Matrix iteration algorithms for solving the generalized Lyapunov matrix equation | Advances in Continuous and Discrete Models

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Matrix iteration algorithms for solving the generalized Lyapunov matrix equation | Advances in Continuous and Discrete Models</title> <meta name="citation_abstract" content="In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms."/> <meta name="journal_id" content="13662"/> <meta name="dc.title" content="Matrix iteration algorithms for solving the generalized Lyapunov matrix equation"/> <meta name="dc.source" content="Advances in Difference Equations 2021 2021:1"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="SpringerOpen"/> <meta name="dc.date" content="2021-04-28"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2021 The Author(s)"/> <meta name="dc.rights" content="2021 The Author(s)"/> <meta name="dc.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="dc.description" content="In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms."/> <meta name="prism.issn" content="1687-1847"/> <meta name="prism.publicationName" content="Advances in Difference Equations"/> <meta name="prism.publicationDate" content="2021-04-28"/> <meta name="prism.volume" content="2021"/> <meta name="prism.number" content="1"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="18"/> <meta name="prism.copyright" content="2021 The Author(s)"/> <meta name="prism.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="prism.url" content="https://link.springer.com/articles/10.1186/s13662-021-03381-1"/> <meta name="prism.doi" content="doi:10.1186/s13662-021-03381-1"/> <meta name="citation_pdf_url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-021-03381-1"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/articles/10.1186/s13662-021-03381-1"/> <meta name="citation_journal_title" content="Advances in Difference Equations"/> <meta name="citation_journal_abbrev" content="Adv Differ Equ"/> <meta name="citation_publisher" content="SpringerOpen"/> <meta name="citation_issn" content="1687-1847"/> <meta name="citation_title" content="Matrix iteration algorithms for solving the generalized Lyapunov matrix equation"/> <meta name="citation_volume" content="2021"/> <meta name="citation_issue" content="1"/> <meta name="citation_publication_date" content="2021/12"/> <meta name="citation_online_date" content="2021/04/28"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="18"/> <meta name="citation_article_type" content="Research"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1186/s13662-021-03381-1"/> <meta name="DOI" content="10.1186/s13662-021-03381-1"/> <meta name="size" content="488930"/> <meta name="citation_doi" content="10.1186/s13662-021-03381-1"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1186/s13662-021-03381-1&api_key="/> <meta name="description" content="In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. 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Finally, we offer several numerical examples to show the effectiveness of the derived algorithms."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Figa_HTML.png"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link rel="stylesheet" media="print" 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xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs40314-021-01523-5/MediaObjects/40314_2021_1523_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s40314-021-01523-5?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/s40314-021-01523-5">Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">08 May 2021</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s11075-017-0432-8?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s11075-017-0432-8">Some iterative methods for the largest positive definite solution to a class of nonlinear matrix equation </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">02 November 2017</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10255-024-1100-0?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s10255-024-1100-0">A Minimum Residual Based Gradient Iterative Method for a Class of Matrix Equations </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">30 November 2023</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732793069, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>In this paper, we consider the generalized Lyapunov equation as follows: </p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} AX + XA^{T}+\sum_{j=1}^{m}N_{j}XN_{j}^{T}+C = 0, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (1) </div></div><p> where <i>A</i>, <span class="mathjax-tex">$N_{j}\in \mathbb{R}^{n\times n}$</span> (<span class="mathjax-tex">$j=1,2,\ldots ,m$</span>), <span class="mathjax-tex">$m\ll n$</span> and <span class="mathjax-tex">$C\in \mathbb{R}^{n\times n}$</span> is symmetric, <span class="mathjax-tex">$X\in \mathbb{R}^{n\times n}$</span> is the symmetric solution of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>).</p><p>The generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) is related to several linear matrix equations displayed in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1186/s13662-021-03381-1#Tab1">1</a>. A large and growing amount of literature has considered the solution for these equations; see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="
					Kurschner, P.: Efficient Low-Rank Solution of Large-Scale Matrix Equations. Dissertation, Otto von Guericke Universitat, Magdeburg (2016)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR1" id="ref-link-section-d522025e586">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="
					Simoncini, V.: Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 37, 1655–1674 (2016)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR2" id="ref-link-section-d522025e589">2</a>] and the references therein for an overview of developments and methods. </p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Table 1 Several linear matrix equations</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1186/s13662-021-03381-1/tables/1" aria-label="Full size table 1"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) often appears in the context of bilinear systems [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="
					Paolo, D.A., Alberto, I., Antonio, R.: Realization and structure theory of bilinear dynamical systems. SIAM J. Control Optim. 12, 517–535 (1974)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR3" id="ref-link-section-d522025e864">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="
					Samir, A., Baiyat, A.L., Bettayeb, M.A., Saggaf, M.A.L.: New model reduction scheme for bilinear systems. Int. J. Syst. Sci. 25, 1631–1642 (1994)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR4" id="ref-link-section-d522025e867">4</a>], stability analysis of linear stochastic systems [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="
					Kleinman, D.L.: On the stability of linear stochastic systems. IEEE Trans. Autom. Control 14, 429–430 (1969)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR5" id="ref-link-section-d522025e870">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="
					Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. 49, 686–711 (2011)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR6" id="ref-link-section-d522025e873">6</a>], special linear stochastic differential equations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="
					Gray, W.S., Mesko, J.: Energy functions and algebraic Gramians for bilinear systems. IFAC Proc. Vol. 31, 101–106 (1998)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR7" id="ref-link-section-d522025e877">7</a>] and other areas. For example, we discuss the origin of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) in bilinear systems. The bilinear system is an interesting subclass of nonlinear control systems that naturally occurs in some boundary control dynamics [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="
					Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. 49, 686–711 (2011)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR6" id="ref-link-section-d522025e883">6</a>]. The bilinear control system has been studied by scholars for many years and has the following state space representation: </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \Sigma : \textstyle\begin{cases} \dot{x}(t)=Ax(t)+\sum_{j=1}^{m}N_{j}x(t)u_{j}(t)+Bu(t), \\ y(t)=Cx(t),\qquad x(0)=x_{0}, \end{cases} $$</span></div><div class="c-article-equation__number"> (2) </div></div><p> where <i>t</i> is the time variable, <span class="mathjax-tex">$x(t)\in \mathbb{R}^{n}$</span>, <span class="mathjax-tex">$u(t)\in \mathbb{R}^{m}$</span>, <span class="mathjax-tex">$y(t)\in \mathbb{R}^{n}$</span> are the stable, input and output vectors, respectively, <span class="mathjax-tex">$u_{j}(t)$</span> is the <i>j</i>th component of <span class="mathjax-tex">$u(t)$</span>. <span class="mathjax-tex">$B \in \mathbb{R}^{n\times m}$</span>, and <i>A</i>, <span class="mathjax-tex">$N_{j}$</span>, <i>C</i> are defined in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>).</p><p>For the bilinear control system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ2">2</a>), define </p><div id="Equa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &P_{1}=e^{At_{1}}B, \\ &P_{i}(t_{1}, \ldots , t_{i})=e^{At_{i}}[N_{1}P_{i-1}, \ldots , N_{m}P_{i-1}], \quad i = 2,3,\ldots . \end{aligned} $$</span></div></div><p> Using the concept of reachability in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="
					Paolo, D.A., Alberto, I., Antonio, R.: Realization and structure theory of bilinear dynamical systems. SIAM J. Control Optim. 12, 517–535 (1974)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR3" id="ref-link-section-d522025e1516">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="
					Dorissen, H.: Canonical forms for bilinear systems. Syst. Control Lett. 13, 153–160 (1989)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR8" id="ref-link-section-d522025e1519">8</a>], the reachability corresponding to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ2">2</a>) is </p><div id="Equb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ P = \sum_{i=1}^{\infty } \int _{0}^{\infty }\cdots \int _{0}^{ \infty }P_{i}P_{i}^{T}\,dt_{1} \cdots dt_{i}, $$</span></div></div><p> where <i>P</i> is the solution of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>).</p><p>Moreover, the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) has wide applications in PDEs. Consider the heat equation subjected to mixed boundary conditions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="
