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Pranesh Srikara - One of the best experts on this subject based on the ideXlab platform.

  • Random Matrices Generating Large Growth in LU Factorization with Pivoting
    2020
    Co-Authors: Higham, Desmond J., Higham, Nicholas J., Pranesh Srikara
    Abstract:

    We identify a class of random, dense $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor typically of size at least $n/(4 \log n)$ for large $n$. The condition number of the matrices can be arbitrarily chosen and large growth also happens for the transpose. No previous matrices with all these properties were known. The matrices can be generated by the MATLAB function \verb"gallery('randsvd',..)", and they are formed as the product of two random orthogonal matrices from the Haar distribution with a diagonal matrix having only one diagonal entry different from $1$, which lies between $0$ and $1$ (the ``one Small Singular Value'' case). Our explanation for the large growth uses the fact that the maximum absolute Value of any element of a Haar distributed orthogonal matrix tends to be relatively Small for large $n$. We verify the behavior numerically, finding that for partial pivoting the actual growth is significantly larger than the lower bound, and much larger than the growth observed for random matrices with elements from the uniform $[0,1$] or standard normal distributions. We show more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions. Finally, we demonstrate that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iterative refinement cannot

  • Random Matrices Generating Large Growth in LU Factorization with Pivoting
    2020
    Co-Authors: Higham, Desmond J., Higham, Nicholas J., Pranesh Srikara
    Abstract:

    We identify a class of random, dense $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor of at least $n/(4 \log n)$ for large $n$ with high probability. The condition number of the matrices can be arbitrarily chosen and large growth also happens for the transpose. No previous matrices with all these properties were known. The matrices can be generated by the MATLAB function \verb"gallery('randsvd',..)", and they are formed as the product of two random orthogonal matrices from the Haar distribution with a diagonal matrix having only one diagonal entry different from $1$, which lies between $0$ and $1$ (the ``one Small Singular Value'' case). Our explanation for the large growth uses the fact that the maximum absolute Value of any element of a Haar distributed orthogonal matrix tends to be relatively Small for large $n$. We verify the behavior numerically, finding that for partial pivoting the actual growth is significantly larger than the lower bound, and much larger than the growth observed for random matrices with elements from the uniform $[0,1$] or standard normal distributions. We show more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions. Finally, we demonstrate that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iterative refinement cannot

Higham, Desmond J. - One of the best experts on this subject based on the ideXlab platform.

  • Random Matrices Generating Large Growth in LU Factorization with Pivoting
    2020
    Co-Authors: Higham, Desmond J., Higham, Nicholas J., Pranesh Srikara
    Abstract:

    We identify a class of random, dense $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor typically of size at least $n/(4 \log n)$ for large $n$. The condition number of the matrices can be arbitrarily chosen and large growth also happens for the transpose. No previous matrices with all these properties were known. The matrices can be generated by the MATLAB function \verb"gallery('randsvd',..)", and they are formed as the product of two random orthogonal matrices from the Haar distribution with a diagonal matrix having only one diagonal entry different from $1$, which lies between $0$ and $1$ (the ``one Small Singular Value'' case). Our explanation for the large growth uses the fact that the maximum absolute Value of any element of a Haar distributed orthogonal matrix tends to be relatively Small for large $n$. We verify the behavior numerically, finding that for partial pivoting the actual growth is significantly larger than the lower bound, and much larger than the growth observed for random matrices with elements from the uniform $[0,1$] or standard normal distributions. We show more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions. Finally, we demonstrate that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iterative refinement cannot

  • Random Matrices Generating Large Growth in LU Factorization with Pivoting
    2020
    Co-Authors: Higham, Desmond J., Higham, Nicholas J., Pranesh Srikara
    Abstract:

    We identify a class of random, dense $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor of at least $n/(4 \log n)$ for large $n$ with high probability. The condition number of the matrices can be arbitrarily chosen and large growth also happens for the transpose. No previous matrices with all these properties were known. The matrices can be generated by the MATLAB function \verb"gallery('randsvd',..)", and they are formed as the product of two random orthogonal matrices from the Haar distribution with a diagonal matrix having only one diagonal entry different from $1$, which lies between $0$ and $1$ (the ``one Small Singular Value'' case). Our explanation for the large growth uses the fact that the maximum absolute Value of any element of a Haar distributed orthogonal matrix tends to be relatively Small for large $n$. We verify the behavior numerically, finding that for partial pivoting the actual growth is significantly larger than the lower bound, and much larger than the growth observed for random matrices with elements from the uniform $[0,1$] or standard normal distributions. We show more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions. Finally, we demonstrate that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iterative refinement cannot

Higham, Nicholas J. - One of the best experts on this subject based on the ideXlab platform.

