Moment-matching method - Revision history

Himpe at 12:55, 25 August 2022

2022-08-25T12:55:54Z

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	[[Category:method]]		[[Category:method]]
	[[Category:~~DAE order unspecified~~]]		[[Category:Differential_algebraic_system]]
	[[Category:linear]]		[[Category:linear]]
	[[Category:first differential order]]		[[Category:first differential order]]

Himpe: Fixed ref

2021-08-27T07:59:36Z

Fixed ref

← Older revision		Revision as of 07:59, 27 August 2021
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	<ref name="feldmann95">P. Feldmann and R.W. Freund, "<span class="plainlinks">[http://dx.doi.org/10.1109/43.384428 Efficient linear circuit analysis by Pade approximation via the Lanczos process]</span>". IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 14:639-649, 1995.</ref>		<ref name="feldmann95">P. Feldmann and R.W. Freund, "<span class="plainlinks">[http://dx.doi.org/10.1109/43.384428 Efficient linear circuit analysis by Pade approximation via the Lanczos process]</span>". IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 14:639-649, 1995.</ref>

	<ref name="grimme97">E.J. Grimme, "<span class="plainlinks">[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.9254&rep=rep1&type=pdf Krylov projection methods for model reduction]</span>. PhD thesis, Univ. Illinois, Urbana-Champaign, 1997.		<ref name="grimme97">E.J. Grimme, "<span class="plainlinks">[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.9254&rep=rep1&type=pdf Krylov projection methods for model reduction]</span>. PhD thesis, Univ. Illinois, Urbana-Champaign, 1997.</ref>

	</references>		</references>

Zhangy: /* Description */

2013-09-27T12:03:47Z

Description

← Older revision		Revision as of 12:03, 27 September 2013
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	::<math>=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B</math>		::<math>=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B</math>

	::<math>=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}[(s_0{ E}-{ A})]^{-1}B</math>		::<math>=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}(s_0{ E}-{ A})^{-1}B</math>

	::<math>=C[{ I}-\sigma(s_0{ E}- A )^{-1}E+\sigma^2[(s_0{ E}-{ A})^{-1}E]^{2}+\ldots]		::<math>=C[{ I}-\sigma(s_0{ E}- A )^{-1}E+\sigma^2[(s_0{ E}-{ A})^{-1}E]^{2}+\ldots]
	s_0{E}-{ A})^{-1}B</math>		(s_0{E}-{ A})^{-1}B</math>

	::<math>=\sum \limits^\infty_{i=0}\underbrace{C[-(s_0{ E}-{A})^{-1}E]^i(s_0{ E}-{ A})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,</math>		::<math>=\sum \limits^\infty_{i=0}\underbrace{C[-(s_0{ E}-{A})^{-1}E]^i(s_0{ E}-{ A})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,</math>

Himpe: /* Description */

2013-05-03T09:03:23Z

Description

Baur at 08:37, 3 May 2013

2013-05-03T08:37:45Z

← Older revision		Revision as of 08:37, 3 May 2013
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	==Description==		==Description==
	The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as		The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as
	''Padé'' approximation methods<ref name="feldmann95"/>. They ~~belongs~~ to the [[Projection based MOR]] methods. These methods are applicable ~~for~~ non-parametric linear time invariant systems, often ~~the~~ descriptor systems, e.g.		''Padé'' approximation methods<ref name="feldmann95"/>. They belong to the [[Projection based MOR]] methods. These methods are applicable to non-parametric linear time invariant systems, often descriptor systems, e.g.

	:<math>		:<math>
	E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad		E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
	y(t)=Cx(t), \quad \quad (1)		y(t)=Cx(t). \quad \quad (1)
	</math>		</math>

Himpe: equation indent

2013-04-30T23:09:30Z

equation indent

Feng: /* Description */

2013-04-29T09:52:05Z

Description

← Older revision		Revision as of 09:52, 29 April 2013
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	<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>		<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

	For the case of one expansion point in (2),(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces		For the case of one expansion point in (2)(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces
	which can easily be computed by Arnoldi or Lanczos methods. The matrices <math>V</math> and <math>W</math> in (4),(5) can be computed with the rational Krylov algorithm in<ref name="grimme97"/> or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting		which can easily be computed by Arnoldi or Lanczos methods. The matrices <math>V</math> and <math>W</math> in (4)(5) can be computed with the rational Krylov algorithm in<ref name="grimme97"/> or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting
	methods is roughly <math>O(n r^2)</math> for a sparse ~~matrix~~ <math>A</math>.		methods is roughly <math>O(n r^2)</math> for sparse matrices <math>A, E</math>.

