<p>The matrix quare root is a very important matrix computation problem. The structure-preserving doubling algorithm (SDA) is quadratically convergent. We demonstrate how the matrix square root is computed by SDA, where the Cayley transform is deployed and the original equation is rewritten in terms of the invariant subspace. Further, we introduce a positive parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> when using the Cayley transform and get SDA-<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$A^{1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. If the appropriate parameter <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is given, SDA-<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$A^{1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> can outperform SDA for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3040_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$A^{1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> in some examples. The convergence properties of the two algorithms are given. The algorithms in this paper are not numerically sensitive when the original matrix is ill-conditioned and the algorithms work when the original matrix is a singular matrix but has a square root.</p>

Computing the matrix square root with the doubling algorithm

期刊论文
发表日期：

2025年9月3日

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Pei-Chang Guo,
Xiao-Xia Guo,
Yi Tang

摘要

The matrix quare root is a very important matrix computation problem. The structure-preserving doubling algorithm (SDA) is quadratically convergent. We demonstrate how the matrix square root is computed by SDA, where the Cayley transform is deployed and the original equation is rewritten in terms of the invariant subspace. Further, we introduce a positive parameter $\alpha $ $α$ when using the Cayley transform and get SDA- $\alpha $ $α$ for $A^{1/2}$ $A^{1 / 2}$ . If the appropriate parameter $\alpha $ $α$ is given, SDA- $\alpha $ $α$ for $A^{1/2}$ $A^{1 / 2}$ can outperform SDA for $A^{1/2}$ $A^{1 / 2}$ in some examples. The convergence properties of the two algorithms are given. The algorithms in this paper are not numerically sensitive when the original matrix is ill-conditioned and the algorithms work when the original matrix is a singular matrix but has a square root.