					Damm, T.: Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equation. Numer. Linear Algebra Appl. 15, 853–871 (2008)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR9" id="ref-link-section-d522025e1640">9</a>] </p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &x_{t}= \Delta x\quad \text{in } \Omega , \\ &n\cdot \nabla x=u(x-1) \quad \text{on } \Gamma _{1}, \\ &x= 0 \quad \text{on } \Gamma _{2},\Gamma _{3},\Gamma _{4}, \end{aligned} $$</span></div><div class="c-article-equation__number"> (3) </div></div><p> where <span class="mathjax-tex">$\Gamma _{1}$</span>, <span class="mathjax-tex">$\Gamma _{2}$</span>, <span class="mathjax-tex">$\Gamma _{3}$</span> and <span class="mathjax-tex">$\Gamma _{4}$</span> are the boundaries of Ω. For example, for a simple <span class="mathjax-tex">$2\times 2$</span> mesh, the state vector <span class="mathjax-tex">$x=[x_{11}, x_{21}, x_{12}, x_{22}]^{T}$</span> contains the temperatures at the inner points and the Laplacian is approximated via </p><div id="Equc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \Delta x_{ij}\thickapprox -\frac{1}{h^{2}} (4x_{ij}-x_{i+1,j}-x_{i,j+1}-x_{i-1,j}-x_{i,j-1}), \end{aligned}$$ </span></div></div><p> with meshsize <span class="mathjax-tex">$h=1/3$</span>. If the Robin condition is imposed on the whole boundary, then we have </p><div id="Equd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &x_{10}\thickapprox x_{11}-h u(x_{11}-1),\qquad x_{20}\thickapprox x_{21}-h u(x_{21}-1), \\ & x_{01}\thickapprox x_{11}-h u(x_{11}-1),\qquad \ldots \end{aligned}$$ </span></div></div><p> Altogether this leads to the bilinear system </p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><math display="block"><mtable xmlns="http://www.w3.org/1998/Math/MathML" columnalign="right left" columnspacing="0.2em"> <mtr> <mtd> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> </mtd> <mtd> <mfrac> <mn>1</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mtable columnalign="center" columnspacing="1em 1em 1em"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd/> <mtd> <mo>+</mo> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> <mrow> <mo>(</mo> <mtable columnalign="center"> <mtr> <mtd> <mrow> <mo>(</mo> <mtable columnalign="center"> <mtr> <mtd> <msub> <mi>x</mi> <mn>11</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mtable columnalign="center"> <mtr> <mtd> <msub> <mi>x</mi> <mn>11</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>12</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mtable columnalign="center"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>22</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mtable columnalign="center"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>22</mn> </msub> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <msub> <mi>u</mi> <mn>4</mn> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mfrac> <mn>1</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> <mrow> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">(</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mn>3</mn> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mn>3</mn> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mn>4</mn> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>4</mn> </msub> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mn>4</mn> </msub> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable></math></div><div class="c-article-equation__number"> (4) </div></div><p> where <span class="mathjax-tex">$E_{j}=e_{j}e_{j}^{T}$</span> with canonical unit vector <span class="mathjax-tex">$e_{j}\in \mathbb{R}^{2}$</span>, and </p><div id="Eque" class="c-article-equation"><div class="c-article-equation__content"><math display="block"><mtable xmlns="http://www.w3.org/1998/Math/MathML" columnalign="right left" columnspacing="0.2em"> <mtr> <mtd/> <mtd> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mtable columnalign="center" columnspacing="1em 1em 1em"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em"/> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd/> <mtd> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>⊗</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em"/> <msub> <mi>A</mi> <mn>3</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em"/> <msub> <mi>A</mi> <mn>4</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>⊗</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd/> <mtd> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>⊗</mo> <mi>e</mi> <mo>,</mo> <mspace width="2em"/> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>⊗</mo> <mi>e</mi> <mo>,</mo> <mspace width="2em"/> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>⊗</mo> <mi>e</mi> <mo>,</mo> <mspace width="2em"/> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>⊗</mo> <mi>e</mi> <mo>,</mo> <mspace width="1em"/> <mi>e</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mtd> </mtr> </mtable></math></div></div><p> Thus, the optimal control problem of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ4">4</a>) reduces to the bilinear control system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ2">2</a>) and we ultimately need solve the generalized Lyapunov equation: </p><div id="Equf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} AX+XA+\sum_{j=1}^{4} A_{j}XA_{j}=-BB^{T}. \end{aligned}$$ </span></div></div><p>Therefore, considering the important applications of the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>), many researchers pay much attention to study the solution for this equation in recent years. Damm showed the direct method to solve the generalized Lyapunov equation [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="
					Damm, T.: Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equation. Numer. Linear Algebra Appl. 15, 853–871 (2008)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR9" id="ref-link-section-d522025e3532">9</a>]. Fan et al. transformed this equation into the generalized Stein equation by generalized Cayley transformation and solved it using GSM [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="
					Fan, H.Y., Weng, P., Chu, E.: Numerical solution to generalized Lyapunov, Stein and rational Riccati equations in stochastic control. Numer. Algorithms 71, 245–272 (2016)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR10" id="ref-link-section-d522025e3535">10</a>]. Dai et al. proposed the HSS algorithm to solve the generalized Lyapunov equation. Li et al. proposed the PHSS iterative method for solving this equation when <i>A</i> is asymmetric positive definite [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="
					Li, S.Y., Shen, H.L., Shao, X.H.: PHSS iterative method for solving generalized Lyapunov equations. Mathematics 7, 1–13 (2019)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR11" id="ref-link-section-d522025e3541">11</a>]. Based on the recent results, we mainly discuss the matrix iteration algorithms for the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>).</p><p>The rest of the paper is organized as follows. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec2">2</a>, we recall some known results on the generalized Lyapunov equation’s solvability and rewrite this equation into the generalized Stein equation by using Cayley transformation. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec5">3</a>, we present the matrix versions and variant forms of the BICR, Bi-CGSTAB, and CRS algorithms. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec9">4</a>, we offer several numerical examples to test the effectiveness of the derived algorithms. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec10">5</a>, we draw some concluding remarks.</p><p>Throughout this paper, we shall adopt the following notations. <span class="mathjax-tex">$\mathbb{R}^{m\times n}$</span> and <span class="mathjax-tex">$\mathbb{Z}^{+}$</span> stand for the set of all <span class="mathjax-tex">$m\times n$</span> real matrices and positive integers. For <span class="mathjax-tex">$A = (a_{ij})=(a_{1},a_{2},\ldots , a_{n})\in \mathbb{R}^{m\times n}$</span>, the symbol <span class="mathjax-tex">$\text{vec}(A)$</span> is a vector defined by <span class="mathjax-tex">$\text{vec}(A) = (a_{1}^{T},a_{2}^{T},\ldots , a_{n}^{T})^{T}$</span>. <span class="mathjax-tex">$A^{T}$</span> and <span class="mathjax-tex">$\|A\|$</span> represent the transpose and 2-norm of matrix <i>A</i>, respectively. The symbol <span class="mathjax-tex">$A\geq 0$</span> means that <i>A</i> is symmetric positive semi-definite. For <span class="mathjax-tex">$B\in \mathbb{R}^{m\times n}$</span>, the Kronecker product and inner product of <i>A</i> and <i>B</i> are defined by <span class="mathjax-tex">$A\otimes B = (a_{ij}B)$</span> and <span class="mathjax-tex">$\langle A, B\rangle = \text{tr}(B^{T}A)$</span>. The open right-half and left-half planes are denoted by <span class="mathjax-tex">$\mathbb{C}_{+}$</span> and <span class="mathjax-tex">$\mathbb{C}_{-}$</span>, respectively.</p></div></div></section><section data-title="Solvability and Cayley transformation"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Solvability and Cayley transformation</h2><div class="c-article-section__content" id="Sec2-content"><h3 class="c-article__sub-heading" id="Sec3"><span class="c-article-section__title-number">2.1 </span>Solvability of the generalized Lyapunov equation</h3><p>This section introduces the solvability for the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>).</p><p>Denote <span class="mathjax-tex">$\sigma (T)\in \mathbb{C}$</span> by the spectrum of a linear operator <i>T</i> and <span class="mathjax-tex">$\rho (T)=\max \{|\lambda | |\lambda \in \sigma (T)\}$</span> by the spectral radius. Define the linear matrix operators <span class="mathjax-tex">$\mathcal{L}_{A}$</span> and <span class="mathjax-tex">$\Pi :\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^{n\times n}$</span> by </p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathcal{L}_{A}=A^{T}X+XA, \qquad \Pi (X)\mapsto \sum _{j=1}^{m}N_{j}XN_{j}^{T}. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (5) </div></div><p> Obviously, <span class="mathjax-tex">$\Pi (X)\geq 0$</span> when <span class="mathjax-tex">$X\geq 0$</span>.</p><p>Therefore, using Theorem 3.9 in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="
					Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. 49, 686–711 (2011)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR6" id="ref-link-section-d522025e4435">6</a>], we immediately get the generalized Lyapunov equation’s stability result.</p> <h3 class="c-article__sub-heading" id="FPar1">Theorem 2.1</h3> <p><i>Let</i> <span class="mathjax-tex">$A\in \mathbb{R}^{n\times n}$</span> <i>and</i> Π <i>be positive</i>. <i>The following conclusions are equivalent</i>:</p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(a)</span> <p><i>For all</i> <span class="mathjax-tex">$Y>0$</span>, <span class="mathjax-tex">$\exists X>0$</span> <i>such that</i> <span class="mathjax-tex">$\mathcal{L}_{A}(X)+\Pi (X)=-Y$</span>;</p> </li> <li> <span class="u-custom-list-number">(b)</span> <p><span class="mathjax-tex">$\exists Y, X>0$</span> <i>such that</i> <span class="mathjax-tex">$\mathcal{L}_{A}(X)+\Pi (X)=-Y$</span>;</p> </li> <li> <span class="u-custom-list-number">(c)</span> <p><span class="mathjax-tex">$\exists Y\geq 0$</span> <i>with</i> <span class="mathjax-tex">$(A,Y)$</span> <i>controllable</i>, <span class="mathjax-tex">$\exists X>0$</span> <i>such that</i> <span class="mathjax-tex">$\mathcal{L}(X)+\Pi (X)=-Y$</span>;</p> </li> <li> <span class="u-custom-list-number">(d)</span> <p><span class="mathjax-tex">$\sigma (\mathcal{L}_{A}(X)+\Pi (X))\subset \mathbb{C}_{-}$</span>;</p> </li> <li> <span class="u-custom-list-number">(e)</span> <p><span class="mathjax-tex">$\sigma (\mathcal{L}_{A}(X))\subset \mathbb{C}_{-}$</span> <i>and</i> <span class="mathjax-tex">$\rho (\mathcal{L}^{-1}_{A}(X)\Pi (X))<1$</span>,</p> </li> </ol><p><i>where the linear matrix operators</i> <span class="mathjax-tex">$\mathcal{L}_{A}$</span> <i>and</i> Π <i>are defined by</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ5">5</a>).</p> <h3 class="c-article__sub-heading" id="FPar2">Remark 2.1</h3> <p>For the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>), we often choose <span class="mathjax-tex">$C=BB^{T}$</span>, i.e., <i>C</i> is symmetric positive semi-definite. Using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar1">2.1</a>, Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) has a positive definite solution <i>X</i> if <i>A</i> is stable, <span class="mathjax-tex">$(A, B)$</span> is controllable, and the norm of the <span class="mathjax-tex">$N_{j}$</span> is sufficiently small.</p> <h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">2.2 </span>Cayley transformation for (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>)</h3><p>In this section, we introduce Cayley transformation for the generalized Lyapunov equation.</p><p>It is well known that Cayley transformation is a link between the classical Lyapunov and Stein equations. Fan et al. have shown that the stability of the Lyapunov and Stein equations is different. Naturally, we wonder if the stability is different and the counterparty method has other effects. It is verified in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec9">4</a> that our iteration methods are more efficient after applying Cayley transformation to the generalized Lyapunov equation. We first recall the definition of Cayley transformation.</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 2.1</h3> <p>(Cayley transformation)</p> <p>Let <span class="mathjax-tex">$M\in \mathbb{R}^{n\times n}$</span> be a skew-symmetric matrix. Then <span class="mathjax-tex">$\mathcal{N}=(I+M)^{-1}(I-M)$</span> is called Cayley transformation of <i>M</i>.</p> <p>Next, we show that the generalized Lyapunov equation can be changed to the generalized Setin equation after Cayley transformation.</p> <h3 class="c-article__sub-heading" id="FPar4">Theorem 2.2</h3> <p><i>For the generalized Lyapunov equation</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>), <i>take the positive parameter</i> <i>γ</i> <i>such that the matrices</i> <span class="mathjax-tex">$(\gamma I+A)$</span> <i>and</i> <span class="mathjax-tex">$(\gamma I+A^{T})$</span> <i>are both nonsingular</i>. <i>Then</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) <i>is equivalent to the generalized Stein equation</i> </p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ X-\hat{A}X\hat{A}^{T}+2\gamma \sum_{j=1}^{m} \hat{N_{j}}X \hat{N_{j}^{T}}+2\gamma \hat{C}=0, $$</span></div><div class="c-article-equation__number"> (6) </div></div><p><i>where</i> </p><div id="Equg" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\hat{A}=(\gamma I+A)^{-1}(\gamma I-A), \\ &\hat{N_{j}}=(\gamma I+A)^{-1}N_{j}, \\ &\hat{C}=(\gamma I+A)^{-1}C(\gamma I+A)^{-T}. \end{aligned}$$ </span></div></div> <h3 class="c-article__sub-heading" id="FPar5">Proof</h3> <p>Introducing the positive parameter <i>γ</i> to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>), we get </p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} (\gamma I+A)X\bigl(\gamma I+A^{T}\bigr)-(\gamma I-A)X\bigl(\gamma I-A^{T}\bigr)+2\gamma \Biggl( \sum_{j=1}^{m}N_{j}XN_{j}^{T} \Biggr) +2\gamma C=0. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (7) </div></div><p> Since <span class="mathjax-tex">$(\gamma I+A)$</span> and <span class="mathjax-tex">$(\gamma I+A^{T})$</span> are both nonsingular, premultiplying <span class="mathjax-tex">$(\gamma I+A)^{-1}$</span> and postmultiplying <span class="mathjax-tex">$(\gamma I+A^{T})^{-1}$</span> on both sides to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ7">7</a>) yield (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ6">6</a>). Thus we complete the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar4">2.2</a>. □</p> <h3 class="c-article__sub-heading" id="FPar6">Remark 2.2</h3> <p>Viewing Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar4">2.2</a>, it involves a positive parameter <i>γ</i>. We offer a practical way to choose <i>γ</i>. Set </p><div id="Equh" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \gamma = \max_{1\leq i\leq n}a_{ii}, \end{aligned}$$ </span></div></div><p> then <span class="mathjax-tex">$(\gamma I+A)$</span> and <span class="mathjax-tex">$(\gamma I+A^{T})$</span> are both nonsingular. Thus the condition of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar4">2.2</a> is satisfied. Appropriate adjustments can be made according to different situations.</p> <h3 class="c-article__sub-heading" id="FPar7">Remark 2.3</h3> <p>Next, we show the relationship between the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) and the generalized Stein equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ6">6</a>) by using the preconditioner method of linear systems.</p> <p>By utilizing the operator vec, the generalized Lyapunov equation can be rewritten as </p><div id="Equi" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathcal{A}_{1}\mathcal{X}= \Biggl(I\otimes A+A\otimes I+\sum _{j=1}^{m}N_{j} \otimes N_{j}\Biggr)\text{vec}(X)=-\text{vec}(C). \end{aligned}$$ </span></div></div><p> The generalized Stein equation can be rewritten as </p><div id="Equj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathcal{A}_{2}\mathcal{X}= \Biggl(I\otimes I+\hat{A}\otimes \hat{A}+\sum_{j=1}^{m} \hat{N}_{j}\otimes \hat{N}_{j}\Biggr)\text{vec}(X) =-2 \gamma \text{vec}(\hat{C}). \end{aligned}$$ </span></div></div><p> Hence, it is not difficult to derive the following relation between <span class="mathjax-tex">$\mathcal{A}_{1}$</span> and <span class="mathjax-tex">$\mathcal{A}_{2}$</span>: </p><div id="Equk" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \mathcal{A}_{2} = 2\gamma \bigl[(\gamma I+A)\otimes (\gamma I+A) \bigr]^{-1} \mathcal{A}_{1}, \end{aligned}$$ </span></div></div><p> where </p><div id="Equl" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ P_{pre}=\frac{1}{2\gamma }\bigl[(\gamma I+A)\otimes (\gamma I+A) \bigr] $$</span></div></div><p> is the preconditioning matrix and the corresponding generalized Stein equation is the preconditioning system.</p> <p>By Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar6">2.2</a>, the operator <span class="mathjax-tex">$\mathcal{P}$</span> can be defined as </p><div id="Equm" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \mathcal{P}(X) = X-\hat{A}X\hat{A}^{T}+2\gamma \sum _{j=1}^{m} \hat{N_{j}}X \hat{N_{j}^{T}}. $$</span></div></div><p>In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec5">3</a>, we apply this operator to derive the variant forms of the BICR, Bi-CGSTAB, and CRS algorithms, respectively. The iteration methods are efficient. Numerical examples address this point in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec9">4</a>.</p></div></div></section><section data-title="Iteration algorithms"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5"><span class="c-article-section__title-number">3 </span>Iteration algorithms</h2><div class="c-article-section__content" id="Sec5-content"><p>This section presents the matrix versions and variant forms of the BICR, Bi-CGSTAB, and CRS algorithms in three subsections, respectively.</p><h3 class="c-article__sub-heading" id="Sec6"><span class="c-article-section__title-number">3.1 </span>BICR algorithm</h3><p>The BiCR method [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="
					Sogabe, T., Sugihara, M., Zhang, S.L.: An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math. 226, 103–113 (2009)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR12" id="ref-link-section-d522025e6906">12</a>] has been proposed as a generalization of the conjugate residual (CR) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="
					Stiefel, E.L.: Relaxationsmethoden bester strategie zur losung linearer gleichungssysteme. Comment. Math. Helv. 29, 157–179 (1955)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR13" id="ref-link-section-d522025e6909">13</a>] method for nonsymmetric matrices. Recently, Abe et al. designed BiCR for symmetric complex matrices (SCBiCR) and analyzed the factor in the loss of convergence speed [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="
					Abea, K., Fujino, S.: Converting BiCR method for linear equations with complex symmetric matrices. Appl. Math. Comput. 321, 564–576 (2018)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR14" id="ref-link-section-d522025e6912">14</a>]. It is easy to see that the BICR algorithm cannot be directly used to solve the generalized Lyapunov equation. Naturally, one can convert this matrix equation into the linear system through Kronecker product and vectorization operators. However, this makes the computational cost especially expensive. When the matrix order becomes larger, as the computer memory is limited, it is hard to implement in practice.</p><p>Therefore, we need to modify the BICR algorithm and ensure that the calculation cost is relatively cheap. In this subsection, we propose the matrix version of the BICR algorithm (Algorithm 1). Then we show the variant version of the BICR algorithm (Algorithm 2). </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-a" data-title="Algorithm 1"><figure><figcaption><b id="Figa" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/a" rel="nofollow"><picture><img aria-describedby="Figa" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Figa_HTML.png" alt="figure a" loading="lazy" width="685" height="673"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-a-desc"><p>Matrix form of the BICR algorithm</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/a" data-track-dest="link:Figurea Full size image" aria-label="Full size image figure a" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-b" data-title="Algorithm 2"><figure><figcaption><b id="Figb" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/b" rel="nofollow"><picture><img aria-describedby="Figb" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Figb_HTML.png" alt="figure b" loading="lazy" width="685" height="495"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-b-desc"><p>The variant form of the BICR algorithm</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/b" data-track-dest="link:Figureb Full size image" aria-label="Full size image figure b" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Using the iteration schemes of Algorithm 1 and Algorithm 2, we can directly solve the generalized Lyapunov equation. Further, we show the bi-orthogonal properties and convergent analysis of Algorithm 1 by Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar8">3.1</a> and Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar9">3.2</a>.