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Leonardo Robol - One of the best experts on this subject based on the ideXlab platform.

  • quasi toeplitz matrix arithmetic a matlab toolbox
    Numerical Algorithms, 2019
    Co-Authors: Dario Andrea Bini, Stefano Massei, Leonardo Robol
    Abstract:

    A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in \mathbb Z^{+}}$ , $E=(e_{i,j})_{i,j\in \mathbb Z^{+}}$ is compact and the norms $\|a\|_{_{\mathcal {W}}}={\sum }_{i\in \mathbb Z}|a_{i}|$ and $\|E\|_{2}$ are finite. These properties allow to approximate any QT matrix, within any given precision, by means of a finite number of parameters. QT matrices, equipped with the norm $\|A\|_{_{\mathcal {Q}\mathcal {T}}}=\alpha {\|a\|}_{_{\mathcal {W}}}+\|E\|_{2}$ , for $\alpha = (1+\sqrt 5)/2$ , are a Banach algebra with the standard arithmetic operations. We provide an Algorithmic Description of these operations on the finite parametrization of QT matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

  • quasi toeplitz matrix arithmetic a matlab toolbox
    arXiv: Numerical Analysis, 2018
    Co-Authors: Dario Andrea Bini, Stefano Massei, Leonardo Robol
    Abstract:

    A Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in\mathbb Z^+}$, $E=(e_{i,j})_{i,j\in\mathbb Z^+}$ is compact and the norms $\lVert a\rVert_{\mathcal W} = \sum_{i\in\mathbb Z}|a_i|$ and $\lVert E \rVert_2$ are finite. These properties allow to approximate any QT-matrix, within any given precision, by means of a finite number of parameters. QT-matrices, equipped with the norm $\lVert A \rVert_{\mathcal QT}=\alpha\lVert a\rVert_{\mathcal{W}} \lVert E \rVert_2$, for $\alpha\ge (1+\sqrt 5)/2$, are a Banach algebra with the standard arithmetic operations. We provide an Algorithmic Description of these operations on the finite parametrization of QT-matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

Martin V. Butz - One of the best experts on this subject based on the ideXlab platform.

Wolfgang Stolzmann - One of the best experts on this subject based on the ideXlab platform.

Sean Devine - One of the best experts on this subject based on the ideXlab platform.

Dario Andrea Bini - One of the best experts on this subject based on the ideXlab platform.

  • quasi toeplitz matrix arithmetic a matlab toolbox
    Numerical Algorithms, 2019
    Co-Authors: Dario Andrea Bini, Stefano Massei, Leonardo Robol
    Abstract:

    A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in \mathbb Z^{+}}$ , $E=(e_{i,j})_{i,j\in \mathbb Z^{+}}$ is compact and the norms $\|a\|_{_{\mathcal {W}}}={\sum }_{i\in \mathbb Z}|a_{i}|$ and $\|E\|_{2}$ are finite. These properties allow to approximate any QT matrix, within any given precision, by means of a finite number of parameters. QT matrices, equipped with the norm $\|A\|_{_{\mathcal {Q}\mathcal {T}}}=\alpha {\|a\|}_{_{\mathcal {W}}}+\|E\|_{2}$ , for $\alpha = (1+\sqrt 5)/2$ , are a Banach algebra with the standard arithmetic operations. We provide an Algorithmic Description of these operations on the finite parametrization of QT matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.

  • quasi toeplitz matrix arithmetic a matlab toolbox
    arXiv: Numerical Analysis, 2018
    Co-Authors: Dario Andrea Bini, Stefano Massei, Leonardo Robol
    Abstract:

    A Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in\mathbb Z^+}$, $E=(e_{i,j})_{i,j\in\mathbb Z^+}$ is compact and the norms $\lVert a\rVert_{\mathcal W} = \sum_{i\in\mathbb Z}|a_i|$ and $\lVert E \rVert_2$ are finite. These properties allow to approximate any QT-matrix, within any given precision, by means of a finite number of parameters. QT-matrices, equipped with the norm $\lVert A \rVert_{\mathcal QT}=\alpha\lVert a\rVert_{\mathcal{W}} \lVert E \rVert_2$, for $\alpha\ge (1+\sqrt 5)/2$, are a Banach algebra with the standard arithmetic operations. We provide an Algorithmic Description of these operations on the finite parametrization of QT-matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.