Distance descending ordering method: An O(n) algorithm for inverting the mass matrix in simulation of macromolecules with long branches

Xiankun Xu, Peiwen Li

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1 Citation (Scopus)

Abstract

Fixman's work in 1974 and the follow-up studies have developed a method that can factorize the inverse of mass matrix into an arithmetic combination of three sparse matrices—one of them is positive definite and needs to be further factorized by using the Cholesky decomposition or similar methods. When the molecule subjected to study is of serial chain structure, this method can achieve O(n) time complexity. However, for molecules with long branches, Cholesky decomposition about the corresponding positive definite matrix will introduce massive fill-in due to its nonzero structure. Although there are several methods can be used to reduce the number of fill-in, none of them could strictly guarantee for zero fill-in for all molecules according to our test, and thus cannot obtain O(n) time complexity by using these traditional methods. In this paper we present a new method that can guarantee for no fill-in in doing the Cholesky decomposition, which was developed based on the correlations between the mass matrix and the geometrical structure of molecules. As a result, the inverting of mass matrix will remain the O(n) time complexity, no matter the molecule structure has long branches or not.

Original languageEnglish (US)
Pages (from-to)253-264
Number of pages12
JournalJournal of Computational Physics
Volume349
DOIs
StatePublished - Nov 15 2017

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Macromolecules
macromolecules
Molecules
matrices
molecules
Decomposition
decomposition
simulation

Keywords

  • Cholesky decomposition
  • Internal coordinates
  • Mass matrix
  • Molecular dynamics
  • O(n) time complexity
  • Sparse matrix

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)
  • Computer Science Applications

Cite this

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abstract = "Fixman's work in 1974 and the follow-up studies have developed a method that can factorize the inverse of mass matrix into an arithmetic combination of three sparse matrices—one of them is positive definite and needs to be further factorized by using the Cholesky decomposition or similar methods. When the molecule subjected to study is of serial chain structure, this method can achieve O(n) time complexity. However, for molecules with long branches, Cholesky decomposition about the corresponding positive definite matrix will introduce massive fill-in due to its nonzero structure. Although there are several methods can be used to reduce the number of fill-in, none of them could strictly guarantee for zero fill-in for all molecules according to our test, and thus cannot obtain O(n) time complexity by using these traditional methods. In this paper we present a new method that can guarantee for no fill-in in doing the Cholesky decomposition, which was developed based on the correlations between the mass matrix and the geometrical structure of molecules. As a result, the inverting of mass matrix will remain the O(n) time complexity, no matter the molecule structure has long branches or not.",
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author = "Xiankun Xu and Peiwen Li",
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AU - Li, Peiwen

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N2 - Fixman's work in 1974 and the follow-up studies have developed a method that can factorize the inverse of mass matrix into an arithmetic combination of three sparse matrices—one of them is positive definite and needs to be further factorized by using the Cholesky decomposition or similar methods. When the molecule subjected to study is of serial chain structure, this method can achieve O(n) time complexity. However, for molecules with long branches, Cholesky decomposition about the corresponding positive definite matrix will introduce massive fill-in due to its nonzero structure. Although there are several methods can be used to reduce the number of fill-in, none of them could strictly guarantee for zero fill-in for all molecules according to our test, and thus cannot obtain O(n) time complexity by using these traditional methods. In this paper we present a new method that can guarantee for no fill-in in doing the Cholesky decomposition, which was developed based on the correlations between the mass matrix and the geometrical structure of molecules. As a result, the inverting of mass matrix will remain the O(n) time complexity, no matter the molecule structure has long branches or not.

AB - Fixman's work in 1974 and the follow-up studies have developed a method that can factorize the inverse of mass matrix into an arithmetic combination of three sparse matrices—one of them is positive definite and needs to be further factorized by using the Cholesky decomposition or similar methods. When the molecule subjected to study is of serial chain structure, this method can achieve O(n) time complexity. However, for molecules with long branches, Cholesky decomposition about the corresponding positive definite matrix will introduce massive fill-in due to its nonzero structure. Although there are several methods can be used to reduce the number of fill-in, none of them could strictly guarantee for zero fill-in for all molecules according to our test, and thus cannot obtain O(n) time complexity by using these traditional methods. In this paper we present a new method that can guarantee for no fill-in in doing the Cholesky decomposition, which was developed based on the correlations between the mass matrix and the geometrical structure of molecules. As a result, the inverting of mass matrix will remain the O(n) time complexity, no matter the molecule structure has long branches or not.

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