</p> <h3 class="c-article__sub-heading" id="FPar8">Theorem 3.1</h3> <p><i>For Algorithm</i> 1, <i>we assume that there exists a positive integer number such that</i> <span class="mathjax-tex">$W(k)\neq 0$</span> <i>and</i> <span class="mathjax-tex">$R(k)\neq 0$</span> <i>for all</i> <span class="mathjax-tex">$k=1, 2, \ldots , r$</span>. <i>Then we get</i> </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \operatorname{tr}\bigl(R(v)^{T}W(u)\bigr)= 0, \quad \textit{for } u,v = 1, 2, \ldots , r, u< v, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (8) </div></div><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \operatorname{tr}\bigl(S(v)^{T}Z(u)\bigr) = 0,\quad \textit{for } u,v = 1, 2, \ldots , r, u< v, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (9) </div></div><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \operatorname{tr}\bigl(Z(v)^{T}Z(u)\bigr) = 0,\quad \textit{for } u,v = 1, 2, \ldots , r, u \neq v, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (10) </div></div><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \operatorname{tr}\bigl(W(v)^{T}W(u)\bigr) = 0, \quad \textit{for } u,v = 1, 2, \ldots , r u\neq v. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (11) </div></div> <p>For the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar8">3.1</a>, refer to the <a data-track="click" data-track-label="link" data-track-action="appendix anchor" href="/article/10.1186/s13662-021-03381-1#App1">Appendix</a>.</p> <h3 class="c-article__sub-heading" id="FPar9">Theorem 3.2</h3> <p><i>For Algorithm</i> 1, <i>the relative residual error has the following property</i>: </p><div id="Equn" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl\Vert R(k+1) \bigr\Vert ^{2}\leq \bigl\Vert R(k) \bigr\Vert ^{2}. $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar10">Proof</h3> <p>Using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar8">3.1</a>, we have </p><div id="Equo" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} \bigl\Vert R(k+1) \bigr\Vert ^{2} &= \operatorname{tr}\bigl(R(k+1)^{T}R(k+1)\bigr) \\ &=\operatorname{tr}\bigl(\bigl(R(k)-\alpha (k)W(k)\bigr)^{T} \bigl(R(k)-\alpha (k)W(k)\bigr)\bigr) \\ &= \bigl\Vert R(k) \bigr\Vert ^{2}+ \alpha (k)^{2} \bigl\Vert W(k) \bigr\Vert ^{2}-2\alpha (k)\operatorname{tr} \bigl(W(k)^{T}R(k)\bigr) \\ &= \bigl\Vert R(k) \bigr\Vert ^{2}-\alpha (k)\operatorname{tr} \bigl(W(k)^{T}R(k)\bigr) \\ &= \bigl\Vert R(k) \bigr\Vert ^{2}-\frac{\operatorname{tr}(W(k)^{T}R(k))^{2}}{\operatorname{tr}(W(k)^{T}W(k))} \\ &\leq \bigl\Vert R(k) \bigr\Vert ^{2}. \end{aligned} $$</span></div></div><p> Hence, the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar9">3.2</a> is completed. □</p> <h3 class="c-article__sub-heading" id="FPar11">Remark 3.1</h3> <p>In terms of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar9">3.2</a>, the property <span class="mathjax-tex">$\|R(k+1)\|\leq \|R(k)\|$</span> ensures that Algorithm 1 possesses fast and smooth convergence.</p> <h3 class="c-article__sub-heading" id="Sec7"><span class="c-article-section__title-number">3.2 </span>Bi-CGSTAB algorithm</h3><p>Sonneveld [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="
					Sonneveld, P.: CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 36–52 (1989)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR15" id="ref-link-section-d522025e8275">15</a>] has shown a variant of BiCG, referred to the conjugate gradient squared (CGS). Van der Vorst [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="
					Vander, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR16" id="ref-link-section-d522025e8278">16</a>] has derived one of the most successful variants of BiCG, known as the Bi-CGSTAB method. The Bi-CGSTAB algorithm is an effective algorithm for solving large sparse linear systems [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="
					Vander, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR16" id="ref-link-section-d522025e8281">16</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="
					Hajarian, M.: Developing Bi-CG and Bi-CR methods to solve generalized Sylvester-transpose matrix equations. Int. J. Autom. Comput. 11, 25–29 (2014)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR17" id="ref-link-section-d522025e8284">17</a>]. Chen et al. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="
					Chen, J., McInnes, L.C., Zhang, H.: Analysis and practical use of flexible BiCGStab. J. Sci. Comput. 68, 803–825 (2016)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR18" id="ref-link-section-d522025e8287">18</a>] proposed a flexible version of the BiCGStab algorithm for solving the linear system. It is easy to see that the Bi-CGSTAB algorithm cannot be directly used to solve the generalized Lyapunov equation. Similarly, we need to modify the Bi-CGSTAB algorithm to the matrix version. The matrix version of the Bi-CGSTAB algorithm is summarized in Algorithm 3. The variant form of the BICR algorithm is shown in Algorithm 4. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-c" data-title="Algorithm 3"><figure><figcaption><b id="Figc" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/c" rel="nofollow"><picture><img aria-describedby="Figc" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Figc_HTML.png" alt="figure c" loading="lazy" width="685" height="616"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-c-desc"><p>Matrix form of the Bi-CGSTAB algorithm</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/c" data-track-dest="link:Figurec Full size image" aria-label="Full size image figure c" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-d" data-title="Algorithm 4"><figure><figcaption><b id="Figd" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/d" rel="nofollow"><picture><img aria-describedby="Figd" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Figd_HTML.png" alt="figure d" loading="lazy" width="685" height="471"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-d-desc"><p>The variant form of the Bi-CGSTAB algorithm</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/d" data-track-dest="link:Figured Full size image" aria-label="Full size image figure d" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Viewing the iteration schemes, we can be seen that Algorithm 3 is a simple matrix form of the Bi-CGSTAB algorithm. Hence, Algorithm 3 has the same properties as the Bi-CGSTAB algorithm. Algorithm 4 is an improved version of the Bi-CGSTAB algorithm, which has high computing efficiency. This point has been addressed by numerical examples in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec9">4</a>.</p><h3 class="c-article__sub-heading" id="Sec8"><span class="c-article-section__title-number">3.3 </span>CRS algorithm</h3><p>Zhang et al. proposed the conjugate residual squared (CRS) method in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="
					Zhang, L.T., Zuo, X.Y., Gu, T.X., Huang, T.Z.: Conjugate residual squared method and its improvement for non-symmetric linear systems. Int. J. Comput. Math. 87, 1578–1590 (2010)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR19" id="ref-link-section-d522025e8339">19</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="
					Zhang, L.T., Huang, T.Z., Gu, T.X., Zuo, X.Y.: An improved conjugate residual squared algorithm suitable for distributed parallel computing. Microelectron. Comput. 25, 12–14 (2008) (in Chinese)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR20" id="ref-link-section-d522025e8342">20</a>] to solve the linear system. The CRS algorithm is mainly aimed to avoid using the transpose of <i>A</i> in the BiCR algorithm and get faster convergence for the same computational cost [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="
					Zhang, L.T., Zuo, X.Y., Gu, T.X., Huang, T.Z.: Conjugate residual squared method and its improvement for non-symmetric linear systems. Int. J. Comput. Math. 87, 1578–1590 (2010)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR19" id="ref-link-section-d522025e8348">19</a>]. Recently, Ma et al. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="
					Chen, L.J., Ma, C.F.: Developing CRS iterative methods for periodic Sylvester matrix equation. Adv. Differ. Equ. 1, 1–11 (2019)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR21" id="ref-link-section-d522025e8351">21</a>] used the matrix CRS iteration method to solve a class of coupled Sylvester-transpose matrix equations. Later, they extended two mathematical equivalent forms of the CRS algorithm to solve the periodic Sylvester matrix equation by applying Kronecker product and vectorization operator [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="
					Chen, L.J., Ma, C.F.: Developing CRS iterative methods for periodic Sylvester matrix equation. Adv. Differ. Equ. 1, 1–11 (2019)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR21" id="ref-link-section-d522025e8355">21</a>]. In fact, in many cases, the CRS algorithm converges twice as fast as the BiCR algorithm [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Zhao, J., Zhang, J.H.: A smoothed conjugate residual squared algorithm for solving nonsymmetric linear systems. In: 2009 Second Int. Confe. Infor. Comput. Sci., vol. 4, pp. 364–367 (2009)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR22" id="ref-link-section-d522025e8358">22</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="
					Sogabe, T., Zhang, S.L.: Extended conjugate residual methods for solving nonsymmetric linear systems. In: International Conference on Numerical Optimization and Numerical Linear Algebra, pp. 88–99 (2003)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR23" id="ref-link-section-d522025e8361">23</a>]. The BiCR method can be derived from the preconditioned conjugate residual (CR) algorithm [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="
					Sogabe, T., Sugihara, M., Zhang, S.L.: An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math. 226, 103–113 (2009)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR24" id="ref-link-section-d522025e8364">24</a>]. In exact arithmetic, they terminate after a limited number of iterations. In short, we can expect that the CRS algorithm will work well in many cases. The numerical examples in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec9">4</a> are shown to address this point.</p><p>It is easy to see that the CRS algorithm cannot be directly used to solve the generalized Lyapunov equation. Similarly, we need to modify the CRS algorithm to the matrix version. The matrix version of the CRS algorithm is summarized in Algorithm 5. The variant version of the CRS algorithm is shown in Algorithm 6. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-e" data-title="Algorithm 5"><figure><figcaption><b id="Fige" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 5</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/e" rel="nofollow"><picture><img aria-describedby="Fige" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fige_HTML.png" alt="figure e" loading="lazy" width="685" height="437"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-e-desc"><p>Matrix form of the CRS algorithm</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/e" data-track-dest="link:Figuree Full size image" aria-label="Full size image figure e" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-f" data-title="Algorithm 6"><figure><figcaption><b id="Figf" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 6</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/f" rel="nofollow"><picture><img aria-describedby="Figf" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Figf_HTML.png" alt="figure f" loading="lazy" width="685" height="389"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-f-desc"><p>The variant form of the CRS algorithm</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/f" data-track-dest="link:Figuref Full size image" aria-label="Full size image figure f" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Viewing the iteration schemes, it can be seen that Algorithm 5 is a simple matrix form of the CRS algorithm. Hence, Algorithm 5 has the same properties as the CRS algorithm. Algorithm 6 is the variant version of the CRS algorithm, which has high computing efficiency. The numerical examples have verified the validity of the iteration algorithms in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1186/s13662-021-03381-1#Sec9">4</a>.