  • Random Matrices Generating Large Growth in LU Factorization with Pivoting
    2020
    Co-Authors: Higham, Desmond J., Higham, Nicholas J., Pranesh Srikara
    Abstract:

    We identify a class of random, dense $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor typically of size at least $n/(4 \log n)$ for large $n$. The condition number of the matrices can be arbitrarily chosen and large growth also happens for the transpose. No previous matrices with all these properties were known. The matrices can be generated by the MATLAB function \verb"gallery('randsvd',..)", and they are formed as the product of two random orthogonal matrices from the Haar distribution with a diagonal matrix having only one diagonal entry different from $1$, which lies between $0$ and $1$ (the ``one Small Singular Value'' case). Our explanation for the large growth uses the fact that the maximum absolute Value of any element of a Haar distributed orthogonal matrix tends to be relatively Small for large $n$. We verify the behavior numerically, finding that for partial pivoting the actual growth is significantly larger than the lower bound, and much larger than the growth observed for random matrices with elements from the uniform $[0,1$] or standard normal distributions. We show more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions. Finally, we demonstrate that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iterative refinement cannot

  • Random Matrices Generating Large Growth in LU Factorization with Pivoting
    2020
    Co-Authors: Higham, Desmond J., Higham, Nicholas J., Pranesh Srikara
    Abstract:

    We identify a class of random, dense $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor of at least $n/(4 \log n)$ for large $n$ with high probability. The condition number of the matrices can be arbitrarily chosen and large growth also happens for the transpose. No previous matrices with all these properties were known. The matrices can be generated by the MATLAB function \verb"gallery('randsvd',..)", and they are formed as the product of two random orthogonal matrices from the Haar distribution with a diagonal matrix having only one diagonal entry different from $1$, which lies between $0$ and $1$ (the ``one Small Singular Value'' case). Our explanation for the large growth uses the fact that the maximum absolute Value of any element of a Haar distributed orthogonal matrix tends to be relatively Small for large $n$. We verify the behavior numerically, finding that for partial pivoting the actual growth is significantly larger than the lower bound, and much larger than the growth observed for random matrices with elements from the uniform $[0,1$] or standard normal distributions. We show more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions. Finally, we demonstrate that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iterative refinement cannot

Ahmed Abdelhadi - One of the best experts on this subject based on the ideXlab platform.

  • PEMWN - Spatial Coexistence of Cooperative Radar and Communication Systems
    2019 8th International Conference on Performance Evaluation and Modeling in Wired and Wireless Networks (PEMWN), 2019
    Co-Authors: Ahmed Abdelhadi
    Abstract:

    Future generations of cellular systems need to meet stringent requirements for bandwidth and latency. This entitles maximum utilization of the available wireless spectrum. Given that some of the spectrum bands are underutilized, e.g. radar band. In this paper, we study the utilization of the radar spectrum for commercial cellular usage for meeting future generations of cellular systems' spectral demands. We propose to avoid destructive high power interference from seaborne radar transmitters with high power that can saturate the cellular base station receivers by using a projection based approach. Additionally, in our design, the radar transmitters will be cooperating with cellular system by transmitting useful broadcast communication signal. Towards that, we study the channel between seaborne multiple input multiple output (MIMO) radar system and MIMO cellular systems. The high power radar signal is projected onto the Small Singular Value subspace of the channel, and therefore, reaches the base station with low power. In our design, the Small Singular Values are selected so that the received power at the cellular base stations is within acceptable cellular system power constraints. Simulation results show that the amount of power transmitted from the radar to the base station increases as the threshold increases, which allows the radar to be cooperatively capable of transmitting communication signals to cellular base stations. Moreover, as the threshold increases, the accuracy of localizing the radar target increases.

  • Spectral Coexistence of MIMO Radar and MIMO Cellular System
    IEEE Transactions on Aerospace and Electronic Systems, 2017
    Co-Authors: Jasmin A. Mahal, Awais Khawar, Ahmed Abdelhadi, T. Charles Clancy
    Abstract:

    This paper details designing the precoder of a MIMO-radar spectrally-coexistent with a MIMO cellular system. Spectrum sharing with zero or minimal interference is achieved by using, respectively, the conventional switched null space projection (SNSP) or the newly proposed switched Small Singular Value space projection (SSSVSP). Loss in radar target localization capability due to precoding can be compensated by using SSSVSP instead of SNSP to some extent but increasing the number of radar antenna elements is more effective.

Zhen Chen - One of the best experts on this subject based on the ideXlab platform.