F\left({x}_{0}\right)$, Do
- Choose ${\eta}_{k}\in [0,1)$. Find ${d}_{k}$ so that $\parallel F\left({x}_{k}\right)+{F}^{\prime}\left({x}_{k}\right){d}_{k}\parallel \le {\eta}_{k}\parallel F\left({x}_{k}\right)\parallel $.
- Set ${x}_{k+1}={x}_{k}+{d}_{k}$; $k=k+1$

**Algorithm HSS**

**[4]**

- Step 1: Choose initial guess ${x}_{0}$, tolerance value $tol$ and $\alpha >0$; Set $l=0$
- Step 2: Set $H=\frac{1}{2}(A+{A}^{*})\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}S=\frac{1}{2}(A-{A}^{*})$, where H is Hermitian and S is skew-Hermitian parts of A.
- Step 3: While $\parallel b-A{x}_{\mathsf{\u0142}}\parallel >tol\times \parallel b-A{x}_{0}\parallel $, Do
- Solve $(\alpha I+H){x}_{l+1/2}=(\alpha I-S){x}_{l}+b$
- Solve $(\alpha I+S){x}_{l}=(\alpha I-H){x}_{\mathsf{\u0142}+1/2}+b$
- Set $l=l+1$

**Algorithm NHSS**

**(The Newton–HSS method [5])**

- Step 1: Choose initial guess ${x}_{0}$, positive constants $\alpha $ and $tol$; Set $k=0$
- Step 2: While $\parallel F\left({x}_{k}\right)\parallel >tol\times \parallel F\left({x}_{0}\right)\parallel $
- -
- Compute Jacobian ${J}_{k}={F}^{\prime}\left({x}_{k}\right)$
- -
- Set$${H}_{k}\left({x}_{k}\right)=\frac{1}{2}({J}_{k}+{J}_{k}^{*})\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}_{k}\left({x}_{k}\right)=\frac{1}{2}({J}_{k}-{J}_{k}^{*}),$$
- -
- Set ${d}_{k,0}=0;\phantom{\rule{3.33333pt}{0ex}}l=0$
- -
- While$$\parallel F\left({x}_{k}\right)+{J}_{k}{d}_{k,\mathsf{\u0142}}\parallel \ge {\eta}_{k}\times \parallel F\left({x}_{k}\right)\parallel \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}({\eta}_{k}\in [0,1))$$Do{
- Solve sequentially:$$\begin{array}{c}\hfill (\alpha I+{H}_{k}){d}_{k,l+1/2}=(\alpha I-{S}_{k}){d}_{k,l}+b\end{array}$$$$\begin{array}{c}\hfill (\alpha I+{S}_{k}){d}_{k,l}=(\alpha I-{H}_{k}){d}_{k,l+1/2}+b\end{array}$$
- Set $l=l+1$

} - -
- Set$${x}_{k+1}={x}_{k}+{d}_{k,l};\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}k=k+1$$
- -
- Compute ${J}_{k}$, ${H}_{k}$ and ${S}_{k}$ for new ${x}_{k}$

- (i)
- The semi-local sufficient convergence criterion provided in (15) of [7] is false. The details are given in Remark 1. Accordingly, Theorem 3.2 in [7] as well as all the followings results based on (15) in [7] are inaccurate. Further, the upper bound function ${g}_{3}$ (to be defined later) on the norm of the initial point is not the best that can be used under the conditions given in [7].
- (ii)
- The convergence domain of NHSS is small in general, even if we use the corrected sufficient convergence criterion (12). That is why, using our technique of recurrent functions, we present a new semi-local convergence criterion for NHSS, which improves the corrected convergence criterion (12) (see also Section 3 and Section 4, Example 4.4).
- (iii)

## 2. Semi-Local Convergence Analysis

- $\mathrm{(\mathcal{A}}$
_{1}) - There exist positive constants $\beta ,\gamma $ and $\delta $ such that$$max\{\parallel H\left({x}_{0}\right)\parallel ,\parallel S\left({x}_{0}\right)\parallel \}\le \beta ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}\parallel \le \gamma ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\parallel F\left({x}_{0}\right)\parallel \le \delta ,$$
- $\mathrm{(\mathcal{A}}$
_{2}) - There exist nonnegative constants ${L}_{h}$ and ${L}_{s}$ such that for all $x,y\in U({x}_{0},r)\subset {\mathsf{\Omega}}_{0},$$$\begin{array}{ccc}\hfill \parallel H\left(x\right)-H\left(y\right)\parallel & \le & {L}_{h}\parallel x-y\parallel \hfill \\ \hfill \parallel S\left(x\right)-S\left(y\right)\parallel & \le & {L}_{s}\parallel x-y\parallel .\hfill \end{array}$$

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Semi-Local Convergence Analysis II

**Lemma**

**1.**

**Proof.**

**Lemma**

**2**

- (i)
- $\parallel {F}^{\prime}\left(x\right)-{F}^{\prime}\left(y\right)\parallel \le L\parallel x-y\parallel $
- (ii)
- $\parallel {F}^{\prime}\left(x\right)\parallel \le L\parallel x-y\parallel +2\beta $
- (iii)
- If $r<\frac{1}{\gamma L},$ then ${F}^{\prime}\left(x\right)$ is nonsingular and satisfies$$\parallel {F}^{\prime}{\left(x\right)}^{-1}\parallel \le \frac{\gamma}{1-\gamma L\parallel x-{x}_{0}\parallel},$$

