On computing the symplectic LLT factorization
We analyze two algorithms for computing the symplectic factorization A = LL T of a given symmetric positive definite symplectic matrix A . The first algorithm W 1 is an implementation of the HH T factorization from Dopico and Johnson ( SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009 ), see Theorem 5....
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| Published in | Numerical algorithms Vol. 93; no. 3; pp. 1401 - 1416 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.07.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 1572-9265 |
| DOI | 10.1007/s11075-022-01472-y |
Cover
| Abstract | We analyze two algorithms for computing the symplectic factorization
A
=
LL
T
of a given symmetric positive definite symplectic matrix
A
. The first algorithm
W
1
is an implementation of the
HH
T
factorization from Dopico and Johnson (
SIAM J. Matrix Anal. Appl.
31(2):650–673,
2009
), see Theorem 5.2. The second one is a new algorithm
W
2
that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in
MATLAB
. A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. |
|---|---|
| AbstractList | We analyze two algorithms for computing the symplectic factorization A = LLT of a given symmetric positive definite symplectic matrix A. The first algorithm W1 is an implementation of the HHT factorization from Dopico and Johnson (SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009), see Theorem 5.2. The second one is a new algorithm W2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. We analyze two algorithms for computing the symplectic factorization A = LL T of a given symmetric positive definite symplectic matrix A . The first algorithm W 1 is an implementation of the HH T factorization from Dopico and Johnson ( SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009 ), see Theorem 5.2. The second one is a new algorithm W 2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB . A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. We analyze two algorithms for computing the symplectic factorization A = LL T of a given symmetric positive definite symplectic matrix A . The first algorithm W 1 is an implementation of the HH T factorization from Dopico and Johnson ( SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009), see Theorem 5.2. The second one is a new algorithm W 2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB . A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. |
| Author | Bujok, Maksymilian Borowik, Grzegorz Smoktunowicz, Alicja |
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| References | Demmel, Higham, Schreiber (CR2) 1995; 2 Lin, Mehrmann, Xu (CR8) 1999; 302–303 Dopico, Johnson (CR3) 2009; 31 Benzi, Razouk (CR1) 2007; 20 Grover, Panwar, Reddy (CR4) 2020; 596 CR6 Xu (CR9) 2003; 368 Tam (CR7) 2006; 19 Higham (CR5) 2002 P Grover (1472_CR4) 2020; 596 NJ Higham (1472_CR5) 2002 TY Tam (1472_CR7) 2006; 19 1472_CR6 W-W Lin (1472_CR8) 1999; 302–303 M Benzi (1472_CR1) 2007; 20 H Xu (1472_CR9) 2003; 368 JM Demmel (1472_CR2) 1995; 2 FM Dopico (1472_CR3) 2009; 31 |
| References_xml | – year: 2002 ident: CR5 publication-title: Accuracy and Stability of Numerical Algorithms doi: 10.1137/1.9780898718027 – volume: 20 start-page: 260 year: 2007 end-page: 265 ident: CR1 article-title: On the Iwasawa decomposition of a symplectic matrix publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2006.04.004 – volume: 596 start-page: 203 year: 2020 end-page: 215 ident: CR4 article-title: Positivity properties of some special matrices publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2020.03.008 – ident: CR6 – volume: 302–303 start-page: 469 year: 1999 end-page: 533 ident: CR8 article-title: Canonical forms for Hamiltonian and symplectic matrices and pencils publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(99)00191-3 – volume: 31 start-page: 650 issue: 2 year: 2009 end-page: 673 ident: CR3 article-title: Parametrization of the matrix symplectic group and applications publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/060678221 – volume: 368 start-page: 1 year: 2003 end-page: 24 ident: CR9 article-title: An SVD-like matrix decomposition and its applications publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(03)00370-7 – volume: 19 start-page: 1421 year: 2006 end-page: 1424 ident: CR7 article-title: Computing Iwasawa decomposition of a symplectic matrix by Cholesky factorization publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2006.03.001 – volume: 2 start-page: 173 issue: 2 year: 1995 end-page: 190 ident: CR2 article-title: Stability of block LU factorization publication-title: Numer. Linear Algebra Appl. doi: 10.1002/nla.1680020208 – volume: 302–303 start-page: 469 year: 1999 ident: 1472_CR8 publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(99)00191-3 – volume: 2 start-page: 173 issue: 2 year: 1995 ident: 1472_CR2 publication-title: Numer. Linear Algebra Appl. doi: 10.1002/nla.1680020208 – volume: 596 start-page: 203 year: 2020 ident: 1472_CR4 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2020.03.008 – volume-title: Accuracy and Stability of Numerical Algorithms year: 2002 ident: 1472_CR5 doi: 10.1137/1.9780898718027 – ident: 1472_CR6 doi: 10.1007/s11075-021-01226-2 – volume: 19 start-page: 1421 year: 2006 ident: 1472_CR7 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2006.03.001 – volume: 368 start-page: 1 year: 2003 ident: 1472_CR9 publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(03)00370-7 – volume: 20 start-page: 260 year: 2007 ident: 1472_CR1 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2006.04.004 – volume: 31 start-page: 650 issue: 2 year: 2009 ident: 1472_CR3 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/060678221 |
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| Snippet | We analyze two algorithms for computing the symplectic factorization
A
=
LL
T
of a given symmetric positive definite symplectic matrix
A
. The first algorithm... We analyze two algorithms for computing the symplectic factorization A = LLT of a given symmetric positive definite symplectic matrix A. The first algorithm W1... |
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| SubjectTerms | Algebra Algorithms Computation Computer Science Decomposition Factorization Floating point arithmetic Lie groups Mathematical analysis Matrices (mathematics) Numeric Computing Numerical Analysis Original Paper Theory of Computation |
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| Title | On computing the symplectic LLT factorization |
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