</p> <h3 class="c-article__sub-heading" id="FPar12">Remark 3.2</h3> <p>The BICGSTAB and CRS algorithms have an orthogonality property similar to that of BICR and thus are omitted.</p> <p>The convergence result of Algorithms 2 to 6 has been summarized in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar13">3.3</a>.</p> <h3 class="c-article__sub-heading" id="FPar13">Theorem 3.3</h3> <p><i>For the generalized Lyapunov equation</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>), <i>if Algorithms</i> 2 <i>to</i> 6 <i>do not break down by zero division</i>, <i>for any initial matrix</i> <span class="mathjax-tex">$X(1)\in \mathbb{R}^{n\times n}$</span>, <i>Algorithms</i> 2 <i>to</i> 6 <i>can compute the solution of</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) <i>within a finite number of iterations in the absence of the roundoff error</i>.</p> </div></div></section><section data-title="Numerical experiments"><div class="c-article-section" id="Sec9-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec9"><span class="c-article-section__title-number">4 </span>Numerical experiments</h2><div class="c-article-section__content" id="Sec9-content"><p>In this section, we give several examples to show the numerical feasibility and effectiveness of Algorithm 1 (BICR), Algorithm 3 (Bi-CGSTAB algorithm), Algorithm 5 (CRS algorithm) and their improved algorithms, including Algorithm 2 (Var-BICR algorithm), Algorithm 4 (Var-Bi-CGSTAB algorithm), Algorithm 6 (Var-CRS algorithm). Set <span class="mathjax-tex">$tol=1.0e-8$</span>. The numerical behavior of iteration methods will be listed with respect to the number of iteration steps (ITs), the computing time (CPU)(s) and relative residual error (Error). All experiments are performed in Matlab (version R2017a) with double precision on a personal computer with 3.20 GHz central processing unit (Inter(R) Core(TM) i5-6500 CPU), 6.00G memory and Windows 7 operating system.</p> <h3 class="c-article__sub-heading" id="FPar14">Example 4.1</h3> <p>Consider the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>) with </p><div id="Equp" class="c-article-equation"><div class="c-article-equation__content"><math display="block"><mtable xmlns="http://www.w3.org/1998/Math/MathML"> <mtr> <mtd columnalign="left"> <mi>A</mi> <mo>=</mo> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mtable> <mtr> <mtd columnalign="left"> <mn>1.6</mn> </mtd> <mtd columnalign="left"> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn>0.3</mn> </mtd> <mtd columnalign="left"> <mo stretchy="false">|</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn>0</mn> </mtd> <mtd columnalign="left"> <mi mathvariant="normal">else</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="2em"/> <mi>N</mi> <mo>=</mo> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mtable> <mtr> <mtd columnalign="left"> <mn>0.05</mn> </mtd> <mtd columnalign="left"> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo>−</mo> <mn>0.01</mn> </mtd> <mtd columnalign="left"> <mo stretchy="false">|</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mn>0</mn> </mtd> <mtd columnalign="left"> <mi mathvariant="normal">else</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>B</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>A</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mtable columnalign="center" columnspacing="1em"> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>×</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mn>0</mn> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>×</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mn>0</mn> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>×</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <msub> <mi>n</mi> <mn>2</mn> </msub> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <msup> <mi>A</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mspace width="2em"/> <mi>C</mi> <mo>=</mo> <mi>B</mi> <msup> <mi>B</mi> <mi>T</mi> </msup> <mo>,</mo> <mspace width="2em"/> <msub> <mi>N</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>×</mo> <mi>j</mi> <mo>×</mo> <mi>N</mi> <mspace width="1em"/> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable></math></div></div> <p>Set the initial value </p><div id="Equq" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ X(1) = 0,\qquad S(1) = I. $$</span></div></div><p> We use Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1186/s13662-021-03381-1#Tab2">2</a> to show the error analysis for this example. </p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-2"><figure><figcaption class="c-article-table__figcaption"><b id="Tab2" data-test="table-caption">Table 2 Numerical results for Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar14">4.1</a></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1186/s13662-021-03381-1/tables/2" aria-label="Full size table 2"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Moreover, when <span class="mathjax-tex">$n=600$</span>, we use Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1186/s13662-021-03381-1#Fig1">1</a> to show the error analysis of Algorithms 1 to 6. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Figure 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fig1_HTML.jpg?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fig1_HTML.jpg" alt="figure 1" loading="lazy" width="685" height="462"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Comparison between the residual error of Algorithms 1 to 6 for Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar14">4.1</a></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>By comparing with these algorithms, it is clear that the algorithms’ efficiency will greatly be improved after using a Cayley transformation. The variant versions of the Bi-CGSTAB and CRS algorithms have the best efficiency.</p> <h3 class="c-article__sub-heading" id="FPar15">Example 4.2</h3> <p>Let <i>P</i> be the block tridiagonal sparse <span class="mathjax-tex">$m^{2}\times m^{2}$</span> matrix, given by a finite difference disretization of the heat equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ3">3</a>) on an <span class="mathjax-tex">$m\times m$</span>-mesh, i.e., </p><div id="Equr" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ P = I\otimes T_{m} + T_{m}\otimes I\in \mathbb{R}^{n\times n} ,\qquad T_{k}= \begin{bmatrix} -2& 1&&\\ 1&-2&\ddots &\\ & \ddots &\ddots &1 \\ &&1&-2 \end{bmatrix}. $$</span></div></div> <p>If the Robin condition is imposed on the whole boundary, then we have </p><div id="Equs" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A = P+E_{1}\otimes I + I \otimes E_{1} + E_{m}\otimes I+ I\otimes E_{m}, $$</span></div></div><p> where <span class="mathjax-tex">$E_{j} = e_{j}e_{j}^{T}$</span> with canonical unit vector <span class="mathjax-tex">$e_{j}$</span>. The coefficient matrices <span class="mathjax-tex">$N_{j}$</span> and the columns <span class="mathjax-tex">$b_{j}$</span> of <i>B</i> corresponding to the left, upper, lower, and right boundaries are given by </p><div id="Equt" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{aligned} &N_{1}=E_{1}\otimes I,\qquad N_{2} = I\otimes E_{1}, \qquad N_{3}=E_{m} \otimes I, \qquad N_{4} = I\otimes E_{m}, \\ &b_{1} = E_{1}\otimes e,\qquad b_{2} = e\otimes E_{1},\qquad b_{3}=E_{m} \otimes e,\qquad b_{4} = e\otimes E_{m}. \end{aligned} $$</span></div></div><p> Then the above heat equation’s optimal control problem reduces to solving the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>).</p><p>We use Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1186/s13662-021-03381-1#Tab3">3</a> to show the residual error analysis. It is obvious that the effect of the Var-Bi-CGSTAB algorithm is optimal compared with other algorithms. </p><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-3"><figure><figcaption class="c-article-table__figcaption"><b id="Tab3" data-test="table-caption">Table 3 Numerical results for Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar15">4.2</a></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1186/s13662-021-03381-1/tables/3" aria-label="Full size table 3"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Further, we use Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1186/s13662-021-03381-1#Fig2">2</a> to show the error analysis when <span class="mathjax-tex">$n = 64$</span>. It can be seen that the variant versions of the algorithms perform better. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Figure 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fig2_HTML.jpg?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fig2_HTML.jpg" alt="figure 2" loading="lazy" width="685" height="462"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Comparison between the residual error of Algorithms 1 to 6 for Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar15">4.2</a></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar16">Example 4.3</h3> <p>Consider the RC trapezoidal circuit with <i>m</i> resistors with <i>g</i> extensions </p><div id="Equu" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \textstyle\begin{cases} \dot{x}(t)=Ax(t)+Nx(t)u(t)+bu(t), \\ y(t)=c^{T}x(t). \end{cases} $$</span></div></div><p> Since the original system is nonlinear, it is linearized by the second-order Carleman bilinear method to obtain a system of order <span class="mathjax-tex">$n= m+m^{2}$</span>.</p> <p>The matrices <i>A</i>, <i>N</i> and <i>b</i> can be referred to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="
					Bai, Z.J., Skoogh, D.: A projection method for model reduction of bilinear dynamical systems. Linear Algebra Appl. 415, 406–425 (2006)