**Lemma**

**3.**

- (i)
- $\parallel {F}^{\prime}\left(x\right)-{F}^{\prime}\left(y\right)\parallel \le L\parallel x-y\parallel $
- (ii)
- $\parallel {F}^{\prime}\left(x\right)\parallel \le {L}_{0}\parallel x-y\parallel +2\beta $
- (iii)
- If $r<\frac{1}{\gamma {L}_{0}},$ then ${F}^{\prime}\left(x\right)$ is nonsingular and satisfies$$\parallel {F}^{\prime}{\left(x\right)}^{-1}\parallel \le \frac{\gamma}{1-\gamma {L}_{0}\parallel x-{x}_{0}\parallel}.$$

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

- (a)
- The point ${s}^{*}$ can be replaced by ${s}^{**}$ (given in closed form by (24)) in Theorem 2.
- (b)
- Suppose there exist nonnegative constants ${L}_{h}^{0},{L}_{s}^{0}$ such that for all $x\in U({x}_{0},r)\subset {\mathsf{\Omega}}_{0}$$$\parallel H\left(x\right)-H\left({x}_{0}\right)\parallel \le {L}_{h}^{0}\parallel x-{x}_{0}\parallel $$$$\parallel S\left(x\right)-S\left({x}_{0}\right)\parallel \le {L}_{s}^{0}\parallel x-{x}_{0}\parallel .$$Set ${L}_{0}={L}_{h}^{0}+{L}_{s}^{0}.$ Define ${\mathsf{\Omega}}_{0}^{1}={\mathsf{\Omega}}_{0}\cap U({x}_{0},\frac{1}{\gamma {L}_{0}}).$ Replace condition (${\mathcal{A}}_{2}$) by(${\mathcal{A}}_{2}^{\prime}$) There exist nonnegative constants ${L}_{h}^{\prime}$ and ${L}_{s}^{\prime}$ such that for all $x,y\in U({x}_{0},r)\subset {\mathsf{\Omega}}_{0}^{1}$$$\parallel H\left(x\right)-H\left(y\right)\parallel \le {L}_{h}^{\prime}\parallel x-y\parallel $$$$\parallel S\left(x\right)-S\left(y\right)\parallel \le {L}_{s}^{\prime}\parallel x-y\parallel .$$Set ${L}^{\prime}={L}_{h}^{\prime}+{L}_{s}^{\prime}.$ Notice that$${L}_{h}^{\prime}\le {L}_{h},\phantom{\rule{0.166667em}{0ex}}{L}_{s}^{\prime}\le Ls\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{L}^{\prime}\le L,$$
- (c)
- The same improvements as in (b) can be made in the case of Theorem 1.

**Proposition**

**1.**

- (i)
- ${s}_{n}\le {t}_{n}$
- (ii)
- ${s}_{n+1}-{s}_{n}\le {t}_{n+1}-{t}_{n}$ and
- (iii)
- ${s}^{*}\le {t}^{*}={lim}_{k\u27f6\infty}{t}_{k}\le {r}_{2}.$

**Remark**

**4.**

## 4. Special Cases and Numerical Examples

**Example**

**1.**

**Lemma**

**4.**

**Proposition**

**2.**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

- (a)
- (b)
- The sufficient convergence criterion (16) given in [7] is false. Therefore, the rest of the results based on (16) do not hold. We have revisited the proofs to rectify this problem. Fortunately, the results can hold if (16) is replaced with (12). This can easily be observed in the proof of Theorem 3.2 in [7]. Notice that the issue related to the criteria (16) is not shown in Example 4.5, where convergence is established due to the fact that the validity of (16) is not checked. The convergence criteria obtained here are not necessary too. Along the same lines, our technique in Section 3 can be used to extend the applicability of other iterative methods discussed in [1,2,3,4,5,6,8,9,12,13,14,15,16].

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Graphs of ${g}_{1}\left(t\right)$ (Violet), ${g}_{2}\left(t\right)$ (Green), ${\overline{g}}_{3}$ (Red).

**Figure 2.**Graphs of ${g}_{1}\left(t\right)$ (Violet), ${g}_{2}\left(t\right)$ (Green), ${\overline{g}}_{3}$ (Red) and ${g}_{4}$ (Blue).

**Figure 3.**Plots of (

**a**) inner iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**b**) outer iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**c**) CPU time vs. $log(\parallel F\left({x}_{k}\right)\parallel )$ for $q=600$ and ${x}_{0}=e$.

**Figure 4.**Plots of (

**a**) inner iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**b**) outer iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**c**) CPU time vs. $log(\parallel F\left({x}_{k}\right)\parallel )$ for $q=2000$ and ${x}_{0}=e$.

**Figure 5.**Plots of (

**a**) inner iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**b**) outer iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**c**) CPU time vs. $log(\parallel F\left({x}_{k}\right)\parallel )$ for $q=600$ and ${x}_{0}=6e$.

**Figure 6.**Plots of (

**a**) inner iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**b**) outer iterations vs. $log(\parallel F\left({x}_{k}\right)\parallel )$, (

**c**) CPU time vs. $log(\parallel F\left({x}_{k}\right)\parallel )$ for $q=2000$ and ${x}_{0}=6e$.

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**MDPI and ACS Style**

Argyros, I.K.; George, S.; Godavarma, C.; Magreñán, A.A.
Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method. *Symmetry* **2019**, *11*, 981.
https://doi.org/10.3390/sym11080981

**AMA Style**

Argyros IK, George S, Godavarma C, Magreñán AA.
Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method. *Symmetry*. 2019; 11(8):981.
https://doi.org/10.3390/sym11080981

**Chicago/Turabian Style**

Argyros, Ioannis K, Santhosh George, Chandhini Godavarma, and Alberto A Magreñán.
2019. "Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method" *Symmetry* 11, no. 8: 981.
https://doi.org/10.3390/sym11080981