Mixed and componentwise condition numbers for matrix decompositions

Abstract We present normwise and componentwise perturbation bounds for the LU , the Cholesky, the L D L T and the QR decompositions by using a new approach. The explicit expressions of mixed and componentwise condition numbers for these matrix decompositions are derived. The condition numbers improve known results of the normwise and componentwise cases and reveal the characterizations of the structured perturbations. The exact explicit perturbation expressions are derived for the factors L and U of the LU decomposition, and the rigorous normwise and componentwise perturbation bounds are presented for the LU decomposition.

[1]  Stephen M. Watt,et al.  QR factoring to compute the GCD of univariate approximate polynomials , 2004, IEEE Transactions on Signal Processing.

[2]  Ulrich Langer,et al.  Numerical and Symbolic Scientific Computing , 2012, Texts & Monographs in Symbolic Computation.

[3]  G. Stewart.  On the perturbation of LU, Cholesky, and QR factorizations , 1993 .

[4]  A. George,et al.  Symbolic factorization for sparse Gaussian elimination with partial pivoting , 1987 .

[5]  Xiao-Wen Chang,et al.  Multiplicative perturbation analysis for QR factorizations , 2011 .

[6]  G. Stewart.  On the perturbation of LU and Cholesky factors , 1997 .

[7]  Z. Drmač,et al.  On the Perturbation of the Cholesky Factorization , 1994 .

[8]  Terence Tao,et al.  Smooth analysis of the condition number and the least singular value , 2008, Math. Comput..

[9]  Ji-guang Sun,et al.  Rounding-error and perturbation bounds for the Cholesky and LDL T factorizations , 1991 .

[10]  Anshul Gupta,et al.  Improved Symbolic and Numerical Factorization Algorithms for Unsymmetric Sparse Matrices , 2002, SIAM J. Matrix Anal. Appl..

[11]  Ji-guang Sun.  Perturbation bounds for the Cholesky andQR factorizations , 1991 .

[12]  J. Demmel.  The Probability That a Numerical, Analysis Problem Is Difficult , 2013 .

[13]  Israel Koltracht,et al.  Mixed componentwise and structured condition numbers , 1993 .

[14]  Siegfried M. Rump,et al.  Structured Perturbations Part II: Componentwise Distances , 2003, SIAM J. Matrix Anal. Appl..

[15]  Erich Kaltofen,et al.  Computer Algebra Handbook : Foundations · Applications · Systems , 2003 .

[16]  Michael T. Heath,et al.  Symbolic Cholesky factorization on a local-memory multiprocessor , 1987, Parallel Comput..

[17]  Damien Stehlé,et al.  Perturbation Analysis of the QR factor R in the context of LLL lattice basis reduction , 2012, Math. Comput..

[18]  Yimin Wei,et al.  Smoothed analysis of some condition numbers , 2006, Numer. Linear Algebra Appl..

[19]  Damien Stehlé,et al.  Rigorous Perturbation Bounds of Some Matrix Factorizations , 2010, SIAM J. Matrix Anal. Appl..

[20]  R. Bhatia.  Matrix factorizations and their perturbations , 1994 .

[21]  G. Stewart,et al.  New perturbation analyses for the Cholesky factorization , 1996 .

[22]  G. Stewart,et al.  Perturbation Analyses for the QR Factorization , 1997, SIAM J. Matrix Anal. Appl..

[23]  H. Zha.  A componentwise perturbation analysis of the QR decomposition , 1993 .

[24]  Claude-Pierre Jeannerod,et al.  Improved Backward Error Bounds for LU and Cholesky Factorizations , 2014, SIAM J. Matrix Anal. Appl..

[25]  Componentwise perturbation bounds for the LU, LDU and LDT -T decompositions , 2000 .

[26]  Shang-Hua Teng,et al.  Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices , 2003, SIAM J. Matrix Anal. Appl..

[27]  Felipe Cucker,et al.  Condition - The Geometry of Numerical Algorithms , 2013, Grundlehren der mathematischen Wissenschaften.

[28]  Perturbation bounds for triangular and full rank factorizations , 2005 .

[29]  Yimin Wei,et al.  Improved rigorous perturbation bounds for the LU and QR factorizations , 2015, Numer. Linear Algebra Appl..

[30]  Yi-min Wei,et al.  New rigorous perturbation bounds for the Cholesky-like factorization of skew-symmetric matrix , 2016 .

[31]  Xiao-Wen Chang,et al.  Perturbation analysis of some matrix factorizations , 1997 .

[32]  G. Stewart.  Perturbation Bounds for the $QR$ Factorization of a Matrix , 1977 .

[33]  J. Demmel.  On condition numbers and the distance to the nearest ill-posed problem , 2015 .

[34]  James Demmel,et al.  Stability of block LU factorization , 1992, Numer. Linear Algebra Appl..

[35]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[36]  Dongming Wang,et al.  Symbolic-Numeric Computation , 2007 .

[37]  Rasmus Brandt,et al.  Polynomial Matrix Decompositions : Evaluation of Algorithms with an Application to Wideband MIMO Communications , 2010 .

[38]  James Demmel,et al.  Parallel Symbolic Factorization for Sparse LU with Static Pivoting , 2007, SIAM J. Sci. Comput..

[39]  Ian Postlethwaite,et al.  A symbolic matrix decomposition algorithm for reduced order linear fractional transformation modelling , 2007, Autom..

[40]  Xiao-Wen Chang,et al.  Componentwise perturbation analyses for the QR factorization , 2001, Numerische Mathematik.

[41]  Ji-guang Sun.  On perturbation bounds for the QR factorization , 1995 .

[42]  Xiao-Qing Jin,et al.  Numerical Linear Algebra and its Applications , 2004 .

[43]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

[44]  John R. Gilbert,et al.  Symbolic and Exact Structure Prediction for Sparse Gaussian Elimination with Partial Pivoting , 2008, SIAM J. Matrix Anal. Appl..

[45]  Ji-guang Sun.  Componentwise perturbation bounds for some matrix decompositions , 1992 .

[46]  Gene H. Golub,et al.  Matrix computations , 1983 .

[47]  X.-W. Chang.  On the perturbation of the Q-factor of the QR factorization , 2012, Numer. Linear Algebra Appl..

[48]  Yimin Wei,et al.  On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems , 2006, Math. Comput..

[49]  Joseph W. H. Liu,et al.  Exploiting Structural Symmetry in Unsymmetric Sparse Symbolic Factorization , 1992, SIAM J. Matrix Anal. Appl..

[50]  Xiao-Wen Chang,et al.  On the sensitivity of the LU factorization , 1998 .

[51]  A. Barrlund.  Perturbation bounds for theLDLH andLU decompositions , 1991 .

[52]  J. Abbott,et al.  Approximate Commutative Algebra , 2009 .