	==References==		==References==

Feng at 09:50, 29 April 2013

2013-04-29T09:50:13Z

← Older revision		Revision as of 09:50, 29 April 2013
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	</math>		</math>

	They are very efficient in many engineering applications, circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..		They are very efficient in many engineering applications, such as circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..

	The basic steps are as follows. First, the transfer function		The basic steps are as follows. First, the transfer function
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	where <math>m_i(s_0)</math> are called the moments of the transfer function about <math>s_0</math> for <math>i=0,1,2,\ldots</math>.		where <math>m_i(s_0)</math> are called the moments of the transfer function about <math>s_0</math> for <math>i=0,1,2,\ldots</math>.
	If the expansion point is chosen as zero then the moments simplify to <math>m_i(0)=C(A^{-1}E)^i(-A^{-1}B)</math>.		If the expansion point is chosen as zero, then the moments simplify to <math>m_i(0)=C(A^{-1}E)^i(-A^{-1}B)</math>.
	~~For <math>s_0=\infty</math> the moments are also called Markov parameters which can be computed by <math>C(E^{-1}A)^i(E^{-1}B)</math> if E is invertable~~.

	The goal in moment-matching model reduction is the construction of a reduced order		The goal in moment-matching model reduction is the construction of a reduced order
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	example, using a single expansion point <math>s_0=0</math>, by		example, using a single expansion point <math>s_0=0</math>, by

	<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\},</math>		<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (2) </math>

	<math>\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots		<math>\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots
	,(E^T{A}^{-T})^{r-1}C^T\},</math>		,(E^T{A}^{-T})^{r-1}C^T\}, \quad \quad (3) </math>

	where <math>\tilde B=-A^{-1}B</math>. ~~The reduced model is in the form as below~~		where <math>\tilde B=-A^{-1}B</math>.

	~~<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>~~

	The transfer function <math>\hat H</math> of the reduced model has good approximation properties around <math>s_0</math>, which matches the first <math>2r</math> moments of <math>H(s)</math> at <math>s_0</math>.		The transfer function <math>\hat H</math> of the reduced model has good approximation properties around <math>s_0</math>, which matches the first <math>2r</math> moments of <math>H(s)</math> at <math>s_0</math>.

	Using a set of <math>k</math> distinct expansion points <math>\{s_1,\cdots,s_k\}</math>, the reduced model ~~can be~~ obtained by, e.g.,		Using a set of <math>k</math> distinct expansion points <math>\{s_1,\cdots,s_k\}</math>, the reduced model obtained by, e.g.,


	<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \}</math>,		<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \} \quad \quad \quad \quad \quad \quad \quad \quad (4)</math>,

			<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\quad \quad \quad \quad \quad (5) </math>

	<math>~~\textrm{range}(W)~~=~~\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T~~,\ldots,~~E^T(A-s_k {E})^{-T}C^T \}~~,</~~math~~>		matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. The reduced model is in the form as below

	~~has order~~ <math>r=~~k</math> and matches the first two moments at each <math>s_j</math>~~, ~~<math>j~~=~~1,\ldots,k~~</math>~~, see <ref name="grimme97"/>.~~		<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

	It can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces		For the case of one expansion point in (2),(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces
	which can easily be computed by Arnoldi or Lanczos methods <~~ref name="freund03"~~/><ref name="~~feldmann95~~"/>. In		which can easily be computed by Arnoldi or Lanczos methods. The matrices <math>V</math> and <math>W</math> in (4),(5) can be computed with the rational Krylov algorithm in<ref name="grimme97"/> or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting
	these algorithms only matrix-vector multiplications are used which		methods is roughly <math>O(n r^2)</math> for a sparse matrix <math>A</math>.
	are simple to implement and the complexity of the resulting
	methods is ~~only~~ <math>O(n r^2)</math>.

