We present the LU decomposition with panel rank revealing pivoting (LU_PRRP),
an LU factorization algorithm based on strong rank revealing QR panel
factorization. LU_PRRP is more stable than Gaussian elimination with partial
pivoting (GEPP). Our extensive numerical experiments show that the new
factorization scheme is as numerically stable as GEPP in practice, but it is
more resistant to pathological cases and easily solves the Wilkinson matrix and
the Foster matrix. We also present CALU_PRRP, a communication avoiding version
of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament
pivoting, with the selection of the pivots at each step of the tournament being
performed via strong rank revealing QR factorization. CALU_PRRP is more stable
than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more
stable in practice and is resistant to pathological cases on which GEPP and
CALU fail.Comment: No. RR-7867 (2012

Demmel, James W.

Grigori, Laura

Gu, Ming

Khabou, Amal

English

arXiv

We present the LU decomposition with panel rank revealing pivoting (LU_PRRP), an LU factorization algorithm based on strong rank revealing QR panel factorization. LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP). Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases and easily solves the Wilkinson matrix and the Foster matrix. We also present CALU_PRRP, a communication avoiding version of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail

Demmel, James,

Hal-Diderot

LU factorization with panel rank revealing pivoting and its communication avoiding version

We present the block LU factorization with panel rank revealing
pivoting (block LU_PRRP), a decomposition algorithm based on strong
rank revealing QR panel factorization.
Block LU_PRRP is more stable than Gaussian elimination with partial
pivoting (GEPP), with a theoretical upper bound of the growth factor
of $(1+ \tau b)^{(n/ b)-1}$, where $b$ is the size of the panel used
during the block factorization, $\tau$ is a parameter of the strong
rank revealing QR factorization, and $n$ is the number of columns of
the matrix. For example, if the size of the panel is $b = 64$, and
$\tau = 2$, then $(1+2b)^{(n/b)-1} = (1.079)^{n-64} \ll 2^{n-1}$, where
$2^{n-1}$ is the upper bound of the growth factor of GEPP.  Our
extensive numerical experiments show that the new factorization scheme
is as numerically stable as GEPP in practice, but it is more resistant
to pathological cases. The block LU_PRRP factorization does only
$O(n^2 b)$ additional floating point operations compared to GEPP.

We also present block CALU_PRRP, a communication avoiding version of block
LU_PRRP that minimizes communication. Block CALU_PRRP is based on
tournament pivoting, with the selection of the pivots at each step of
the tournament being performed via strong rank revealing QR
factorization. Block CALU_PRRP is more stable than CALU, the communication avoiding
version of GEPP, with a theoretical upper bound of the growth factor
of $(1+ \tau b)^{{n\over b}(H+1)-1}$, where $H$ is the height of the reduction tree used during tournament
pivoting.  The upper bound of the growth factor of CALU is
$2^{n(H+1)-1}$. Block CALU_PRRP is also more stable in practice and
is resistant to pathological cases on which GEPP and CALU fail

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LU FACTORIZATION WITH PANEL RANK REVEALING PIVOTING AND ITS COMMUNICATION AVOIDING VERSION

International audienceWe present block LU factorization with panel rank revealing pivoting (block LU PRRP), a decomposition algorithm based on strong rank revealing QR panel factorization. Block LU PRRP is more stable than Gaussian elimination with partial pivoting (GEPP), with a theoretical upper bound of the growth factor of (1 + τb)(n/b)−1, where b is the size of the panel used during the block factorization, τ is a parameter of the strong rank revealing QR factorization, n is the number of columns of the matrix, and for simplicity we assume that n is a multiple of b. We also assume throughout the paper that 2 ≤ b n. For example, if the size of the panel is b = 64, and τ = 2, then (1+2b)(n/b)−1 = (1.079)n−64 2n−1, where 2n−1 is the upper bound of the growth factor of GEPP. Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases. The block LU PRRP factorization does only O(n2b) additional floating point operations compared to GEPP. We also present block CALU PRRP, a version of block LU PRRP that minimizes communication and is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. Block CALU PRRP is more stable than CALU, the communication avoiding version of GEPP, with a theoretical upper bound of the growth factor of (1 + τb) nb (H+1)−1, where H is the height of the reduction tree used during tournament pivoting. The upper bound of the growth factor of CALU is 2n(H+1)−1. Block CALU PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail

Demmel, James, W.

HAL-CentraleSupelec

LU Factorization with Panel Rank Revealing Pivoting and its Communication Avoiding Version

INRIA a CCSD electronic archive server

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We present block LU factorization with panel rank revealing pivoting (block LU_PRRP), a decomposition algorithm based on strong rank revealing QR panel factorization. Block LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP), with a theoretical upper bound of the growth factor of $(1+ \tau b)^{(n/ b)-1}$, where $b$ is the size of the panel used during the block factorization, $\tau$ is a parameter of the strong rank revealing QR factorization, $n$ is the number of columns of the matrix, and for simplicity we assume that n is a multiple of b. We also assume throughout the paper that $2\leq b \ll n$. For example, if the size of the panel is $b = 64$, and $\tau = 2$, then $(1+2b)^{(n/b)-1} = (1.079)^{n-64} \ll 2^{n-1}$, where $2^{n-1}$ is the upper bound of the growth factor of GEPP. Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases. The block LU_PRRP factorization does only $O(n^2 b)$ additional floating point operations compared to GEPP. We also present block CALU_PRRP, a version of block LU_PRRP that minimizes communication and is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. Block CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP, with a theoretical upper bound of the growth factor of $(1+ \tau b)^{{n\over b}(H+1)-1}$, where $H$ is the height of the reduction tree used during tournament pivoting. The upper bound of the growth factor of CALU is $2^{n(H+1)-1}$. Block CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail

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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.232.7079

LU factorization with panel rank revealing pivoting and its communication avoiding version

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