				" href="/article/10.1186/s13662-021-03381-1#ref-CR25" id="ref-link-section-d522025e10826">25</a>]. The corresponding generalized Lyapunov equation is </p><div id="Equv" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} AX + XA^{T}+NXN^{T}+C = 0. \end{aligned}$$ </span></div></div><p>We use Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/article/10.1186/s13662-021-03381-1#Tab4">4</a> to show the residual error analysis. Further, we use Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1186/s13662-021-03381-1#Fig3">3</a> to show the error analysis when <span class="mathjax-tex">$n = 8$</span>. It can be seen that the Var-Bi-CGSTAB algorithm performs best. </p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Figure 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Figure 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fig3_HTML.jpg?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1186%2Fs13662-021-03381-1/MediaObjects/13662_2021_3381_Fig3_HTML.jpg" alt="figure 3" loading="lazy" width="685" height="459"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Comparison between the residual error of Algorithms 3 to 6 for Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar16">4.3</a></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1186/s13662-021-03381-1/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-4"><figure><figcaption class="c-article-table__figcaption"><b id="Tab4" data-test="table-caption">Table 4 Numerical results for Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar16">4.3</a></b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/article/10.1186/s13662-021-03381-1/tables/4" aria-label="Full size table 4"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar17">Remark 4.1</h3> <p>From the three numerical examples above, it can be seen that the variant algorithms proposed in this paper will greatly improve the operating efficiency. In other words, the conjugate gradient-like methods are more efficient than the generalized Setin equation.</p> </div></div></section><section data-title="Conclusions"><div class="c-article-section" id="Sec10-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec10"><span class="c-article-section__title-number">5 </span>Conclusions</h2><div class="c-article-section__content" id="Sec10-content"><p>This paper has proposed the matrix versions of the BICR algorithm, Bi-CGSTAB algorithm, and CRS algorithm to solve the generalized Lyapunov equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ1">1</a>). Then we have introduced the variant versions of these three algorithms. Finally, we have provided numerical examples to illustrate the feasibility and effectiveness of the derived algorithms.</p></div></div></section> </div> <section data-title="Availability of data and materials"><div class="c-article-section" id="availability-of-data-and-materials-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="availability-of-data-and-materials">Availability of data and materials</h2><div class="c-article-section__content" id="availability-of-data-and-materials-content"> <p>Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1"> Kurschner, P.: Efficient Low-Rank Solution of Large-Scale Matrix Equations. Dissertation, Otto von Guericke Universitat, Magdeburg (2016) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=Efficient%20Low-Rank%20Solution%20of%20Large-Scale%20Matrix%20Equations&publication_year=2016&author=Kurschner%2CP."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2"> Simoncini, V.: Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations. SIAM J. Matrix Anal. Appl. <b>37</b>, 1655–1674 (2016) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/16M1059382" data-track-item_id="10.1137/16M1059382" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2F16M1059382" aria-label="Article reference 2" data-doi="10.1137/16M1059382">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3570279" aria-label="MathSciNet reference 2">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 2" href="http://scholar.google.com/scholar_lookup?&title=Analysis%20of%20the%20rational%20Krylov%20subspace%20projection%20method%20for%20large-scale%20algebraic%20Riccati%20equations&journal=SIAM%20J.%20Matrix%20Anal.%20Appl.&doi=10.1137%2F16M1059382&volume=37&pages=1655-1674&publication_year=2016&author=Simoncini%2CV."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3"> Paolo, D.A., Alberto, I., Antonio, R.: Realization and structure theory of bilinear dynamical systems. SIAM J. Control Optim. <b>12</b>, 517–535 (1974) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/0312040" data-track-item_id="10.1137/0312040" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2F0312040" aria-label="Article reference 3" data-doi="10.1137/0312040">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=424307" aria-label="MathSciNet reference 3">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=Realization%20and%20structure%20theory%20of%20bilinear%20dynamical%20systems&journal=SIAM%20J.%20Control%20Optim.&doi=10.1137%2F0312040&volume=12&pages=517-535&publication_year=1974&author=Paolo%2CD.A.&author=Alberto%2CI.&author=Antonio%2CR."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4"> Samir, A., Baiyat, A.L., Bettayeb, M.A., Saggaf, M.A.L.: New model reduction scheme for bilinear systems. Int. J. Syst. Sci. <b>25</b>, 1631–1642 (1994) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1080/00207729408949302" data-track-item_id="10.1080/00207729408949302" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1080%2F00207729408949302" aria-label="Article reference 4" data-doi="10.1080/00207729408949302">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1293469" aria-label="MathSciNet reference 4">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 4" href="http://scholar.google.com/scholar_lookup?&title=New%20model%20reduction%20scheme%20for%20bilinear%20systems&journal=Int.%20J.%20Syst.%20Sci.&doi=10.1080%2F00207729408949302&volume=25&pages=1631-1642&publication_year=1994&author=Samir%2CA.&author=Baiyat%2CA.L.&author=Bettayeb%2CM.A.&author=Saggaf%2CM.A.L."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5"> Kleinman, D.L.: On the stability of linear stochastic systems. IEEE Trans. Autom. Control <b>14</b>, 429–430 (1969) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1109/TAC.1969.1099206" data-track-item_id="10.1109/TAC.1969.1099206" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1109%2FTAC.1969.1099206" aria-label="Article reference 5" data-doi="10.1109/TAC.1969.1099206">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=252067" aria-label="MathSciNet reference 5">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 5" href="http://scholar.google.com/scholar_lookup?&title=On%20the%20stability%20of%20linear%20stochastic%20systems&journal=IEEE%20Trans.%20Autom.%20Control&doi=10.1109%2FTAC.1969.1099206&volume=14&pages=429-430&publication_year=1969&author=Kleinman%2CD.L."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6"> Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. <b>49</b>, 686–711 (2011) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/09075041X" data-track-item_id="10.1137/09075041X" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2F09075041X" aria-label="Article reference 6" data-doi="10.1137/09075041X">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2801215" aria-label="MathSciNet reference 6">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 6" href="http://scholar.google.com/scholar_lookup?&title=Lyapunov%20equations%2C%20energy%20functionals%2C%20and%20model%20order%20reduction%20of%20bilinear%20and%20stochastic%20systems&journal=SIAM%20J.%20Control%20Optim.&doi=10.1137%2F09075041X&volume=49&pages=686-711&publication_year=2011&author=Benner%2CP.&author=Damm%2CT."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7"> Gray, W.S., Mesko, J.: Energy functions and algebraic Gramians for bilinear systems. IFAC Proc. Vol. <b>31</b>, 101–106 (1998) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/S1474-6670(17)40318-1" data-track-item_id="10.1016/S1474-6670(17)40318-1" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2FS1474-6670%2817%2940318-1" aria-label="Article reference 7" data-doi="10.1016/S1474-6670(17)40318-1">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Energy%20functions%20and%20algebraic%20Gramians%20for%20bilinear%20systems&journal=IFAC%20Proc.%20Vol.&doi=10.1016%2FS1474-6670%2817%2940318-1&volume=31&pages=101-106&publication_year=1998&author=Gray%2CW.S.&author=Mesko%2CJ."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8"> Dorissen, H.: Canonical forms for bilinear systems. Syst. Control Lett. <b>13</b>, 153–160 (1989) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/0167-6911(89)90032-7" data-track-item_id="10.1016/0167-6911(89)90032-7" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2F0167-6911%2889%2990032-7" aria-label="Article reference 8" data-doi="10.1016/0167-6911(89)90032-7">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1014241" aria-label="MathSciNet reference 8">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Canonical%20forms%20for%20bilinear%20systems&journal=Syst.%20Control%20Lett.&doi=10.1016%2F0167-6911%2889%2990032-7&volume=13&pages=153-160&publication_year=1989&author=Dorissen%2CH."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9"> Damm, T.: Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equation. Numer. Linear Algebra Appl. <b>15</b>, 853–871 (2008) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1002/nla.603" data-track-item_id="10.1002/nla.603" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1002%2Fnla.603" aria-label="Article reference 9" data-doi="10.1002/nla.603">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2464173" aria-label="MathSciNet reference 9">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 9" href="http://scholar.google.com/scholar_lookup?&title=Direct%20methods%20and%20ADI-preconditioned%20Krylov%20subspace%20methods%20for%20generalized%20Lyapunov%20equation&journal=Numer.%20Linear%20Algebra%20Appl.&doi=10.1002%2Fnla.603&volume=15&pages=853-871&publication_year=2008&author=Damm%2CT."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10"> Fan, H.Y., Weng, P., Chu, E.: Numerical solution to generalized Lyapunov, Stein and rational Riccati equations in stochastic control. Numer. Algorithms <b>71</b>, 245–272 (2016) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/s11075-015-9991-8" data-track-item_id="10.1007/s11075-015-9991-8" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/s11075-015-9991-8" aria-label="Article reference 10" data-doi="10.1007/s11075-015-9991-8">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3452930" aria-label="MathSciNet reference 10">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 10" href="http://scholar.google.com/scholar_lookup?&title=Numerical%20solution%20to%20generalized%20Lyapunov%2C%20Stein%20and%20rational%20Riccati%20equations%20in%20stochastic%20control&journal=Numer.%20Algorithms&doi=10.1007%2Fs11075-015-9991-8&volume=71&pages=245-272&publication_year=2016&author=Fan%2CH.Y.&author=Weng%2CP.&author=Chu%2CE."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11"> Li, S.Y., Shen, H.L., Shao, X.H.: PHSS iterative method for solving generalized Lyapunov equations. Mathematics <b>7</b>, 1–13 (2019) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 11" href="http://scholar.google.com/scholar_lookup?&title=PHSS%20iterative%20method%20for%20solving%20generalized%20Lyapunov%20equations&journal=Mathematics&volume=7&pages=1-13&publication_year=2019&author=Li%2CS.Y.&author=Shen%2CH.L.&author=Shao%2CX.H."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12"> Sogabe, T., Sugihara, M., Zhang, S.L.: An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math. <b>226</b>, 103–113 (2009) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/j.cam.2008.05.018" data-track-item_id="10.1016/j.cam.2008.05.018" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2Fj.cam.2008.05.018" aria-label="Article reference 12" data-doi="10.1016/j.cam.2008.05.018">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2501885" aria-label="MathSciNet reference 12">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 12" href="http://scholar.google.com/scholar_lookup?