	==References==		==References==

Feng: /* Description */

2013-04-29T09:24:34Z

Description

← Older revision		Revision as of 09:24, 29 April 2013
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	==Description==		==Description==
	The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as		The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as
	''Padé'' approximation methods<ref name="feldmann95"/>. They ~~are~~ [[Projection based MOR]] methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.		''Padé'' approximation methods<ref name="feldmann95"/>. They belongs to the [[Projection based MOR]] methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.

	<math>		<math>

Feng: /* Description */

2013-04-29T09:20:41Z

Description

← Older revision		Revision as of 09:20, 29 April 2013
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	==Description==		==Description==
	The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as		The moment-matching methods are also called the ''Krylov'' subspace methods<ref name="freund03"/>, as well as
	''Padé'' approximation methods<ref name="feldmann95"/>. They are [[Projection based MOR]]methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.		''Padé'' approximation methods<ref name="feldmann95"/>. They are [[Projection based MOR]] methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.

	<math>		<math>

← Older revision		Revision as of 09:03, 3 May 2013
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	''Padé'' approximation methods<ref name="feldmann95"/>. They belong to the [[Projection based MOR]] methods. These methods are applicable to non-parametric linear time invariant systems, often descriptor systems, e.g.		''Padé'' approximation methods<ref name="feldmann95"/>. They belong to the [[Projection based MOR]] methods. These methods are applicable to non-parametric linear time invariant systems, often descriptor systems, e.g.

	:<math>		:<math>E \dot{x}(t)=A x(t)+B u(t),</math>
	E \~~frac~~{dx(t)~~}{dt}~~=A x(t)+B u(t), ~~\quad~~		:<math>y(t)=Cx(t).</math>
	y(t)=Cx(t). ~~\quad \quad (1)~~
	</math>

	They are very efficient in many engineering applications, such as circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..		They are very efficient in many engineering applications, such as circuit simulation, Microelectromechanical systems (MEMS) simulation, etc..
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	example, using a single expansion point <math>s_0=0</math>, by		example, using a single expansion point <math>s_0=0</math>, by

	:<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (2) </math>		:<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (1) </math>

	:<math>\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots		:<math>\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots
	,(E^T{A}^{-T})^{r-1}C^T\}, \quad \quad (3) </math>		,(E^T{A}^{-T})^{r-1}C^T\}, \quad \quad (2) </math>

	where <math>\tilde B=-A^{-1}B</math>.		where <math>\tilde B=-A^{-1}B</math>.
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	:<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \}, \quad \quad \quad \quad \quad \quad \quad \quad (4)</math>		:<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \}, \quad \quad \quad \quad \quad \quad \quad \quad (3)</math>

	:<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\quad \quad \quad \quad \quad (5) </math>		:<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\quad \quad \quad \quad \quad (4) </math>

	matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. The reduced model is in the form as below		matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. The reduced model is in the form as below
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	:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>		:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

	For the case of one expansion point in (2)(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces		For the case of one expansion point in (1)(2), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces
	which can easily be computed by Arnoldi or Lanczos methods. The matrices <math>V</math> and <math>W</math> in (4)(5) can be computed with the rational Krylov algorithm in<ref name="grimme97"/> or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting		which can easily be computed by Arnoldi or Lanczos methods. The matrices <math>V</math> and <math>W</math> in (3)(4) can be computed with the rational Krylov algorithm in<ref name="grimme97"/> or with the modified Gram-Schmidt process. In these algorithms only a few number of linear systems need to be solved, where matrix-vector multiplications are only used if using iterative solvers, which are simple to implement and the complexity of the resulting
	methods is roughly <math>O(n r^2)</math> for sparse matrices <math>A, E</math>.		methods is roughly <math>O(n r^2)</math> for sparse matrices <math>A, E</math>.

← Older revision		Revision as of 23:09, 30 April 2013
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	''Padé'' approximation methods<ref name="feldmann95"/>. They belongs to the [[Projection based MOR]] methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.		''Padé'' approximation methods<ref name="feldmann95"/>. They belongs to the [[Projection based MOR]] methods. These methods are applicable for non-parametric linear time invariant systems, often the descriptor systems, e.g.

	<math>		:<math>
	E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad		E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
	y(t)=Cx(t), \quad \quad (1)		y(t)=Cx(t), \quad \quad (1)
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	The basic steps are as follows. First, the transfer function		The basic steps are as follows. First, the transfer function

	<math>H(s)=Y(s)/U(s)=C(sE-A)^{-1}B</math>		:<math>H(s)=Y(s)/U(s)=C(sE-A)^{-1}B</math>

	is expanded into a power series at an expansion point <math>s_0\in\mathbb{C}\cup \infty</math>.		is expanded into a power series at an expansion point <math>s_0\in\mathbb{C}\cup \infty</math>.
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	Let <math>s=s_0+\sigma</math>, then, within the convergence radius of the series, we have		Let <math>s=s_0+\sigma</math>, then, within the convergence radius of the series, we have

	<math>H(s_0 + \sigma)= C[(s_{0}+\sigma){E}-A]^{-1}B</math>		:<math>H(s_0 + \sigma)= C[(s_{0}+\sigma){E}-A]^{-1}B</math>

	<math>=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B</math>		::<math>=C[\sigma { E}+(s_{0}{ E}-{ A})]^{-1}B</math>

	<math>=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}[(s_0{ E}-{ A})]^{-1}B</math>		::<math>=C[{ I}+\sigma(s_0{ E}-{ A})^{-1}E]^{-1}[(s_0{ E}-{ A})]^{-1}B</math>

	<math>=C[{ I}-\sigma(s_0{ E}- A )^{-1}E+\sigma^2[(s_0{ E}-{ A})^{-1}E]^{2}+\ldots]		::<math>=C[{ I}-\sigma(s_0{ E}- A )^{-1}E+\sigma^2[(s_0{ E}-{ A})^{-1}E]^{2}+\ldots]
	s_0{E}-{ A})^{-1}B</math>		s_0{E}-{ A})^{-1}B</math>

	<math>=\sum \limits^\infty_{i=0}\underbrace{C[-(s_0{ E}-{A})^{-1}E]^i(s_0{ E}-{ A})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,</math>		::<math>=\sum \limits^\infty_{i=0}\underbrace{C[-(s_0{ E}-{A})^{-1}E]^i(s_0{ E}-{ A})^{-1}B}_{:= m_i(s_0)} \, \sigma^i,</math>

	where <math>m_i(s_0)</math> are called the moments of the transfer function about <math>s_0</math> for <math>i=0,1,2,\ldots</math>.		where <math>m_i(s_0)</math> are called the moments of the transfer function about <math>s_0</math> for <math>i=0,1,2,\ldots</math>.
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	example, using a single expansion point <math>s_0=0</math>, by		example, using a single expansion point <math>s_0=0</math>, by

	<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (2) </math>		:<math>\textrm{range}(V)=\textrm{span}\{\tilde B,({ A}^{-1}E)^2 \tilde B, \ldots,({ A}^{-1}E)^{r}{\tilde B}\}, \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \ (2) </math>

	<math>\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots		:<math>\textrm{range}(W)=\textrm{span}\{C^T, E^T{ A}^{-T}C^T,(E^T{A}^{-T})^2C^T, \ldots
	,(E^T{A}^{-T})^{r-1}C^T\}, \quad \quad (3) </math>		,(E^T{A}^{-T})^{r-1}C^T\}, \quad \quad (3) </math>

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	<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \} \quad \quad \quad \quad \quad \quad \quad \quad (4)</math>,		:<math>\textrm{range}(V)=\textrm{span}\{(A-s_1 {E})^{-1}E\tilde B,\ldots,(A-s_k {E})^{-1}E\tilde B \}, \quad \quad \quad \quad \quad \quad \quad \quad (4)</math>

	<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\quad \quad \quad \quad \quad (5) </math>		:<math>\textrm{range}(W)=\textrm{span}\{E^T(A-s_1 {E})^{-T}C^T,\ldots,E^T(A-s_k {E})^{-T}C^T \},\quad \quad \quad \quad \quad (5) </math>

	matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. The reduced model is in the form as below		matches the first two moments at each <math>s_j</math>, <math>j=1,\ldots,k</math>, see <ref name="grimme97"/>. The reduced model is in the form as below

	<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>		:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

	For the case of one expansion point in (2)(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces		For the case of one expansion point in (2)(3), it can be seen that the columns of <math>V</math>, <math>W</math> span Krylov subspaces