&title=An%20extension%20of%20the%20conjugate%20residual%20method%20to%20nonsymmetric%20linear%20systems&journal=J.%20Comput.%20Appl.%20Math.&doi=10.1016%2Fj.cam.2008.05.018&volume=226&pages=103-113&publication_year=2009&author=Sogabe%2CT.&author=Sugihara%2CM.&author=Zhang%2CS.L."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13"> Stiefel, E.L.: Relaxationsmethoden bester strategie zur losung linearer gleichungssysteme. Comment. Math. Helv. <b>29</b>, 157–179 (1955) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/BF02564277" data-track-item_id="10.1007/BF02564277" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/BF02564277" aria-label="Article reference 13" data-doi="10.1007/BF02564277">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=71119" aria-label="MathSciNet reference 13">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 13" href="http://scholar.google.com/scholar_lookup?&title=Relaxationsmethoden%20bester%20strategie%20zur%20losung%20linearer%20gleichungssysteme&journal=Comment.%20Math.%20Helv.&doi=10.1007%2FBF02564277&volume=29&pages=157-179&publication_year=1955&author=Stiefel%2CE.L."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14"> Abea, K., Fujino, S.: Converting BiCR method for linear equations with complex symmetric matrices. Appl. Math. Comput. <b>321</b>, 564–576 (2018) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3732398" aria-label="MathSciNet reference 14">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1426.65045" aria-label="MATH reference 14">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 14" href="http://scholar.google.com/scholar_lookup?&title=Converting%20BiCR%20method%20for%20linear%20equations%20with%20complex%20symmetric%20matrices&journal=Appl.%20Math.%20Comput.&volume=321&pages=564-576&publication_year=2018&author=Abea%2CK.&author=Fujino%2CS."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15"> Sonneveld, P.: CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. <b>10</b>, 36–52 (1989) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/0910004" data-track-item_id="10.1137/0910004" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2F0910004" aria-label="Article reference 15" data-doi="10.1137/0910004">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=976160" aria-label="MathSciNet reference 15">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 15" href="http://scholar.google.com/scholar_lookup?&title=CGS%2C%20a%20fast%20Lanczos-type%20solver%20for%20nonsymmetric%20linear%20systems&journal=SIAM%20J.%20Sci.%20Stat.%20Comput.&doi=10.1137%2F0910004&volume=10&pages=36-52&publication_year=1989&author=Sonneveld%2CP."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16"> Vander, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. <b>13</b>, 631–644 (1992) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1137/0913035" data-track-item_id="10.1137/0913035" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1137%2F0913035" aria-label="Article reference 16" data-doi="10.1137/0913035">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1149111" aria-label="MathSciNet reference 16">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 16" href="http://scholar.google.com/scholar_lookup?&title=Bi-CGSTAB%3A%20a%20fast%20and%20smoothly%20converging%20variant%20of%20Bi-CG%20for%20the%20solution%20of%20nonsymmetric%20linear%20systems&journal=SIAM%20J.%20Sci.%20Stat.%20Comput.&doi=10.1137%2F0913035&volume=13&pages=631-644&publication_year=1992&author=Vander%2CH.A."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17"> Hajarian, M.: Developing Bi-CG and Bi-CR methods to solve generalized Sylvester-transpose matrix equations. Int. J. Autom. Comput. <b>11</b>, 25–29 (2014) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/s11633-014-0762-0" data-track-item_id="10.1007/s11633-014-0762-0" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/s11633-014-0762-0" aria-label="Article reference 17" data-doi="10.1007/s11633-014-0762-0">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=Developing%20Bi-CG%20and%20Bi-CR%20methods%20to%20solve%20generalized%20Sylvester-transpose%20matrix%20equations&journal=Int.%20J.%20Autom.%20Comput.&doi=10.1007%2Fs11633-014-0762-0&volume=11&pages=25-29&publication_year=2014&author=Hajarian%2CM."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18"> Chen, J., McInnes, L.C., Zhang, H.: Analysis and practical use of flexible BiCGStab. J. Sci. Comput. <b>68</b>, 803–825 (2016) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/s10915-015-0159-4" data-track-item_id="10.1007/s10915-015-0159-4" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/s10915-015-0159-4" aria-label="Article reference 18" data-doi="10.1007/s10915-015-0159-4">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3519201" aria-label="MathSciNet reference 18">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=Analysis%20and%20practical%20use%20of%20flexible%20BiCGStab&journal=J.%20Sci.%20Comput.&doi=10.1007%2Fs10915-015-0159-4&volume=68&pages=803-825&publication_year=2016&author=Chen%2CJ.&author=McInnes%2CL.C.&author=Zhang%2CH."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="19."><p class="c-article-references__text" id="ref-CR19"> Zhang, L.T., Zuo, X.Y., Gu, T.X., Huang, T.Z.: Conjugate residual squared method and its improvement for non-symmetric linear systems. Int. J. Comput. Math. <b>87</b>, 1578–1590 (2010) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1080/00207160802401029" data-track-item_id="10.1080/00207160802401029" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1080%2F00207160802401029" aria-label="Article reference 19" data-doi="10.1080/00207160802401029">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2665576" aria-label="MathSciNet reference 19">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 19" href="http://scholar.google.com/scholar_lookup?&title=Conjugate%20residual%20squared%20method%20and%20its%20improvement%20for%20non-symmetric%20linear%20systems&journal=Int.%20J.%20Comput.%20Math.&doi=10.1080%2F00207160802401029&volume=87&pages=1578-1590&publication_year=2010&author=Zhang%2CL.T.&author=Zuo%2CX.Y.&author=Gu%2CT.X.&author=Huang%2CT.Z."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="20."><p class="c-article-references__text" id="ref-CR20"> Zhang, L.T., Huang, T.Z., Gu, T.X., Zuo, X.Y.: An improved conjugate residual squared algorithm suitable for distributed parallel computing. Microelectron. Comput. <b>25</b>, 12–14 (2008) (in Chinese) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 20" href="http://scholar.google.com/scholar_lookup?&title=An%20improved%20conjugate%20residual%20squared%20algorithm%20suitable%20for%20distributed%20parallel%20computing&journal=Microelectron.%20Comput.&volume=25&pages=12-14&publication_year=2008&author=Zhang%2CL.T.&author=Huang%2CT.Z.&author=Gu%2CT.X.&author=Zuo%2CX.Y."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="21."><p class="c-article-references__text" id="ref-CR21"> Chen, L.J., Ma, C.F.: Developing CRS iterative methods for periodic Sylvester matrix equation. Adv. Differ. Equ. <b>1</b>, 1–11 (2019) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3919491" aria-label="MathSciNet reference 21">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?07040256" aria-label="MATH reference 21">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 21" href="http://scholar.google.com/scholar_lookup?&title=Developing%20CRS%20iterative%20methods%20for%20periodic%20Sylvester%20matrix%20equation&journal=Adv.%20Differ.%20Equ.&volume=1&pages=1-11&publication_year=2019&author=Chen%2CL.J.&author=Ma%2CC.F."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="22."><p class="c-article-references__text" id="ref-CR22"> Zhao, J., Zhang, J.H.: A smoothed conjugate residual squared algorithm for solving nonsymmetric linear systems. In: 2009 Second Int. Confe. Infor. Comput. Sci., vol. 4, pp. 364–367 (2009) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 22" href="http://scholar.google.com/scholar_lookup?&title=A%20smoothed%20conjugate%20residual%20squared%20algorithm%20for%20solving%20nonsymmetric%20linear%20systems&pages=364-367&publication_year=2009&author=Zhao%2CJ.&author=Zhang%2CJ.H."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="23."><p class="c-article-references__text" id="ref-CR23"> Sogabe, T., Zhang, S.L.: Extended conjugate residual methods for solving nonsymmetric linear systems. In: International Conference on Numerical Optimization and Numerical Linear Algebra, pp. 88–99 (2003) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 23" href="http://scholar.google.com/scholar_lookup?&title=Extended%20conjugate%20residual%20methods%20for%20solving%20nonsymmetric%20linear%20systems&pages=88-99&publication_year=2003&author=Sogabe%2CT.&author=Zhang%2CS.L."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="24."><p class="c-article-references__text" id="ref-CR24"> Sogabe, T., Sugihara, M., Zhang, S.L.: An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math. <b>226</b>, 103–113 (2009) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/j.cam.2008.05.018" data-track-item_id="10.1016/j.cam.2008.05.018" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2Fj.cam.2008.05.018" aria-label="Article reference 24" data-doi="10.1016/j.cam.2008.05.018">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2501885" aria-label="MathSciNet reference 24">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 24" href="http://scholar.google.com/scholar_lookup?&title=An%20extension%20of%20the%20conjugate%20residual%20method%20to%20nonsymmetric%20linear%20systems&journal=J.%20Comput.%20Appl.%20Math.&doi=10.1016%2Fj.cam.2008.05.018&volume=226&pages=103-113&publication_year=2009&author=Sogabe%2CT.&author=Sugihara%2CM.&author=Zhang%2CS.L."> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="25."><p class="c-article-references__text" id="ref-CR25"> Bai, Z.J., Skoogh, D.: A projection method for model reduction of bilinear dynamical systems. Linear Algebra Appl. <b>415</b>, 406–425 (2006) </p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/j.laa.2005.04.032" data-track-item_id="10.1016/j.laa.2005.04.032" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2Fj.laa.2005.04.032" aria-label="Article reference 25" data-doi="10.1016/j.laa.2005.04.032">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2227782" aria-label="MathSciNet reference 25">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 25" href="http://scholar.google.com/scholar_lookup?&title=A%20projection%20method%20for%20model%20reduction%20of%20bilinear%20dynamical%20systems&journal=Linear%20Algebra%20Appl.&doi=10.1016%2Fj.laa.2005.04.032&volume=415&pages=406-425&publication_year=2006&author=Bai%2CZ.J.&author=Skoogh%2CD."> Google Scholar</a> </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1186/s13662-021-03381-1?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>The research is supported by Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University.</p></div></div></section><section data-title="Funding"><div class="c-article-section" id="Fun-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Fun">Funding</h2><div class="c-article-section__content" id="Fun-content"><p>The work was supported in part by National Natural Science Foundation of China (11771368, 11771370) and the Project of Education Department of Hunan Province (19A500).</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411105, P.R. China</p><p class="c-article-author-affiliation__authors-list">Juan Zhang, Huihui Kang & Shifeng Li</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Juan-Zhang-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Juan Zhang</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=Juan%20Zhang" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Juan%20Zhang" data-track="click" data-track-action="author link - 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pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Shifeng%20Li%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p>All authors read and approved the final manuscript.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:zhangjuan@xtu.edu.cn">Juan Zhang</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar18">Competing interests</h3> <p>The authors declare that they have no competing interests.</p> </div></div></section><section aria-labelledby="appendices"><div class="c-article-section" id="appendices-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="appendices">Appendix: Proof of Theorem 3.1 </h2><div class="c-article-section__content" id="appendices-content"><h3 class="c-article__sub-heading u-visually-hidden" id="App1">Appendix: Proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar8">3.1</a> </h3><p>We prove Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1186/s13662-021-03381-1#FPar8">3.1</a> by mathematical induction to <i>v</i> and <i>u</i>. It is enough to prove (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ8">8</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ11">11</a>) for <span class="mathjax-tex">$1\leq u < v \leq r$</span>. </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(i)</span> <p>If <span class="mathjax-tex">$v=2$</span>, <span class="mathjax-tex">$u=1$</span>, then we have </p><div id="Equw" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\operatorname{tr}\bigl(R(2)^{T}W(1)\bigr)=\operatorname{tr}\bigl( \bigl(R(1)-\alpha (1)-W(1)\bigr)^{T}W(1)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(R(2)^{T}W(1)\bigr)}=\operatorname{tr}\bigl(R(1)^{T}W(1)\bigr)-\operatorname{tr} \bigl(W(1)^{T}R(1)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(R(2)^{T}W(1)\bigr)}=0, \\ &\operatorname{tr}\bigl(S(2)^{T}Z(1)\bigr)=\operatorname{tr}\bigl( \bigl(S(1)-\beta (1)Z(1)\bigr)^{T}Z(1)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(S(2)^{T}Z(1)\bigr)}=\operatorname{tr}\bigl(S(1)^{T}Z(1)\bigr)-\operatorname{tr} \bigl(Z(1)^{T}S(1)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(S(2)^{T}Z(1)\bigr)}=0, \\ &\operatorname{tr}\bigl(Z(2)^{T}Z(1)\bigr) \\ &\quad =\operatorname{tr}\Biggl( \Biggl(A^{T}R(2)+R(2)A+\sum_{j=1}^{m}N_{j}^{T}R(2)N_{j}- \eta (1)Z(1)\Biggr)^{T}Z(1)\Biggr) \\ &\quad =\operatorname{tr}\bigl(\bigl(A^{T}R(2)\bigr)^{T}Z(1) \bigr) + \operatorname{tr}\bigl(\bigl(R(2) A\bigr)^{T} Z(1)\bigr) + \operatorname{tr}\Biggl(\Biggl( \sum_{j=1}^{m}N_{j}^{T}R(2)N_{j} \Biggr)^{T}Z(1)\Biggr) \\ &\qquad {}-\operatorname{tr}\bigl(Z(1)^{T}\bigl(A^{T}R(2)\bigr) \bigr)-\operatorname{tr}\bigl(Z(1)^{T}\bigl(R(2)A\bigr)\bigr)- \operatorname{tr}\Biggl(Z(1)^{T}\Biggl( \sum _{j=1}^{m}N_{j}^{T}R(2)N_{j} \Biggr)\Biggr) \\ &\quad =0, \end{aligned}$$ </span></div></div><p> and </p><div id="Equx" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \operatorname{tr}\bigl(W(2)^{T}W(1)\bigr)={}&\operatorname{tr}\Biggl( \Biggl(AS(2)+S(2)A^{T}+\sum_{j=1}^{m}N_{j}S(2)N_{j}^{T}- \gamma (1)W(1)\Biggr)^{T}W(1)\Biggr) \\ ={}&\Biggl(\Biggl(AS(2)+S(2)A^{T}+\sum_{j=1}^{m}N_{j}S(2)N_{j}^{T} \Biggr)^{T}W(1)\Biggr) \\ &{}-\operatorname{tr}\Biggl(W(1)^{T}\Biggl(AS(2)+S(2)A^{T}+ \sum_{j=1}^{m}N_{j}S(2)N_{j}^{T} \Biggr)\Biggr) \\ ={}&0. \end{aligned}$$ </span></div></div><p> Thus when <span class="mathjax-tex">$u=1$</span>, <span class="mathjax-tex">$v=2$</span>, (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ8">8</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ11">11</a>) is true.</p> </li> <li> <span class="u-custom-list-number">(ii)</span> <p>Now for <span class="mathjax-tex">$u< w< r$</span>, we assume that </p><div id="Equy" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \operatorname{tr}\bigl(R(w)^{T}W(u)\bigr)=0, \\& \operatorname{tr}\bigl(S(w)^{T}Z(u)\bigr)=0, \\& \operatorname{tr}\bigl(Z(w)^{T}Z(u)\bigr)=0, \\& \operatorname{tr}\bigl(W(w)^{T}W(u)\bigr)=0. \end{aligned}$$ </span></div></div> </li> <li> <span class="u-custom-list-number">(iii)</span> <p>Next, we will prove (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ8">8</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ11">11</a>) for <span class="mathjax-tex">$w+1$</span>. Using the induction hypothesis, we get </p><div id="Equz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned}& \operatorname{tr}\bigl(R(w+1)^{T}W(u)\bigr)= \operatorname{tr}\bigl( \bigl(R(w)-\alpha (w)W(w)\bigr)^{T}W(u)\bigr)=0 \\& \operatorname{tr}\bigl(S(w+1)^{T}Z(u)\bigr)= \operatorname{tr}\bigl( \bigl(S(w)-\beta (w)Z(w)\bigr)^{T}Z(u)\bigr)=0, \end{aligned}$$ </span></div></div><p> and </p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\operatorname{tr}\bigl(Z(w+1)^{T}Z(u)\bigr) \\ &\quad =\operatorname{tr}\Biggl(\Biggl(A^{T}R(w+1)+R(w+1)A+\sum _{j=1}^{m}N_{j}^{T}R(w+1)N_{j}- \eta (w)Z(w)\Biggr)^{T}Z(u)\Biggr) \\ &\quad = \frac{1}{\beta (u)}\Biggl[\operatorname{tr}\bigl(R(w+1)^{T} \bigl(A\bigl(S(u)-S(u+1)\bigr)\bigr)\bigr) \\ &\qquad {}+\operatorname{tr}\bigl(R(w+1)^{T} \bigl(S(u)-S(u+1)A^{T}\bigr)\bigr) \\ &\qquad {}+\operatorname{tr}\Biggl(R(w+1)^{T}\Biggl(\sum _{j=1}^{m}N_{j}^{T} \bigl(S(u)-S(u+1)\bigr)N_{j}\Biggr)\Biggr)\Biggr] \\ &\quad =\frac{1}{\beta (u)}\bigl[\operatorname{tr}\bigl(R(w+1)^{T} \bigl(W(u)+\gamma (u-1)W(u-1)\bigr)\bigr) \\ &\qquad {}-\operatorname{tr}(R(w+1)^{T}\bigl(W(u+1)+\gamma (u)W(u) \bigr)\bigr] \\ &\quad =-\frac{1}{\beta (u)}\bigl[\operatorname{tr}\bigl(R(w+1)^{T}W(u+1) \bigr)\bigr], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (12) </div></div><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\operatorname{tr}\bigl(W(w+1)^{T}W(u)\bigr) \\ &\quad =\operatorname{tr}\Biggl(\Biggl(AS(w+1)+S(w+1)A^{T}+\sum _{j=1}^{m}N_{j}S(w+1)N_{j}^{T} \\ &\qquad {}-\gamma (w)W(w)\Biggr)^{T}w(u)\Biggr) \\ &\quad =\operatorname{tr}\bigl(S(w+1)^{T}\bigl(AS(w+1) \bigr)^{T}\bigr)+\operatorname{tr}\bigl(S(w+1)^{T} \bigl(S(w+1)A^{T}\bigr)^{T}\bigr) \\ &\qquad {}+\operatorname{tr}\Biggl(S(w+1)^{T}\Biggl(\sum _{j=1}^{m}N_{j}S(w+1)N_{j}^{T} \Biggr)^{T}\Biggr) \\ &\quad = \frac{1}{\alpha (u)}\Bigg[\operatorname{tr}\bigl(S(w+1)^{T} \bigl(A\bigl(R(u)-R(u+1)\bigr)\bigr)\bigr) \\ &\qquad {}+\operatorname{tr}\bigl(S(w+1)^{T} \bigl(R(u)-R(u+1)A^{T}\bigr)\bigr) \\ &\qquad {}+\operatorname{tr}\Biggl(S(w+1)^{T}\Biggl(\sum _{j=1}^{m}N_{j}^{T} \bigl(R(u)-R(u+1)\bigr)N_{j}\Biggr)\Biggr)\Bigg] \\ &\quad =\frac{1}{\alpha (u)}[\operatorname{tr}\bigl(S(w+1)^{T} \bigl(Z(u)+\eta (u-1)Z(u-1)\bigr)-Z(u+1)- \eta (u)Z(u)\bigr) \\ &\quad =-\frac{1}{\alpha (u)}\bigl[\operatorname{tr}\bigl(S(w+1)^{T}Z(u+1) \bigr)\bigr]. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (13) </div></div><p> For <span class="mathjax-tex">$u=w$</span>, again from the induction hypothesis we can obtain </p><div id="Equaa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\operatorname{tr}\bigl(R(w+1)^{T}W(w)\bigr) = \operatorname{tr} \bigl(\bigl(R(w)-\alpha (w)W(w)^{T}\bigr)W(w)\bigr), \\ &\operatorname{tr}\bigl(R(w)^{T}W(w)\bigr)-\operatorname{tr} \bigl(W(w)^{T}R(w)\bigr)= 0, \\ &\operatorname{tr}\bigl(S(w+1)^{T}Z(w)\bigr) = \operatorname{tr} \bigl(\bigl(S(w)-\beta (w)Z(w)\bigr)^{T} Z(w)\bigr) \\ & \hphantom{\operatorname{tr}\bigl(S(w+1)^{T}Z(w)\bigr)}= \operatorname{tr}\bigl(S(w)^{T}Z(w)\bigr)-\operatorname{tr} \bigl(Z(w)^{T}S(w)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(S(w+1)^{T}Z(w)\bigr)} =0, \end{aligned}$$ </span></div></div><p> and </p><div id="Equab" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)=\operatorname{tr} \Biggl(\Biggl(A^{T}R(w+1)+R(w+1)A \\ &\hphantom{\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)=}{}+\sum_{j=1}^{m}N_{j}^{T}R(w+1)N_{j}- \eta (w)Z(w)\Biggr)^{T}Z(w)\Biggr) \\ &\hphantom{\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)}=\operatorname{tr}\bigl(\bigl(A^{T}R(w+1)\bigr)^{T}Z(w) \bigr) + \operatorname{tr}\bigl(\bigl(R(w+1) A\bigr)^{T} Z(w)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)=}{}+\operatorname{tr}\Biggl(\Biggl(\sum_{j=1}^{m}N_{j}^{T}R(w+1)N_{j} \Biggr)^{T}Z(w)\Biggr) \\ &\hphantom{\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)=}{}-\operatorname{tr}\bigl(Z(w)^{T}\bigl(A^{T}R(w+1)\bigr) \bigr)-\operatorname{tr}\bigl(Z(w)^{T}\bigl(R(w+1)A\bigr)\bigr) \\ &\hphantom{\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)=}{}-\operatorname{tr}\Biggl(Z(w)^{T}\Biggl(\sum _{j=1}^{m}N_{j}^{T}R(w+1)N_{j} \Biggr)\Biggr) \\ &\hphantom{\operatorname{tr}\bigl(Z(w+1)^{T}Z(w)\bigr)}=0, \\ &\operatorname{tr}\bigl(W(w+1)^{T}W(w)\bigr) \\ &\quad =\operatorname{tr} \Biggl(\Biggl(AS(w+1)+S(w+1)A^{T} \\ &\qquad {}+\sum_{j=1}^{m}N_{j}S(w+1)N_{j}^{T}- \gamma (w)W(w)\Biggr)^{T}W(w)\Biggr) \\ &\quad =\Biggl(\Biggl(AS(w+1)+S(w+1)A^{T}+\sum _{j=1}^{m}N_{j}S(w+1)N_{j}^{T} \Biggr)^{T}W(w)\Biggr) \\ &\qquad {}-\operatorname{tr}\Biggl(W(w)^{T}\Biggl(AS(w+1)+S(w+1)A^{T}+ \sum_{j=1}^{m}N_{j}S(w+1)N_{j}^{T} \Biggr)\Biggr) \\ &\quad =0. \end{aligned}$$ </span></div></div> </li> </ol><p>Noting that <span class="mathjax-tex">$\operatorname{tr}(Z(w)^{T}Z(u)) = 0$</span>, <span class="mathjax-tex">$\operatorname{tr}(R(w+1)^{T}W(w)) = 0$</span> with (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ12">12</a>) we deduce that </p><div id="Equac" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \operatorname{tr}\bigl(Z(w+1)^{T}Z(u)\bigr)=0. $$</span></div></div><p> Similarly from <span class="mathjax-tex">$\operatorname{tr}(W(w)^{T}W(u))=0$</span>, <span class="mathjax-tex">$\operatorname{tr}(S(w)^{T}Z(w))=0$</span> and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ13">13</a>), it can be seen that </p><div id="Equad" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \operatorname{tr}\bigl(W(w+1)^{T}W(u)\bigr)= 0. \end{aligned}$$ </span></div></div><p> Hence, (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ8">8</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1186/s13662-021-03381-1#Equ11">11</a>) hold true for <span class="mathjax-tex">$w+1$</span>. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Zhang, J., Kang, H. & Li, S. 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c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Generalized%20Lyapunov%20equation&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Generalized Lyapunov equation</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Cayley%20transformation&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Cayley transformation</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=BICR%20algorithm&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">BICR algorithm</a></span></li><li class="c-article-subject-list__subject"><span><a 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Matrix iteration algorithms for solving the generalized Lyapunov matrix equation | Advances in Continuous and Discrete Models